Rational Quartic Symmetroids · 2017. 12. 7. · 1.1. Basic Properties isomorphic,sotheseareallsymmetroids. ♥ Genus 3 Sextics The(3×3)-minorsofa(3×4)-matrixoflinearformsinC[x0,x

Rational Quartic Symmetroids

Martin HelsøMaster’s Thesis, Spring 2016

Abstract

The study of generic quartic symmetroids in projective 3-space dates backto Cayley, but little is known about the special specimens. This thesis setsout to survey the possibilities in the non-generic case. We prove that theSteiner surface is a symmetroid. We also find quartic symmetroids thatare double along a line and have two to eight isolated nodes, symmetroidsthat are double along two lines and have zero to four isolated nodes andsymmetroids that are double along a smooth conic section and have two tofour isolated nodes. In addition, we present degenerated symmetroids withfewer isolated singularities. A symmetroid that is singular along a smoothconic section and a line is given. This culminates in a 21-dimensionalfamily of rational quartic symmetroids.

Acknowledgements

I am beholden to my advisor, Kristian Ranestad, for introducing me to a deeplyfascinating thesis topic. His perseverance in the most daunting stages of theresearch has been crucial. Even though I still feel a novice at them, I havedeveloped a fondness for geometrical arguments working with him.

For the terrors of errors, I have employed a meticulous posse of proofreaders:Håkon Kolderup, Jonas Kylling, Simen Lønsethagen and Fredrik Meyer. Eachof them deserve the deepest of thanks. Together they have combated misprintsand made the presentation more perspicuous.

Fredrik is also deserving of special mention and gratitude for serving as aMacaulay2 reference manual.

To my partners in crime, all who have shared the study hall B601 with methese past nine semesters, I thank you for distracting me.

Martin HelsøOslo, May 2016

Contents

Abstract i

Acknowledgements iii

Contents iv

List of Figures vi

List of Tables vii

1 Introduction 11.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Veronese Varieties 72.1 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Veronese Embedding . . . . . . . . . . . . . . . . . . . . . . . . 92.3 The Degree of a Veronese Variety . . . . . . . . . . . . . . . . . 102.4 Projection of the Veronese Surface to P3 . . . . . . . . . . . . . 11

3 Space of Quadrics in P3 173.1 Moduli Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Web of Quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Some Unusual Symmetroids . . . . . . . . . . . . . . . . . . . . 20

4 Symmetroids with a Double Conic 254.1 Mimicking the Pillow . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Menagerie of Projected Del Pezzo Surfaces . . . . . . . . . . . . 304.3 Quadrics Through Four General Points . . . . . . . . . . . . . . 384.4 Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5 Symmetroids with a Double Line 455.1 Quadrics Through Four Coplanar Points . . . . . . . . . . . . . 455.2 Linear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.3 Double Line Not Contained in the Rank 2 Locus . . . . . . . . 565.4 Dimensions of Families of Rational Symmetroids . . . . . . . . 57

6 Degenerated Symmetroids 61

Contents

6.1 Symmetroids with No Isolated Singularities . . . . . . . . . . . 616.2 Quartic Symmetroids with a Triple Point . . . . . . . . . . . . 636.3 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

A Gallery of Symmetroids 67A.1 Symmetroids Singular along a Line . . . . . . . . . . . . . . . . 67A.2 Symmetroids Singular along a Conic . . . . . . . . . . . . . . . 70A.3 Symmetroid Singular along a Reducible Cubic . . . . . . . . . . 72

B Code and Computations 73B.1 Computations for Example 3.3.1 . . . . . . . . . . . . . . . . . 73B.2 Computations for Example 3.3.2 . . . . . . . . . . . . . . . . . 75B.3 Computations for Example 4.1.6 . . . . . . . . . . . . . . . . . 76B.4 Computations for Example 4.1.7 . . . . . . . . . . . . . . . . . 77B.5 Computations for Example 4.2.4 . . . . . . . . . . . . . . . . . 81B.6 Computations for Example 4.2.6 . . . . . . . . . . . . . . . . . 82B.7 Computations for Section 4.3 . . . . . . . . . . . . . . . . . . . 83B.8 Computations for Example 5.2.1 . . . . . . . . . . . . . . . . . 84B.9 Computations for Example 5.2.2 . . . . . . . . . . . . . . . . . 87

C Rational Double Points 89C.1 ADE Singularities . . . . . . . . . . . . . . . . . . . . . . . . . 89

Bibliography 93

Glossary of Symbols 97

Index 98

v

List of Figures

2.1 Projection of the Veronese surface to the Steiner surface . . . . . . 132.2 Pairwise intersection of conics and lines . . . . . . . . . . . . . . . 142.3 The Steiner surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Pencils of conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Projected Veronese surface . . . . . . . . . . . . . . . . . . . . . . 16

3.1 The pillow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Toeplitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1 Ramification locus for a symmetroid with two double lines and fournodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Ramification locus for a symmetroid with a double conic and fournodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3 Ramification loci for a symmetroid with two double lines and threenodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4 Ramification locus for a symmetroid with two double lines and twonodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.5 Intersection of components in rank 2 locus . . . . . . . . . . . . . . 394.6 Ramification locus for a symmetroid with a double conic and three

nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.7 Ramification locus for a symmetroid with a double conic and two

nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.8 Ramification locus for a symmetroid with two double lines and one

node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.1 Ramification locus for a symmetroid with a double line and six nodes 485.2 Ramification locus for a symmetroid with a double line and seven

nodes, projected from a rank-2-point . . . . . . . . . . . . . . . . . 485.3 Ramification locus for a symmetroid with a double line and seven

nodes, projected from a rank-3-node . . . . . . . . . . . . . . . . . 485.4 Ramification locus for a symmetroid with a double line and five nodes 495.5 Ramification locus for a symmetroid with a double line, five rank-

2-nodes and one rank-3-node, projected from a rank-2-node . . . . 505.6 Ramification locus for a symmetroid with a double line, five rank-

2-nodes and one rank-3-node, projected from the rank-3-node . . . 505.7 Ramification locus for a symmetroid with a double line and four nodes 51

5.8 Ramification locus for a symmetroid with a double line, three nodesand no rank-1-point . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.9 Ramification locus for a symmetroid with a double line, three nodesand a rank-1-point . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.10 Ramification locus for a symmetroid with a double line and two nodes 535.11 Ramification locus for a symmetroid with a double line that is not

contained in the rank 2 locus . . . . . . . . . . . . . . . . . . . . . 56

A.1 Rational quartic symmetroids singular along a line, with seven toeight additional nodes . . . . . . . . . . . . . . . . . . . . . . . . . 67

A.2 Rational quartic symmetroids singular along a line, with six addi-tional nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A.3 Rational quartic symmetroids singular along a line, with four to fiveadditional nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A.4 Rational quartic symmetroids singular along a line, with zero tothree additional nodes . . . . . . . . . . . . . . . . . . . . . . . . . 69

A.5 Rational quartic symmetroids singular along two intersecting lines 70A.6 Rational quartic symmetroids singular along a smooth conic section 71A.7 Quartic symmetroid singular along a smooth conic section and a line 72

All figures are drawn by the author using TikZ or surf.

List of Tables

4.1 Possible singularities on quartic del Pezzo surfaces . . . . . . . . . 274.2 Differences resulting from dividing the ramification locus into dif-

ferent cubics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3 Differences in rank 2 loci . . . . . . . . . . . . . . . . . . . . . . . . 35

C.1 ADE classification . . . . . . . . . . . . . . . . . . . . . . . . . . . 89C.2 Fundamental cycles for ADE curves . . . . . . . . . . . . . . . . . 91

CHAPTER 1

Introduction

It is my experience that proofs involvingmatrices can be shortened by 50% if onethrows the matrices out.

Emil Artin

Asymmetroid is a hypersurface V(f) in Pn whose defining equation f can bewritten as the determinant of a symmetric matrix A(x) of linear forms.In this thesis we will concern ourselves with quartic symmetroids in

CP3, that is,

A(x) = A0x0 +A1x1 +A2x2 +A3x3, (1.1)

where the Ai are (4 × 4)-matrices with entries in C. We will mostly omit theargument in the notation and denote A(x) by simply A. It should be clear fromthe context at which point the matrix is evaluated.

The term symmetroid was coined by Cayley in the memoir [Cay69] from 1869,where it is related to the Jacobian surface of four quadric surfaces. It appearslater in classical treatises such as [Jes16] and [Cob29]. It is shown in [Cos83]that the K3 surfaces that arise as the étale double cover of the generic Reyecongruences are quartic symmetroids. Symmetroids are also discussed in [Dol12],which is Dolgachev’s monumental effort to preserve classical algebraic geometry.Nevertheless, it is fair to say that symmetroids have received little attentionduring their long history. However, they have recently obtained renewed interestwith the study of spectrahedra.

A spectrahedron is the intersection of the cone of positive semidefinitematrices with an affine subspace of the space of real symmetric (n×n)-matrices.The motivation for studying spectrahedra comes from semidefinite program-ming, which is the subfield of convex optimisation concerned with optimisinga linear function over a spectrahedron. The algebraic boundary of a spectra-hedron in R3 is a symmetroid in CP3. If a symmetroid has a representation(1.1) such that each Ai is a real matrix, then its associated spectrahedron isthe set

(x0, x1, x2) ∈ R3 | A0x0 +A1x1 +A2x2 +A3 is semidefinite.

If the Ai are (n × n)-matrices, we say that the spectrahedron has degree n.Ottem, Ranestad, Sturmfels and Vinzant gave a new and elementary proof of

1. Introduction

the characterisation of the types of transversal quartic spectrahedra in [Ott+14].Some of the key tools and ideas used in this thesis stem from that article, suchas Algorithm 1.1.4 and the pillow in Example 3.3.1, which inspired Chapter 4.

In this thesis we find several novel examples of quartic symmetroids thatare double along a line or a conic section. These are necessarily rational. Wedescribe techniques for finding families of such surfaces. The methods employedalso indicate heuristically to what degree the examples are generic or special.This is not a complete classification of rational quartic symmetroids, but it givesa good overview of the land. Hopefully the thesis has laid useful groundworkfor future works on symmetroids or spectrahedra. Some existing results aboutquartic symmetroids are unsatisfactory in that they require the surfaces to benodal. The explicit examples in this thesis may contribute to discern whichstatements can be generalised.

1.1 Basic Properties

Having identified the surface with a matrix, we are able to talk about the rank ofa point. Namely, a point x is a rank-k-point if rankA(x) = k. For a symmetroidof degree d, the rank-(d− 2)-points are singular on V(f), and generically theyare nodes [Ott+14, Section 1]. The set of points where the rank is at most k iscalled the rank k locus. The rank (d− 2) locus is not necessarily equal to thesingular locus; the inclusion is strict in Example 4.1.7.

Generically, there are(

d+13)nodes on V(f). If the symmetroid is nodal, that

is, all its singularities are isolated nodes, then it has exactly(

d+13)rank-(d− 2)-

points. The symmetroid is called transversal if it does not have additional nodes.The terminology is motivated by Theorem 1.1.2. Kummer surfaces are examplesof nodal symmetriods that are not transversal [Jes16, Article 105].

Our main object of study, the quartic symmetroids, has therefore generically10 nodes. They have generically no rank-1-points and the rank 2 locus is notcontained in a quadric surface.

Moreover, we will concern ourselves with symmetroids that are rational, thatis, birationally equivalent to P2. If the only singularities on a quartic surface arefinitely many double points, then they can be resolved. Therefore, the quartichas a relatively minimal model with trivial canonical bundle. Hence it is notrational. Thus to find rational symmetroids, we need to look at quartic surfacesthat either have nodes along a curve or have a triple point.

Example 1.1.1 (Cayley’s symmetroid). Rational cubic symmetroids have beenknown for a long time. Cayley’s 4-nodal cubic can be realised as the blow-upof P2 in the six intersection points of four general lines. With a suitable choiceof coordinates, it can be taken to have

x0x1x2 + x0x1x3 + x0x2x3 + x1x2x3 = 0as its equation. This is the determinant ofx0 + x3 x3 x3

x3 x1 + x3 x3x3 x3 x2 + x3

.Cubic surfaces with four singularities of type1 A1 that span P3 are projectively

1See Appendix C for a note on this notation.

2

1.1. Basic Properties

isomorphic, so these are all symmetroids. ♥

Genus 3 Sextics

The (3×3)-minors of a (3×4)-matrix of linear forms in C[x0, x1, x2, x3]1 definein general a sextic curve in CP3 with arithmetic genus 3. Let

A :=

l00 l01 l02 l03l10 l11 l12 l13l20 l21 l22 l23l30 l31 l32 l33

be a (4× 4)-matrix of linear forms in C[x0, x1, x2, x3]1. Let S be the quartic inCP3 given by det(A) = 0. The (3× 3)-minors of any (3× 4)-submatrix A1 andany (4× 3)-submatrix A2 define such curves, C1 and C2, lying on S. Say

A1 :=

l10 l11 l12 l13l20 l21 l22 l23l30 l31 l32 l33

and A2 :=

l00 l01 l03l10 l11 l13l20 l21 l23l30 l31 l33

.The curves C1 and C2 both vanish on the common (3× 3)-minor of A1 and A2.In our example, that is ∣∣∣∣∣∣

l10 l11 l13l20 l21 l23l30 l31 l33

∣∣∣∣∣∣ = 0.

The common minor defines a cubic surface S′ in CP3 intersecting S along C1and C2. The intersection S ∩ S′ has degree 4 · 3 = 12 by Bézout’s theorem.Since C1, C2 ⊂ S ∩S′ both have degree 6, there is nothing else in S ∩S′, exceptpossibly some points.

Now, assume that A is symmetric and A1, A2 are obtained by removing thei-th row and column, respectively. Then C1 = C2. By a continuity argument, ordirect computation, one can show that S ∩ S′ equals two times C1, plus somepoints.

In other words, if S ⊂ CP3 is a quartic symmetroid, then there exist cubicsurfaces S′ that are tangent to S along a sextic curve with arithmetic genus3. This is a rare property among quartic surfaces. In [Ble+12, Section 5], analgorithm is given for producing a representation (1.1) for a 10-nodal quarticwith such a curve, using the Hilbert-Burch theorem [Eis95, Theorem 20.15].

Projection From a Node

One of the classic tools for examining a quartic surface S := V(f) ⊂ P3 withat least one node p, is to study the projection S \ p → P2 from p. This is atwo-to-one map, which extends to a morphism πp : S → P2 on the blow-up S ofS with centre p. If p := [1 : 0 : 0 : 0], then f is on the form ax2

0 + bx0 + c wherea, b, c ∈ C[x1, x2, x3]. The fibres of πp are given by the roots of f , so the pointswhere f has a double root, known as the ramification locus, are described bythe discriminant Rp :=

(b2 − 4ac

)⊆ P2.

3

1. Introduction

A singular point on Rp is the image under πp of either a singularity on S orof a line through p contained in S. Conversely, singular points on S and lineson S through p are mapped to singularities on Rp.

Cayley gave a characterisation of quartic symmetroids using the sextic curveRp [Cay69], but we refer to a more modern source for the proofs:

Theorem 1.1.2 (Cayley, [Ott+14, Theorem 1.2]). If S is a quartic symmetroidin CP3 and p ∈ S is a rank-2-point, then the ramification locus is the union oftwo cubic curves Rp = R1 ∪R2. Moreover, if no line through p is contained inS, then S is transversal if and only if R1 and R2 are smooth and intersectingtransversally in 9 points.

Cayley was also able to provide a converse for nodal quartics.

Theorem 1.1.3 (Cayley, [Ott+14, Theorem 1.2]). Let p be a node on a nodal,quartic surface S ⊂ CP3. If the ramification locus Rp splits into two cubics,then S is a symmetroid.

In their proof of [Ott+14, Theorem 1.2], Ottem et al. gave an algorithm forconstructing a matrix with the correct determinant from the ramification locus.We omit most of the details in the proof and boil it down to the steps in thealgorithm:

Algorithm 1.1.4. Suppose that S := V(f) is a nodal quartic surface witha node in p := [1 : 0 : 0 : 0]. Then f = −qx2

0 + 2gx0 + ∆ for polynomialsq, g,∆ ∈ C[x1, x2, x3]. Assume also that the ramification locus Rp factors intocubics V(F11) and V(F22). Up to rescaling of F11, we have that

F11F22 = g2 + q∆

in C[x1, x2, x3]. Since q∆ is a square modulo F11, there is a subscheme Z∆ oflength 6 in V(F11) such that V(F11,∆) = 2Z∆.

There are no conics containing Z∆, thus the space of cubics in C[x1, x2, x3]3vanishing on Z∆ is 4-dimensional. Extend F11 and F12 := g to a basis

F11, F12, F13, F14

for that space. For 2 ≤ j ≤ k ≤ 4, we can find a cubic Fjk ∈ C[x1, x2, x3]3 anda quadric qjk ∈ C[x1, x2, x3]2 satisfying

F1jF1k = F11Fjk + qjk∆.

Let

F :=

F11 F12 F13 F14F12 F22 F23 F24F13 F23 F33 F34F14 F24 F34 F44

.It follows from the construction that ∆3 divides det(F ), and since they bothhave degree 12, there exists a constant c ∈ C such that det(F ) = c ·∆3. It is aconsequence that c 6= 0, because Z∆ is not contained in a conic.

4

1.2. Outline

The entries in the adjoint matrix F adj are divisible by ∆2. Define the (4×4)-matrix

M := 1∆2 · F

adj

with linear entries mjk. A computation shows that

A :=

m11 cx0 +m12 m13 m14

cx0 +m12 m22 m23 m24m13 m23 m33 m34m14 m24 m34 m44

has determinant det(A) = c3 · f . Thus A is a determinantal representation ofS, since c 6= 0.

Remark 1.1.5. An important property to note is that the rank 2 locus in therepresentation obtained from Algorithm 1.1.4 is mapped to V(F11)∩V(F22) viathe projection πp. ♣

Classically, V(F11) is called a contact curve for V(∆), meaning that all thepoints in their intersection have even multiplicity.Remark 1.1.6. The first part of Algorithm 1.1.4 is to use Dixon’s method tofind a symmetric, determinantal representation of the ternary quartic ∆. Ingeneral, it is hard to apply Dixon’s method; the first application was published36 years after Dixon’s proof [Edge38]. Plaumann, Sturmfels and Vinzant notedin 2012 that it is difficult to construct suitable contact curves to use as inputin the algorithm [PSV12]. However, in the particular case of Algorithm 1.1.4,all the input is given. ♣Remark 1.1.7. We will exclusively use Algorithm 1.1.4 on surfaces which arenot nodal. Despite the fact that the algorithm is not proven to always workin that case we certainly encountered instances where it did not it hasbeen one of our most valuable tools for producing explicit examples of rationalquartic symmetroids. ♣

For a transversal symmetroid, the matrix representation (1.1) is unique upto conjugation by an invertible matrix [Ble+12, Proposition 11]. With Kummersymmetroids, only 10 of the 16 nodes are rank-2-points. However, for each nodethere exists a representation such that the node has rank 2 [Ott+14]. In [Jes16,Article 9], there is an 11-nodal quartic symmetroid with a node p that has rank3 in every representation. The projection from p has a ramification locus thatcannot be decomposed into cubics.

1.2 Outline

The rest of the thesis is organised as follows:

Chapter 2 presents the Veronese varieties with particular emphasis on theVeronese surface, which is the rank 1 locus of a symmetroid. We show inChapter 6 that the general projection of the Veronese surface to P3 is asymmetroid.

5

1. Introduction

Chapter 3 develops the theory of quadric hypersurfaces in P3, employingthe moduli space as an abstract tool for analysing symmetroids. This isthen applied superficially to two non-generic examples, but we revisit thetechnique more thoroughly later in Sections 4.3 and 5.1.

Chapter 4 is concerned with symmetroids that are singular along a conicsection. These are projections of del Pezzo surfaces and therefore rational.The surfaces are first studied through the projection itself and then viathe quadrics.

Chapter 5 serves to bridge the gap between symmetroids having finitely manydouble points and symmetroids that are double along a curve of degree2. It uses the same type of analysis of the quadrics as in Chapter 4 onsymmetroids that are double along a single line.

Chapter 6 features symmetroids that are degenerations of the surfaces fromthe previous chapters. It includes symmetroids that are singular along acubic curve, have a triple point or are cones. Among them is the Steinersurface. The chapter ends with a discussion of our work.

Appendix A is a means to reduce printing costs by collecting colour imagesthat would otherwise be scattered throughout the text.

Appendix B contains the majority of the computational work. It should beread in parallel with the other chapters. Readers who are only interestedin the results may skip this section entirely.

Appendix C gives a brief account of the ADE notation used in the text. Thecontent of this appendix is largely disjoint from the rest of the thesis andcan be ignored by readers who already have an understanding of rationaldouble points.

Throughout the thesis we assume familiarity with linear systems of divisorsand blow-ups. An excellent reference is [Har77, V, Section 4]. We also refer to[Ott+14] for basic facts about symmetroids.

6

CHAPTER 2

Veronese Varieties

The Italian geometers have erected, on somewhat shakyfoundations, a stupendous edifice: The theory of algebraicsurfaces. It is the main object of modern algebraic geometryto strengthen, preserve, and further embellish this edifice.

Oscar Zariski

HERE we review Veronese varieties, because the image V of a quadraticVeronese map is a quintessential example of a determinantal variety. Itssecant variety Sec(V ) is a symmetroid, with V being the rank 1 locus of

Sec(V ). We showcase some classical arguments involved in the study of thesekinds of varieties.

2.1 Quadratic Forms

Before we plunge into the examination of the Veronese embedding, we allowourselves a detailed digression. The relationship between quadratic forms andthe rank of the corresponding matrix is vital throughout the thesis.

Proposition 2.1.1. Let C be the conic in CP2 given by

xTAx = ax2 + 2bxy + cy2 + 2dxz + 2eyz + fz2 = 0,

where

A :=

a b db c ed e f

.Then we have that

(i) rankA = 1 if and only if C is a double line,

(ii) rankA = 2 if and only if C consists of two distinct lines,

(iii) rankA = 3 if and only if C is smooth.

2. Veronese Varieties

Proof. Since C is a conic, A cannot be the zero matrix.

(i) Assume that C is a double line. Then its equation is on the form

(ax+ by + cz)2 = a2x2 + 2abxy + b2y2 + 2acxz + 2bcyz + c2z2 = 0.

Thus

A =

a2 ab acab b2 bcac bc c2

,which has rank 1.Conversely, if rankA = 1, then in particular the (2× 2)-minors∣∣∣∣a b

b c

∣∣∣∣ = ac− b2∣∣∣∣a dd f

∣∣∣∣ = af − d2∣∣∣∣c ee f

∣∣∣∣ = cf − e2

vanish. Hence b =√ac, d =

√af and e =

√cf . Substituting a :=

√a,

c :=√c and f :=

√f , the equation defining C reduces to

a2x2 + 2acxy + c2y2 + 2afxy + 2cfyz + f2z2 =(ax+ cy + f z

)2 = 0.

In conclusion, C is a double line.

(iii) Let

f(x, y, z) := ax2 + 2bxy + cy2 + 2dxz + 2eyz + fz2,

then C is nonsingular if and only if the system of equations of partialderivatives,

∂f

∂x= 2ax+ 2by + 2dz = 0,

∂f

∂y= 2bx+ 2cy + 2ez = 0,

∂f

∂z= 2dx+ 2ey + 2fz = 0,

has no nontrivial solution. This is equivalent to the determinant∣∣∣∣∣∣2a 2b 2d2b 2c 2e2d 2e 2f

∣∣∣∣∣∣ = det(2A) 6= 0.

This occurs precisely when A has full rank.

The case (ii) now follows because we have exhausted the list of possible ranksand conic sections.

Remark 2.1.2. Proposition 2.1.1 extends easily to an (n × n)-matrix A. Thequadratic form is a square if and only if rankA = 1, and xTAx is singular ifand only if A is singular. Moreover, since the partial derivatives of xTAx vanishprecisely when Ax = 0, the Rank-Nullity theorem implies that the projectivedimension of the singular locus of the quadratic form is n− rankA− 1. ♣

8

2.2. Veronese Embedding

2.2 Veronese Embedding

Recall that the Veronese surface means the image of the map ν2 : P2 → P5

given by

[x0 : x1 : x2] 7→[x2

0 : x0x1 : x0x2 : x21 : x1x2 : x2

2].

If we let P5 have coordinates [y00 : y01 : y02 : y11 : y12 : y22], we see that thepolynomials

q1 := y00y11 − y201

q4 := y01y12 − y02y11

q2 := y00y12 − y01y02

q5 := y01y22 − y02y12

q3 := y00y22 − y202

q6 := y11y22 − y212

disappear on the Veronese surface. In fact, the Veronese surface is precisely thezero locus of the qi [Har77, I, Exercise 2.12]. This is equal to the rank 1 locusof the symmetric matrix

M :=

y00 y01 y02y01 y11 y12y02 y12 y22

.In light of Proposition 2.1.1 this can be interpreted as the set of points

[y00 : . . . : y22] ∈ P5

such that the quadratic form associated to M defines a double line.We note two properties about the Veronese surface:

Observation 2.2.1. Let V ⊆ P5 be the Veronese surface. Then

(i) if C is a curve on V , then there exists a hypersurface H ⊂ P5 such thatH ∩ V = C [Har77, I, Exercise 2.13],

(ii) there are no lines on V [Har92, Lecture 19].

More generally, a Veronese map of degree d is a morphism νd : Pn → PN ,where N =

(n+d

d

)− 1, defined by sending a point [a0 : . . . : an] to

[M0(a0, . . . , an) : . . . : MN (a0, . . . , an)].

The Mi are the different monomials of degree d in x0, . . . , xn. The map νd is anisomorphism onto its image, which is called a Veronese variety. The image ofa variety X ⊆ Pn under νd is a subvariety of PN [Har92, Lecture 2]. The imageof the special case νd : P1 → Pd is called a rational, normal curve.

Let PN have coordinates [y0 : . . . : yN ]. Then νd(Pn) equals the intersectionof the quadratic hypersurfaces V(yiyj − ykyl) for multi-indices (i, j, k, l) suchthat MiMj = MkMl. In the case of the quadratic Veronese map,

ν2 : Pn → P12 (n+1)(n+2)−1,

the image can be expressed as the zero locus of the (2×2)-minors of a symmetric(n+ 1)× (n+ 1)-matrix.

Any projective variety is isomorphic to an intersection of a Veronese varietywith a linear space [Har92, Exercise 2.9].

9


2.3 The Degree of a Veronese Variety

Before we can compute the degree of a Veronese variety, we need to recollectsome well-known results.

Definition 2.3.1. Let X ⊆ Pn be an irreducible, k-dimensional variety. IfΩ is a general (n − k)-plane, then the degree deg(X) of X, is the number ofpoints of intersection of Ω and X. If X is reducible and X =

⋃i Xi is its

decomposition into irreducible components, then the definition is extendedlinearly by deg(X) :=

∑i deg(Xi). ♠

Definition 2.3.2. Let X,Y ⊆ Pn be subvarieties. We say that X and Yintersect transversally at a point P ∈ X ∩ Y , if they are smooth at P and theirtangent spaces span TPPn. Let Zi be the irreducible components of X ∩ Y .Then X and Y intersect generically transversally if they for each i intersecttransversally at a general point Pi ∈ Zi. ♠

If dim(X) + dim(Y ) = n, then saying that X and Y intersect genericallytransversally is the same as saying that they intersect transversally.

Theorem 2.3.3 (Bézout, [Har92, Theorem 18.3]). Let X,Y ⊆ Pn be subvari-eties of pure dimensions k and l with k+ l ≥ n, and suppose that they intersectgenerically transversally. Then

deg(X ∩ Y ) = deg(X) · deg(Y ).

In particular, if k + l = n, then X ∩ Y consists of deg(X) · deg(Y ) points.

Theorem 2.3.4 (Bertini, [Har92, Theorem 17.16]). Let X be a quasi-projectivevariety, f : X → Pn a regular map, H ⊂ Pn a general hyperplane and putY := f−1(H). Then Ysing = Xsing ∩ Y .

One can use Bertini’s theorem to show that k general hypersurfaces inPn, for k < n, intersect transversally in a smooth, (n− k)-dimensional varietyX ⊂ Pn [Har92, Exercise 17.17].

Now we can turn to the task of computing the degree. Let

P := [a00 : a01 : a02 : a11 : a12 : a22] := [a20 : a0a1 : a0a2 : a2

1 : a1a2 : a22]

be the point on the Veronese surface V ⊂ P5 coming from P ′ := [a0 : a1 : a2]via the embedding ν2 : P2 → P5. Saying that a hyperplane H ⊂ P5 given by

b00y00 + b01y01 + b02y02 + b11y11 + b12y12 + b22y22 = 0

intersects V in P , amounts to saying that a certain curve vanish at P ′. Indeed,∑bijaij = 0

if and only if (∑bijxixj

)(a0, a1, a2) = 0.

In other words, there is a bijection between hyperplane sections of the Veronesesurface and conic sections in P2.

10

2.4. Projection of the Veronese Surface to P3

In precisely the same manner, for a Veronese map νd : Pn → PN of degreed, the inverse image ν−1

d (H) of a general hyperplane H ⊂ PN is a generalhypersurface of degree d in Pn. The degree of the Veronese variety νd(Pn) isequal to the number of intersection points of νd(Pn) and n general hyperplanesHi ⊂ PN . Let Yi := ν−1

d (Hi) ⊂ Pn be the inverse images. Bézout’s theoremgives that

deg(Y1 ∩ . . . ∩ Yn) = deg(Y1) · · · deg(Yn) = dn.

Bertini’s theorem implies that the intersection of the Yi has dimension 0. An(n− 0)-plane in Pn is just Pn itself, so there are dn points in

(Y1 ∩ . . . ∩ Yn) ∩ Pn = Y1 ∩ . . . ∩ Yn.

Subsequently, νd(Pn) has degree dn since νd is injective. A similar argumentshows that if Y ⊆ Pn is a variety of dimension k and degree e, then νd(Y ) ⊂ PN

has degree dke.In particular, ν2(P2) ⊂ P5 has degree 4. Its intersection with a general

hyperplane is a rational, normal curve in P4. This is the only incidence wherethe intersection of a Veronese variety with a general hyperplane is anotherVeronese variety [Har92, Exercise 18.14].

2.4 Projection of the Veronese Surface to P3

Projection from P5 to P3 can be done in two steps: First projecting from apoint p1 ∈ P5 to P4, and then projecting further from another point p2 ∈ P4 toP3. The composition P5 → P3 is said to be projection from the line L betweenp1 and p2. We shall consider the image of the Veronese surface when L is intwo different positions.

General Projection

The general projection of the Veronese surface V to P3 is called a Steiner surfaceS. In this section we will show that a Steiner surface has nodes along threelines which meet in a triple point. First, we need to make a few notes aboutthe secant variety Sec(V ) of V, starting with a simple definition.

Definition 2.4.1. The span ΓΦ of two subsets Γ,Φ ⊆ Pn is defined as thesmallest linear subspace of Pn containing Γ∪Φ. If Γ := P(W1) and Φ := P(W2)are projectivisations of subspaces W1,W2 ⊆ kn+1, then ΓΦ = P(W1 +W2). ♠

For linear subspaces Λ1,Λ2 ⊆ Pn, we have the inequality

dim(Λ1Λ2

)≤ dim(Λ1) + dim(Λ2) + 1.

Equality holds if and only if Λ1 and Λ2 are disjoint. In general,

dim(Λ1Λ2

)≤ dim(Λ1) + dim(Λ2)− dim(Λ1 ∩ Λ2),

with the convention that dim(∅) = −1 [Har92, Example 1.1].Recall that for an irreducible variety X ⊆ Pn, the secant line map is a

rational map s : X × X 99K G(1, n), where the Grassmannian G(1, n) is the

11


lines in Pn. The secant line map is defined on the complement (X ×X) \∆ ofthe diagonal ∆ by sending a pair of points (p, q) to the line pq connecting them.A point l ∈ im s is called a secant line of X. The secant variety of X,

Sec(X) :=⋃

l∈im s

l ⊆ Pn,

is the union of all secant lines of X.In general, it is hard to compute secant varieties, but for determinantal

varieties we have the following result:

Proposition 2.4.2 ([Har92, Exercise 11.29]). Let M be the projective space of(m×n)-matrices and let Mk ⊆M be the subvariety of matrices of rank at mostk. Assume that 2k < minm,n. Then the secant variety Sec(Mk) is equal tothe subvariety M2k ⊆M of matrices of rank at most 2k.

We know that the Veronese surface V is the rank 1 locus of

M =

y00 y01 y02y01 y11 y12y02 y12 y22

.However, M only parametrises the symmetric (3 × 3)-matrices, so Proposi-tion 2.4.2 is not directly applicable. Nevertheless, we shall see that the conclusionthat Sec(V ) is the rank 2 locus of M still holds true.

Any secant line of V can be written as ν2(p)ν2(q) for some points p, q ∈ P2.The line pq is mapped to a conic C ⊂ V ⊂ P5 under ν2. Then C spans a planeΠ. A general line in Π intersects C in two points, hence is a secant line to Cand thus to V . It follows that Sec(V ) is the union of the planes intersecting Vin a conic.

Explicitly, the planes are parametrised by the column space or equivalentlythe row space of M . For each [a : b : c] ∈ P2, the equation

a

y00y01y02

+ b

y01y11y12

+ c

y02y12y22

=

000

defines a plane which intersects V in the conic ν2(l), where l ⊂ P2 is the linegiven by ax0 + bx1 + cx2 = 0. This expresses that the columns are linearlydependent, so rank(M) is at most 2. Thus the equation for Sec(V ) is det(M) = 0.Note that V is the singular locus of Sec(V ).

As Sec(V ) is a hypersurface in P5, it has dimension 4. A k-dimensionalvariety X ⊆ Pn is said to have a deficient secant variety if

dim(Sec(X)) < min2k + 1, n.

The Veronese surface is the only smooth surface with a deficient secant variety[Har92, Lecture 11].

We are now ready to study what happens when V is projected from a generalline L to P3. Since

Sec(V ) = V(detM)

12


has degree 3, it will meet L in three distinct points P1, P2 and P3. Each Pi

defines a plane Πi ⊂ Sec(V ) intersecting V in a conic Ci. The projectionπL : P5 → P3 maps Πi to a line Li ⊂ P3 and surjects Ci two-to-one onto Li.This explains the promised double points along three lines. The situation isdisplayed in Figure 2.1.

Sec(V )

L

C1

C2

C3

P1

P2

P3

S

L1 L2

L3

πL

Figure 2.1: Under the projection from a general line L in P5 to P3, three conics onthe Veronese surface V are mapped two-to-one onto lines in a Steiner surface S.

The conics Ci can be thought of as ν2(li) for lines li ⊂ P2. Since the lines limeet pairwise, so must the Ci and therefore also the Li. Hence there are twopossible configurations, as shown in Figure 2.2 on the next page. Either all theLi lie in a plane or they form three axes. Suppose for contradiction that theLi lie in a plane. Then this L together with this plane spans a P4 ⊂ P5. Theintersection of this P4 with V will contain C1 + C2 + C3, which has degree 6.But deg

(P4 ∩ V

)= 4. This is impossible, so the lines Li meet in a single point

with multiplicity 3.Choose coordinates [z0 : z1 : z2 : z3] for P3 such that the singular lines of

the Steiner surface S are z1 = z2 = 0, z1 = z3 = 0 and z2 = z3 = 0. Then wemay assume by [Dol12, Section 2.1.1] that the equation for S is

z0z1z2z3 + z21z

22 + z2

1z23 + z2

2z23 = 0, (2.1)

after scaling the coordinates. We will henceforth refer to all general projectionsof the Veronese surface to P3 as the Steiner surface. The affine piece z0 = 1 isplotted in Figure 2.3 on the following page. Moreover, an explicit birationalmap P2 99K S can be given by sending [x0 : x1 : x2] to[

(−x0 + x1 + x2)2 : (x0 − x1 + x2)2 : (x0 + x1 − x2)2 : (x0 + x1 + x2)2].This map is surjective [Dol12, Section 2.1.1].Remark 2.4.3. In Section 6.1, we find a symmetric matrix having (2.1) as itsdeterminant, showing that the Steiner surface is a symmetroid. ♣

13


C1

C2 C3

πL or

Figure 2.2: Pairwise intersection of conics and lines.

Figure 2.3: The Steiner surface from different angles.

Duality and Special Projection

Later, when hunting for rational symmetroids, we shall be interested in quarticsthat are singular along a curve. As an exercise, we find a line L ⊂ P5 such thatthe dual of the Veronese surface V intersected with the orthogonal subspaceL⊥ of L, is singular along a curve. We begin by clarifying the terminology.

Definition 2.4.4. The dual projective space (Pn)∨ consists of all hyperplanesin Pn. That is, the point [a0 : . . . : an] ∈ (Pn)∨ corresponds to the hyperplaneV(a0x0 + · · ·+ anxn) ⊂ Pn. ♠

Definition 2.4.5. Let X ⊂ Pn be an irreducible, projective variety. A hyper-plane H ⊂ Pn is said to be tangent to X if H contains an embedded tangentspace TPX at some smooth point P ∈ X. ♠

Definition 2.4.6. For an irreducible, projective variety X ⊂ Pn, we define thedual variety X∗ ⊂ (Pn)∨ to be the closure of the set of all tangent hyperplanesof X. ♠

Theorem 2.4.7 (Reflexivity Theorem, [Tev03, Theorem 1.7]). Let X ⊆ Pn bean irreducible, projective variety. Then

(i) X∗∗ = X.

(ii) More precisely, if x ∈ X and H ∈ X∗ are smooth points, then H is tangentto X at x if and only if x, regarded as a hyperplane in (Pn)∨, is tangentto X∗ at H.

14


Definition 2.4.8. For a projective subspace X ⊂ Pn, let the orthogonal sub-space X⊥ ⊂ (Pn)∨ be the set of hyperplanes that contain X. ♠

To motivate our undertaking, consider a line L ⊂ P5. Then L⊥ ⊂ (P5)∨is a P3. Denote the projection πL(V ) of V from L by V . In the irreduciblecase, projecting from L and then taking the dual is the same as taking thedual first and then intersecting with L⊥, that is V ∗ = V ∗∩ L⊥. Then we coulduse Theorem 2.4.7 to deduce V . We will however shortly see that V ∗∩ L⊥ isreducible in our case.

Let us now turn to the task of finding the dual of the Veronese surface. Ahyperplane is tangent to V if and only if its intersection with V is singular.From Section 2.3, we know that a hyperplane section of V is the image ν2(C)of a conic C ⊂ P2. Thus the only singular hyperplane sections are those thatbreak up into pairs of conics, meaning they are the image of either a line-pairor double line in P2. Thus Proposition 2.1.1 implies that V ∗ is parametrised bythe rank 2 locus of y00 y01 y02

y01 y11 y12y02 y12 y22

.This is precisely the same description as the one we gave for Sec(V ). Note thatTheorem 2.4.7 implies that Sec(V )∗ ∼= V .

For simpler notation, let V1 := V , Y1 := Sec(V1), Y2 := V ∗1 and V2 := Y ∗1 ,so that Y2 = Sec(V2). Strict notational propriety dictates that V should bedenoted V1, but we shall keep the original symbol in this case.

Since Sing Y2 = V2, we have that Sing(Y2 ∩ L⊥

)= V2 ∩ L⊥. Thus finding a

line L ⊂ P5 such that Y2 ∩ L⊥ is singular along a curve, amounts to finding aP3 ⊂ P5 such that P3 ∩ V2 contains a curve.

What are the possible curves in the intersection of the Veronese surface anda P3? As noted above, the intersection of V2 and a hyperplane is the imageν2(C) of a conic C ⊂ P2. Because P3 ⊂ P5 is contained in a hyperplane, wededuce that V2 ∩ P3 = ν2(X) for some subset X ⊆ C of a conic C ⊂ P2. A P3

in P5 is contained in a pencil of hypersurfaces, so X must be the base locus ofa pencil of conics. Generically, X will consist of 4 points, but specially it canbe a line and a point. See Figure 2.4 on the next page. In the latter case, ν2(X)consists of a conic C and a point P , where P is either an embedded componentin C or P is not in the plane spanned by C, since ν2(X) span P3.

Consequently, if L ⊂ P5 is such that V ∗1 ∩L⊥ is singular along a curve, thenV2 ∩ L⊥ contains a conic C. The conic C spans a plane Π. Because Π ⊂ L⊥,we have that L ⊂ Π⊥. In addition, Π is a plane intersecting V2 in a conic; inSection 2.4 we demonstrated that Π ⊂ Sec(V2) = Y2 = V ∗1 . Thus points in Πcorrespond to hyperplanes that are tangent to V1. It follows that Π⊥ is thetangent plane TqV1 at some point q ∈ V1. In summary, if Y2 ∩ L⊥ is singularalong a curve, then L lies in a tangent plane of V1.

Suppose that L lies in the tangent plane TqV1 at some point q ∈ V1. There isa pencil of lines in TqV1 passing through q. This corresponds to conic sections inV1 meeting in q via the embedding ν2 : TqV1 → P5. Any such line l ⊂ TqV1 willintersect L in a point q′. The projection πL factors through the projection from q′.It follows that the conic ν2(l) is projected two-to-one onto a line. Consequently,πL : V1 → V is a double cover and V is a union of concurrent lines; a cone.

15


We now give an account of V ∗1 ∩L⊥. First, note that Y2∩L⊥ = Sec(V2)∩P3

is a cubic. If Sing(Y2 ∩ L⊥

)consists of the conic C and the point P , then the

plane Π spanned by C is contained in both Sec(V2) and P3. Hence Y2 ∩ L⊥is reducible, consisting of a plane and some quadric component. We deducethat the quadric is a cone with base C and apex P . There is a depiction inFigure 2.5.

(a) Pencil of ellipses.

l

(b) Pencil of line-pairswhere a line l and a pointnot on l is fixed.

l

(c) Pencil of line-pairswhere a line l and a pointon l is fixed.

Figure 2.4: The base locus of a pencil of conics consists generically (a) of 4points, but consist in special cases (b, c) of a line and a point.

(a) V (b) V ∗∩ L⊥

Figure 2.5: The projection of the Veronese surface V to P3 from a line lyingin a tangent plane of V (a) and the dual of V intersected with the orthogonalsubspace of the line (b).

16

CHAPTER 3

Space of Quadrics in P3

I am a great fan of science, but Icannot do a quadratic equation.

Terry Pratchett

EVERY point x on a symmetroid V(det(M)) is associated with a symmetricmatrix M(x). To M(x) we associate the quadratic form aTM(x)a. Inorder to fully understand symmetroids, we must explore the quadrics. We

begin in full generality, describing the moduli space of quadrics in P3, beforewe specialise to pencils and webs of quadrics.

3.1 Moduli Space

Consider P9 as the space of quadratic surfaces in P3. We stratify this spaceaccording to the ranks of the symmetric matrices representing the quadraticforms. More precisely, parametrise P9 with the matrix

M :=

y00 y01 y02 y03y01 y11 y12 y13y02 y12 y22 y23y03 y13 y23 y33

and let Qi denote the set of points where M has rank at most i.

Then Q4 = P9 and Q3 is the discriminant quartic hypersurface det(M) = 0.The rank 2 locus Q2 has dimension 6 and degree 10. Finally, Q1 equals theimage of the Veronese map ν2 : P3 → P9, so it is a threefold of degree 8.

The same argument as in Section 2.4 shows that Sec(Q1) is the union ofthe planes intersecting Q1 in a conic. Furthermore, if the conic is ν2(l), for aline l ⊂ P3 with ideal

〈a0x0 + a1x1 + a2x2 + a3x3, b0x0 + b1x1 + b2x2 + b3x3〉,

then the plane Π ⊂ Sec(Q1) intersecting Q1 in ν2(l) is given by

a0m0 + a1m1 + a2m2 + a3m3 = 0,b0m0 + b1m1 + b2m2 + b3m3 = 0,

(3.1)

3. Space of Quadrics in P3

where mi are the columns of M . This shows that M has nullity at least 2 in Π,so the rank is at most 2. Ergo Sec(Q1) = Q2 and thus it is deficient.

Since any point in (P3 × P3) \ ∆ yields one of these planes, we have de-scribed Q2 as a 6-dimensional family of planes. However, we can also give ita characterisation as a 3-dimensional family of 3-spaces. We begin with someterminology.

Definition 3.1.1. Let X ⊂ Pn be a smooth variety. Then its tangential varietyTan(X) is defined as the closure of the union of all embedded tangent spacesTPX for P ∈ X. ♠

Notation 3.1.2. If f is the polynomial of a quadric in P3, let [f ] denote itscorresponding point. ♦

Remark 3.1.3. Unfortunately, Notation 3.1.2 is vague because the same symbolis used for two different, albeit similar, concepts. In this chapter, [f ] means thecorresponding point in P9, with the exception of Lemma 3.2.2. In Lemma 3.2.2and Chapter 4, [Q] is the point with the property that aTM([Q])a = Q. ♣

Take a point[l2]∈ Q1, where l is a linear form. We have from [Dol12,

Exercise 1.18] that the embedded tangent space T[l2]Q1 is the set

T[l2]Q1 = l · l′ | l′ is a linear form. (3.2)

Furthermore, it is a consequence of the Fulton–Hansen connectedness theoremthat Sec(X) is deficient if and only if Tan(X) = Sec(X) for a variety X [FH79,Corollary 4]. Together, this proves the claim that Q2 is a 3-dimensional familyof 3-spaces.

The purpose of cogitating P9 is to give us another tool for analysing quarticsymmetroids. Given a quartic symmetroid S ⊂ P3 with equation∣∣∣∣∣∣∣∣

l00 l01 l02 l03l01 l11 l12 l13l02 l13 l22 l23l03 l13 l23 l33

∣∣∣∣∣∣∣∣ = 0,

where the lij are linear forms in the variables x0, x1, x2 and x3, then S can beviewed as the intersection of Q3 and the 3-space

H :=

[l00(x0, x1, x2, x3) : . . . : l33(x0, x1, x2, x3)][x0 : x1 : x2 : x3] ∈ P3.

Conversely, any P3 ⊂ P9 gives rise to a quartic symmetroid: It arises as thediscriminant hypersurface of the web of quadrics. The symmetroid defined bythe determinant of a matrix which parametrises a space of quadrics, is oftencalled the discriminant hypersurface. The term originates from the following:The quadratic form ax2

0 + bx0x1 + cx21 can be written as xTAx, where

A :=[a 1

2b12b c

].

Then det(A) is the discriminant of the dehomogenised polynomial ax20 +bx0 +c,

up to multiplication with −4.

18

3.2. Web of Quadrics

Remark 3.1.4. If the symmetroid S is a cone, then H only defines a net ofquadrics. It degenerates further if S is defined by a polynomial in only one ortwo variables. ♣

Bézout’s theorem implies that a P3 will generically meet Q2 in 10 pointsand miss Q1. If the 3-space is chosen to be fully contained in Q3 for instancea tangent space T[l2]Q1 then the polynomial defining the symmetroid willbe the zero polynomial, since the matrix is rank deficient at all points. Fromthe description (3.2) of T[l2]Q1, it is clear that quadrics in this space have theplane V(l) in common. Observe that the equations (3.1) show that quadrics ina secant plane Π are pairs of planes with a fixed line of intersection, since thematrices in Π have the same null space.

3.2 Web of Quadrics

Let S ⊂ P3 be a quartic symmetroid given by the symmetric matrixM := M(x)in the variables x0, x1, x2 and x3. Denote the web of quadrics aTMa byW . Theunion of all the singularities of the quadrics inW form an algebraic set, called thevariety of singular loci1 of W . It is easy to describe: Recall from Remark 2.1.2that the singular locus of the quadratic form aTMa is the solutions of Ma = 0.Let A := A(a) be the matrix in the variables a0, a1, a2 and a3 such thatMa = Ax. Then Ma = 0 if and only if Ax = 0, so the variety of singular lociis given by det(A) = 0.

Let S be a generic quartic symmetroid and S′ the associated variety ofsingular loci. Then S and S′ are birationally equivalent. Indeed, it follows fromRemark 2.1.2 that the 10 rank-2-points on S are replaced by lines in S′; onecan obtain S′ by blowing up S in these points.

The connection between the symmetroid and its variety of singular locimight not be as clear in general, but we see some patterns in Chapter 4. Therewe also compute the quadrics Q ∈W that have singularities in the base locusof W , so we make a general note about this first.

Notation 3.2.1. The base locus of a linear system of divisors d is denotedBl(d). ♦

Lemma 3.2.2. Let W be the linear space of quadrics in Pn given by yTMy forsome symmetric (n+1)×(n+1)-matrixM of linear forms in k ≤ 1

2 (n+1)(n+2)variables. Let D be the discriminant hypersurface V(det(M)). If [QP ] ∈W is apoint such that the quadric QP is singular at P ∈ Bl(W ), then [QP ] ∈ SingD.

The proof is due to Kristian Ranestad.

Proof. We claim that the tangent space to D in W at a point [QP ], such thatQP is singular at a point P ∈ Pn not necessarily in the base locus is thehyperplane of quadrics that vanish at P .

In order to see this, choose coordinates such that [QP ] := [1 : 0 : . . . : 0] andsuch that the hyperplane of quadrics vanishing at P passes through the points[0 : 1 : 0 : . . . : 0], . . . , [0 : . . . : 0 : 1 : 0]. Thus if

M(x) := M0x0 + · · ·+Mkxk,

1Occasionally singular loci is replaced by singularities or singular points.

19


then QP = yTM0y and yTMiy vanish at P for i = 1, . . . , k− 1. In addition, thehyperplane of quadrics vanishing at P has equation xk = 0.

Moreover, choose coordinates such that P := [1 : 0 : . . . : 0]. Then M0 is onthe form

0 0 · · · 00 m11 · · · m1n

...... . . . ...

0 m1n · · · mnn

and the (0, 0)-entry in Mi is 0 for i = 1, . . . , k − 1. This implies that there areno xn

0xi terms in F (x) := det(M) for i = 0, . . . , k − 1. Hence

∂F

∂xi(P ) = 0,

for i = 0, . . . , k − 1. Thus the tangent space to D at [QP ] has equation xk = 0,as claimed.

Finally, if P ∈ Bl(W ), then all quadrics inW vanish at P . Hence the tangentspace of D at [QP ] equals W , so D is singular at [QP ].

Remark 3.2.3. The converse to Lemma 3.2.2 does not hold. In Example 4.1.7we see that there are quadrics Q, for [Q] in the singular locus of the symmetroid,that are not singular in any of the base points. ♣

3.3 Some Unusual Symmetroids

We give here an exposition of two known symmetroids that are not nodal.

Example 3.3.1 (Pillow, [Ott+14, Example 5.2]). The symmetroid defined bythe matrix

P :=

x3 x0 0 x0x0 x3 x1 00 x1 x3 x2x0 0 x2 x3

is called the pillow for its shape, which is shown in Figure 3.1 on the next page.

It is irreducible and has rank 2 along the two lines L1 := V(x0, x3) andL2 := V(x1 − x2, x3), as well as in the four coplanar corners of the pillow.Therefore the surface has the unusual property that the rank 2 locus is containedin a quadric. Consult Appendix B.1 for the Macaulay2 code for the calculationsin this section.

Consider

Pa =

x3 x0 0 x0x0 x3 x1 00 x1 x3 x2x0 0 x2 x3

a0a1a2a3

=

a1 + a3 0 0 a0a0 a2 0 a10 a1 a3 a2a0 0 a2 a3

x0x1x2x3

=: Ax.

So Pa = 0 is equivalent to Ax = 0. In other words,

det(A) = a2(a1 + a3)(−a20 + a2

1 − a22 + a2

3) = 0

20

3.3. Some Unusual Symmetroids

describes the variety of the singularities of all the quadrics defined by the pillow.It is the union of two planes and a smooth quadric Q.

Each of the quadrics defined by the corners of the pillow are singular along aline on Q. The quadratic form of a point [0 : x1 : x2 : 0] ∈ L1 is 2a2(x1a1+x2a3).The base locus of the pencil of quadrics parametrised by L1, is the union ofthe plane V(a2) and the line V(a1, a3). Hence the singular loci of the pairs ofplanes in this pencil can be summarised as all lines in V(a2) passing throughthe intersection point V(a2) ∩ V(a1, a3). The situation is completely analogousfor L2, substituting V(a1 + a3) for V(a2) and V(a0, a2) for V(a1, a3).

We can formulate this in light of the pillow being the intersection of Q3 anda P3 in P9. Since all the quadrics along Li share a plane, the P3 intersects twotangent spaces T[l2

i ]Q1 in a line each. ♥

Figure 3.1: The affine piece x3 = 1 of the pillow. The double lines lie in theplane at infinity.

Example 3.3.2 (Toeplitz, [Ott+14, Example 5.1]). Consider the surface definedby the symmetric Toeplitz matrix

T :=

x0 x1 x2 x3x1 x0 x1 x2x2 x1 x0 x1x3 x2 x1 x0

.The determinant det(T ) factors as a product of two irreducible quadratic forms.Each of the components are singular in a single point, wherein the Toeplitzsymmetroid has rank 1. The components intersect in the union of the twistedcubic curve

C := V(x2

0 − 2x21 + x0x2, x0x1 − 2x1x2 + x0x3, x

20 − x0x2 − 2x2

2 + 2x1x3)

and the line L := V(x0−x2, x1−x3). See Figure 3.2 on the following page. Thisintersection forms the rank 2 locus of the symmetroid, with the rank-1-points

21


being C ∩L. The different rank k loci are computed by the Macaulay2 code inAppendix B.2.

The variety of the singular loci of the quadrics aTTa of the Toeplitz surfaceis given by the asymmetric matrix

A :=

a0 a1 a2 a3a1 a0 + a2 a3 0a2 a1 + a3 a0 0a3 a2 a1 a0

.Its determinant is the product

det(A) = (a0 + a1 + a2 + a3)(a0 − a1 + a2 − a3)(a2

0 − a0a2 + a1a3 − a23).

It should be noted that quadrics from the rank-1-points are double along thetwo planes Π1 := V(a0+a1+a2+a3) and Π2 := V(a0−a1+a2−a3), respectively.

Because L is a secant line of Q1, we know that the quadrics along L makeup a pencil of pairs of planes with a common intersection line, Π1 ∩Π2. We canalso see this by computing the quadrics along L. The quadratic form aTTa ata point [x0 : x1 : x0 : x1] ∈ L is given by

x0(a2

0 + a21 + a2

2 + a23 + 2(a0a2 + a1a3)

)+ 2x1(a0a1 + a0a3 + a1a2 + a2a3).

This splits as ((a0 + a2)c+ (a1 + a3)d

)((a0 + a2)d+ (a1 + a3)c

),

where c and d are complex numbers satisfying

cd = x0,

c2 + d2 = x1.

We see that the base locus is indeed the line V(a0 +a2, a1 +a3) = Π1∩Π2. ♥

(a) (b)

Figure 3.2: The Toeplitz symmetroid (a) and its two components (b).

Pencils of Quadrics with Rank at Most 2

In both of the above examples there were pencils consisting of quadrics thathad rank at most 2, so we make a general note about such pencils.

22

3.3. Some Unusual Symmetroids

According to Remark 2.1.2, if the symmetric matrix defining a quadraticform has rank 2, then the singular locus of the quadric is a linear space of co-dimension 2. It follows that the quadric is the union of two distinct hyperplanes.Moreover, the hyperplanes coincide if and only if the rank is 1.

Consider a pencil of quadrics al1l2 + bl3l4, where [a : b] ∈ P1 and the li arelinear forms, such that each quadric in the pencil has at most rank 2. Thenthere are two possibilities: Either l3 is a scalar multiple of l1, or there existscalars c1, c2, d1 and d4 such that

l3 = c1l1 + c2l2

l4 = d1l1 + d2l2(3.3)

are linear combinations of l1 and l2. The pencil would otherwise contain aquadric of higher rank.

In the first case, the hyperplane V(l1) and the subspace V(l2, l4) is fixed inthe pencil. For us, this is a plane and a line, as in Example 3.3.1. In the secondcase, the base locus of the pencil is the double linear subspace V(l1, l2). In ourcase, this is a double line. Substituting (3.3), an element in the pencil becomes

bc1d1l21 + (a+ bc1d2 + bc2d1)l1l2 + bc2d2l

22,

which is the square(√c1d1l1 ±

√c2d2l2

)2 for[±2√c1c2d1d2 − c1d2 − c2d1 : 1

]∈ P1.

This is two distinct points unless c1c2d1d2 = 0. Hence the second kind of pencilshas generically two rank-1-points, as in Example 3.3.2. If c1c2d1d2 = 0, thenthe pencil is also of the first kind.

23

CHAPTER 4

Symmetroids with a Double Conic

A good stock of examples, as large as possible, isindispensable for a thorough understanding of anyconcept, and when I want to learn something new,I make it my first job to build one.

Paul Halmos

THE singular locus of the pillow in Example 3.3.1 has a description similarto that of a projected quartic del Pezzo surfaces, so we are led to pursuethese surfaces. A quartic surface in P3 with a double conic is the projection

of a del Pezzo surface. It is well-known that del Pezzo surfaces are rational.Pictures of some of the surfaces in this chapter can be found in Appendix A.

4.1 Mimicking the Pillow

A variety X ⊂ Pn is called nondegenerate if it is not contained in a proper,linear subspace. The classical definition of a del Pezzo surface is as follows:

Definition 4.1.1. A del Pezzo surface is a nondegenerate, irreducible surfaceof degree d in Pd that is not a cone and not isomorphic to a projection of asurface of degree d in Pd+1. ♠

It can be showed that surfaces satisfying this definition are also del Pezzosurfaces in the more general definition:

Definition 4.1.2. A normal, algebraic surface S is called a del Pezzo surfaceif its canonical sheaf ωS is invertible, ω−1

S is ample and all its singularities arerational double points. ♠

One of the reasons we are interested in del Pezzo surfaces is the followingresult:

Theorem 4.1.3 ([Dol12, Theorem 8.1.15]). Let X be a minimal resolution ofa del Pezzo surface. Then, either X ∼= F0, X ∼= F2 or X is obtained from P2 byblowing up N ≤ 8 points.

Here Fn denotes the Hirzebruch surfaces. The importance of Theorem 4.1.3is that it shows that del Pezzo surfaces are rational. Before we can describe theprojection of a quartic del Pezzo surface, we note that they are Segre surfaces:

4. Symmetroids with a Double Conic

Theorem 4.1.4 ([Dol12, Theorem 8.6.2]). Let S be a del Pezzo surface ofdegree 4. Then S is a complete intersection of two quadrics in P4.

The converse also holds [Cas41]. Hence S is the base locus of a pencil ofquadrics.

We are now ready to look at the projections of del Pezzo surfaces. Take theprojection centre p to be a point in P4 not on S. Let Qp be the unique quadricin the pencil that contains p.

Theorem 4.1.5 ([Dol12, Theorem 8.6.4]). Assume that the quadric Qp isnonsingular. Then the projection X of S from p is a quartic surface in P3

which is singular along a nonsingular conic. Any irreducible quartic surface inP3 which is singular along a nonsingular conic arise in this way from a Segresurface S in P4. The surface S is nonsingular if and only if X is nonsingularoutside the conic.

Similarly, surfaces with nodes along singular conics can be obtained bytaking the projection centre to be a singular or nonsingular point on a singularquadric Q from the pencil. They have been classified by Segre [Seg84]. It is alsoknown that the second statement of the theorem irreducible quartic surfacesthat are singular along a conic are projected del Pezzo surfaces is true evenwhen the conic is singular. In fact, Dolgachev’s proof of the second statementis still correct if the word nonsingular is omitted. We repeat it here with moredetails.

Proof. Let F := V(f) ⊂ P3 be an irreducible quartic surface with a conic sectionC ⊆ SingF . Choose coordinates such that C spans the plane V(x0). Then theideal IC of C is generated by x0 and a quadratic form Q. We may assume thatQ is a linear combination L of the monomials in

B1 :=x2

1, x1x2, x1x3, x22, x2x3, x

23.

We take B2 :=x2

0, x0x1, x0x2, x0x3, Q

as a basis for the vector space ofquadrics containing C. This gives rise to a rational map ϕ : P3 99K P4, sendinga point p to

[x2

0(p) : x0x1(p) : x0x2(p) : x0x3(p) : Q(p)].

Consider the quartic x20Q. It is the linear combination L of the monomials

inx2

0s | s ∈ B1

= q1q2 | q1, q2 ∈ B2 \ x0, Q. Change coordinates suchthat B2 are the new coordinates in P4. Then x2

0Q − L is a quadratic relationin these coordinates. It is satisfied by all points in the image of ϕ. Henceϕ(P3) ⊆ V

(x2

0Q− L)is contained in a quadric.

Since C ⊆ SingF , we have f ∈ I2C , so there exists a quadric q, a linear form

L′ and a constant a such that f = qx20 + L′x0Q+ aQ2. Now f is written as a

quadratic form in B2, so it defines a quadric hypersurface in P4.Let S := VP4

(x2

0Q− L)∩ VP4(f). It is the complete intersection of two

quadrics, and thus a quartic del Pezzo surface. Note that ϕ maps VP3(x0) tothe point P := [0 : 0 : 0 : 0 : 1]. The projection from P is the inverse of ϕ. Itfollows that F is the image of S under the projection.

We automatically get that the pillow is a projected del Pezzo surface andtherefore rational. We want to see if other projections of del Pezzo surfaces canbe realised as symmetroids. The pillow is singular along a conic section, but

26

4.1. Mimicking the Pillow

it also has four nodes outside of the two lines. To mimic this, we require theoriginal del Pezzo surface in P4 to have four nodes.

Dolgachev gives a complete list of the possible singularities on quartic delPezzo surfaces and the configuration of the five points needed to construct themas blow-ups of P2 [Dol12, Section 8.6.3]. For convenience, this is repeated witha minor change1 in Table 4.1.

Singularities Position of pointsA1 P1, P2, P3 are collinear

A1 +A2 P3 P2 P1 and P5 P4

A1 +A3 P3 P2 P1; P5 P4 and P1, P4, P5 are collinear

2A1P1, P2, P3 and P1, P4, P5 are collinear, orP2 P1 and P3, P4, P5 are collinear

2A1 +A2 P3 P2 P1; P5 P4

2A1 +A3P3 P2 P1; P5 P4;

P1, P2, P3 and P1, P4, P5 are collinear3A1 P2 P1; P4 P3 and P1, P2, P5 are collinear4A1 P2 P1; P4 P3; P1, P2, P5 and P3, P4, P5 are collinearA2 P3 P2 P1

A3P4 P3 P2 P1, or

P3 P2 P1 and P1, P2, P4 are collinearA4 P5 P4 P3 P2 P1

D4 P4 P3 P2 P1 and P1, P2, P5 are collinearD5 P5 P4 P3 P2 P1 and P1, P2, P3 are collinear

Table 4.1: Possible singularities on quartic del Pezzo surfaces and how to obtainthem by blowing up P2 in five points. Here Pi Pj denotes that Pi is a pointon the exceptional divisor over Pj , and that Pi, Pj , Pk are collinear means thatthe linear equivalence class l − ei − ej − ek is effective.

First we compute the ideal of a 4-nodal del Pezzo surface S. The calculationsuses the fact that S is the image of the map induced by the linear system ofcubics passing through the points Pi.

Example 4.1.6 (4-nodal quartic del Pezzo surface). Following the 4A1 entryin Table 4.1, take P1, P3 and P5 to be distinct points in P2. For i = 1, 3, let Pi+1be the points in the exceptional divisor Ei over Pi, corresponding to the tangentdirection at Pi given by the line PiP5. Then the linear equivalence classes

e1 − e2, l − e1 − e2 − e5,

e3 − e4, l − e3 − e4 − e5

are effective. Here ei denotes the linear equivalence class of Ei and l is theclass of the pullback of a line in P2. Moreover, the intersection with one of

1In later tables, Dolgachev utilises that there are inequivalent linear systems giving rise toquartic del Pezzo sufaces with two A1 singularities. However, the particular table in questionlists two equivalent systems.

27


these curves and a general hyperplane section 3l −∑5

i=1 ei is empty, so theyare contracted to points.

In Appendix B.3 we use [1 : 0 : 0], [0 : 1 : 0] and [0 : 0 : 1] as P1, P3 and P5,respectively, and

x20x1, x0x

21, x0x1x2, x0x

22, x1x

22

as a basis for the linear system of cubics passing through the Pi. This linearsystem induces a rational map ϕ : P2 99K P4. The image of ϕ is the del Pezzosurface S with ideal

〈x1x3 − x0x4, x22 − x0x4〉.

We see that it is indeed contained in a pencil of quadrics. The surface has nodesin [1 : 0 : 0 : 0 : 0], [0 : 1 : 0 : 0 : 0], [0 : 0 : 0 : 1 : 0] and [0 : 0 : 0 : 0 : 1]. ♥

Now that we have an explicit del Pezzo surface, we can study its projections.

Example 4.1.7. We continue with the linear system from Example 4.1.6. Thetwo families of conic sections l − e5 and 2l − e1 − e2 − e3 − e4 both avoid thesingularities. Also, their intersection number is

(l − e5) · (2l − e1 − e2 − e3 − e4) = 2.

Hence we can take a curve from each family and they will intersect in twopoints, which spans a line L. We choose the projection centre p to be a pointon L, but not on the del Pezzo surface S. If the target P3 does not containthe two conics, then they are collapsed to double lines under the projectionmap πp : P4 99K P3. With a suitable choice of p and P3, the four nodes are notmapped onto the double lines. Also, if p is contained in the P3 spanned by thenodes, their images are coplanar.

In Appendix B.4 we do this for a concrete example and find the quartic Xwith equation

f(x0, x1, x2, x3) = x20x

22 + 2x0x1x

22 + x2

1x22 − x2

0x1x3 − x0x21x3

+ 2x0x22x3 + 2x1x

22x3 − x0x1x

23 − x2

1x23 + x2

2x23.

It is singular along two intersecting lines, L1 and L2, and in four coplanar points,p1, p2, p3 and p4. In Appendix B.1 we noted that there are embedded points inthe singular locus of the pillow in addition to the intersection between the lines.The same is true for this projected del Pezzo surface, but now we are able toexplain them as the branch locus of πp of the two conics on S.

For Algorithm 1.1.4, we consider the further projection to P2 from the nodep1 := [1 : 0 : 0 : 0]. The ramification locus is given by

x21x

23(x1 + 2ix2 − x3)(x1 − 2ix2 − x3) = 0.

In other words, the branch curve consists of two double lines and two reducedlines, l1 and l2. Since the projection centre is in the plane spanned by the othernodes on X, their images are collinear in P2. One of them, p4, is mapped to theintersection between the two reduced lines, whereas the other two are pointson either of the double lines. The remaining intersection points in the branchlocus are the images of the embedded points. This is evinced in Figure 4.1.

28

4.1. Mimicking the Pillow

L1

L2

l1

l2

p4

p3

p2

q2

q′2

q1q′1

Figure 4.1: The ramification locus of the surface X in Example 4.1.7 projectedfrom a node. It consists of two double lines and two reduced lines. The dashedline is not part of the ramification locus. The points marked with • are theimages of the embedded points and indicates the image of a node.

There are several ways of dividing the ramification locus into two cubics.If we let F11 in Algorithm 1.1.4 be x3(x1 + 2ix2 − x3)(x1 − 2ix2 − x3), whereL1 := V(x3) is the projected image of L1, we find the symmetric matrix

A1 :=

0 4x0 4x1 4x2

4x0 −4x3 2x1 + 2x3 04x1 2x1 + 2x3 −4x1 −2x24x2 0 −2x2 −x3

,whose determinant is det(A1) = 64f . Although we constructed X to imitatethe pillow, there is a seemingly remarkable difference between the two. Therank 2 locus of the pillow is the whole singular locus, apart from the embeddedpoints. However, for X the rank 2 locus is only L1, the two embedded pointsq2, q

′2 ∈ L2 \ L1 and two of the four nodes, p1 and p2.If we instead define F11 := x1(x1 + 2ix2 − x3)(x1 − 2ix2 − x3), where V(x1)

is the image of L2, we get

A2 :=

0 4x0 + 8x3 −4x3 4x2

4x0 + 8x3 4x1 − 8x3 −2x1 + 6x3 −4x2−4x3 −2x1 + 6x3 −4x3 2x24x2 −4x2 2x2 −x1

.Since A1 and A2 have different eigenvalues at the point [0 : 1 : 0 : 0], they arenot conjugate. For A2 the situation is opposite of A1: It has rank 2 along L2,the two embedded points q1, q

′1 ∈ L1 \ L2 and the two nodes p1 and p3.

29


While the remaining node p4 is a rank-3-point for both A1 and A2, theprojection from p4 splits into cubics, so it is possible that it has rank 2 in anotherrepresentation. We suspect that there might exist a symmetric matrix A3 withdet(A3) = f , where the rank 2 locus equals the singular locus. Unfortunately,since we have to work over Q with Macaulay2, we have been unable to separatethe complex conjugate lines l1 and l2 in the computations, thereby not checkingall the ways to divide the branch curve into cubics.

Let W1 denote the web of quadrics aTA1a. The matrix

M1 :=

4a1 4a2 4a3 04a0 2a2 0 −4a1 + 2a20 4a0 + 2a1 − 4a2 −2a3 2a10 0 4a0 − 2a2 −a3

is such that M1x = A1a. The variety of singular loci of the quadrics in W1 isgiven by det(M1) = 0. It is the union of a plane Π1 and a 3-nodal cubic C1.The base locus Bl(W1) is five points spanning P3: The singularities on C1 andtwo more points in C1 ∩ Π1. One of the nodes on C1 is contained in C1 ∩ Π1and has multiplicity 2 in Bl(W1).

For P ∈ Bl(W1), we look at the quadrics QP which are singular at P andconsider the corresponding points [QP ] ∈ P3. From Lemma 3.2.2, we know that[QP ] ∈ SingX. For the double point P ∈ Bl(W1), the set of [QP ] is the lineL2. For the other points P ∈ Bl(W1), [QP ] are the points p3, p4, q1 and q′1,respectively.

The situation is completely analogous for A2, albeit reversed. The mainaspects are condensed into Table 4.2.

The surface X is shown in Figure A.5a on page 70. ♥

F11 F22 rank 2 [QP ] for P ∈ Bl(W )

L1 + l1 + l2 L1 + 2L2 L1, p1, p2, q2, q′2 L2, p3, p4, q1, q

′1

L2 + l1 + l2 2L1 + L2 L2, p1, p3, q1, q′1 L1, p2, p4, q2, q

′2

Table 4.2: Differences resulting from dividing the ramification locus into differentcubics F11 and F22 in Algorithm 1.1.4.

Remark 4.1.8. In [Ott+14, Remark 5.4], Kummer symmetroids are interpretedin terms of quadrics, showing that the base locus consists of six distinct points.The base locus in Example 4.1.7 is a non-reduced scheme of length six. In thisregard, X can be considered as a degenerated Kummer surface. ♣

4.2 Menagerie of Projected Del Pezzo Surfaces

We found some more rational symmetroids using the procedure in Example 4.1.7by considering different projection centres and del Pezzo surfaces.

By a slight variation of projection centre, we can ensure that the four isolatednodes are not coplanar:

30

4.2. Menagerie of Projected Del Pezzo Surfaces

Example 4.2.1. Take the same del Pezzo surface S as in Example 4.1.7 andagain let the projection centre p := [1 : −1 : −2 : −1 : 1] be a point on the lineL, but such that p is not contained in the P3 spanned by the four nodes.

Projecting from p onto V(x4) yields a symmetroid X, which has two inter-secting double lines and four nodes that span P3. Projecting further from oneof the nodes on X, the ramification locus has exactly the same description asin Example 4.1.7, except that the nodes are not projected onto a line.

Choosing F11 in the two ways as in Example 4.1.7 we obtain the dissimilarmatrices

−64x2 − 64x3 4x0 − 32x3 −4x1 − 48x2 − 64x3 4x2 + 8x34x0 − 32x3 −4x3 −2x1 − 34x3 4x3

−4x1 − 48x2 − 64x3 −2x1 − 34x3 −4x1 − 32x2 − 64x3 2x2 + 8x34x2 + 8x3 4x3 2x2 + 8x3 −x3

and

A(x) := A0x0 +A1x1 +A2x2 +A3x3,

where

A0 :=

0 256 0 0

256 0 0 00 0 0 00 0 0 0

, A1 :=

−16 16 0 4

16 240 128 −40 128 0 04 −4 0 −1

,

A2 :=

−128 128 64 0

128 −128 −64 064 −64 0 −160 0 −16 8

, A3 :=

256 128 0 0128 0 −128 32

0 −128 −256 640 32 64 −16

.In Example 4.1.7, the rank 2 locus was contained in the union of the planesspanned by the two double lines and the four nodes, respectively. The rank2 locus here is also contained in a quadric, but it is smooth. Otherwise thedescription of the rank 2 locus and the base locus of the web of quadrics is justas in Example 4.1.7. ♥

We now consider projected del Pezzo surfaces that are double along a smoothconic section.

Example 4.2.2. Again, start with the del Pezzo surface S from Example 4.1.6,but let the projection centre p := [1 : 1 : 0 : −1 : 1] be a general point inthe P3 spanned by the nodes on S. That is, p is not on any line which is theintersection between any planes spanned by two conic sections on S.

The projection of S from p is the symmetroid X in Figure A.6a on page 71,which is double along a smooth conic section and has four coplanar nodes.There are four embedded points on the double conic in the singular locus. Theprojection to P2 from one of the nodes on X is branched along a double smoothconic and two reduced lines. The configuration of the projected images of thenodes and the embedded points are displayed in Figure 4.2 on the next page.

31


Figure 4.2: The ramification locus of the surface X in Example 4.2.2 projectedfrom a node. It consists of a double, smooth conic section and two reduced lines.The dashed line is not part of the ramification locus. The points marked with• are the images of the embedded points and indicates the image of a node.

The two different ways of defining F11 in Algorithm 1.1.4 give the dissimilarmatrices

A1 :=

8x1 + 16x2 + 8x3 4x0 − 4x1 − 8x2 2x1 + 4x2 + 2x3 2x34x0 − 4x1 − 8x2 2x1 + 4x2 + 2x3 −x1 − 4x2 − x3 −2x2 − x32x1 + 4x2 + 2x3 −x1 − 4x2 − x3 x1 + 2x2 + x3 x2 + x3

2x3 −2x2 − x3 x2 + x3 x3

and

A2 :=

8x1 − 16x2 + 8x3 4x0 − 4x1 + 8x2 −2x1 + 4x2 − 2x3 −2x34x0 − 4x1 + 8x2 2x1 − 4x2 + 2x3 x1 − 4x2 + x3 −2x2 + x3−2x1 + 4x2 − 2x3 x1 − 4x2 + x3 x1 − 2x2 + x3 −x2 + x3

−2x3 −2x2 + x3 −x2 + x3 x3

.For both representations, the rank 2 locus equals the whole singular locus, sansthe embedded points.

The variety of singular loci of the web Wi of quadrics aTAia is the unionof two smooth quadrics, Qi and Q′i. The base locus Bl(Wi) consists of fourpoints in Qi ∩Q′i. The points [QP ] such that the quadric QP ∈Wi is singularat P ∈ Bl(Wi), are the four embedded points in the double conic section onX. ♥

Again, the four isolated nodes need not be coplanar.

Example 4.2.3. Take that same del Pezzo surface S from Example 4.1.6 asbefore, and let the projection centre p := [2 : 1 : 3 : 8 : 2] be a generic point asin Example 4.2.2, but such that p is not contained in the P3 spanned by thenodes on S.

Then S is projected onto a symmetroid X, which has a double smooth conicsection and four nodes that span P3. The ramification locus for the projectionfrom one of the nodes on X is just as in Example 4.2.2, except the nodes arenot mapped onto a line. The dissimilar matrices

A(x) := A0x0 +A1x1 +A2x2 +A3x3,

B(x) := B0x0 +B1x1 +B2x2 +B3x3,

where

A0 :=

0 36 0 036 0 0 00 0 0 00 0 0 0

, A1 :=

576 −432 72 0−432 144 36 0

72 36 −36 00 0 0 0

,

32


A2 :=

−288 216 −126 0

216 −144 84 0−126 84 −48 6

0 0 6 0

, A3 :=

−279 261 −72 18

261 −244 68 −17−72 68 −16 4

18 −17 4 −1

,and

B0 :=

0 400 0 0

400 0 0 00 0 0 00 0 0 0

,

B1 :=

−8000 16000 2000 016000 −32000 −4000 02000 −4000 −400 0

0 0 0 0

,

B2 :=

−24400 36800 1780 120

36800 −54400 −3040 −1601780 −3040 40 −20120 −160 −20 0

,

B3 :=

−6725 13950 −85 8513950 −28400 170 −170−85 170 −1 1

85 −170 1 −1

,have the correct determinant.

The rank 2 locus in Example 4.2.2 is contained in the union of the twoplanes spanned by the nodes and the conic, respectively. The rank 2 locus ofthis symmetroid is also contained in a quadric, but it is smooth. ♥

Now we turn to symmetroids with fewer isolated nodes. We will use thesame technique of projecting a del Pezzo surface. We begin by constructing adel Pezzo surface with three nodes:

Example 4.2.4 (3-nodal quartic del Pezzo surface). We shall find a 3-nodaldel Pezzo surface S by blowing up P2 in five points. Start with distinct pointsP1, P3, P5 ∈ P2 and let P2 be the point on the exceptional divisor E1 thatcorresponds to the line P1P5 and let P4 be any point in E3 that does notcorrespond to P3P5. Then the linear equivalence classes

e1 − e2, e3 − e4, l − e1 − e2 − e5

are effective and contracted to points.Note that the line e4 between the nodes e1 − e2 and l − e1 − e2 − e5 is

contained in the blow-up. The same is true for the line l − e1 − e3 betweene1 − e2 and e3 − e4.

In Appendix B.5 we compute the equation of the del Pezzo surface for aconcrete example. ♥

We consider the projections of the 3-nodal del Pezzo surface.

33


Example 4.2.5. Let S be the del Pezzo surface given in Appendix B.5. Thetwo families of conic sections in Example 4.1.7 avoid the three singularities andwe construct the line L in the same manner.

Let the projection centre p := [0 : 3 : 1 : 0 : 0] be a point on L and projectonto the P3 given by x1 = 0. The image is the symmetroid X in Figure A.5b onpage 70, which is double along two lines L1 and L2, and has three additionalnodes p1, p2 and p3. The singular locus also contains five embedded points: Theintersection L1 ∩ L2 and two other points each on Li. Number the nodes suchthat p3 is the image of e1 − e2. Then the lines p1p3 and p2p3 are contained inX.

If we project to P2 from p3, the ramification locus is the union of two doublelines, L1, L2, and a smooth conic section C. If we project from p1 or p2, theramification locus is the union of two double lines, also called L1 and L2, andtwo reduced lines, l1 and l2. In either case, the ramification locus, which isshowed in Figure 4.3 on page 36, is decomposable into cubics.

The two different ways of defining F11 for the projection from p3, results inthe matrices

A(x) := A0x0 +A1x1 +A2x2 +A3x3,

A′(x) := A′0x0 +A′1x1 +A′2x2 +A′3x3,

where

A0 :=

0 0 8 00 0 4 08 4 −4 00 0 0 0

, A1 :=

−384 −96 72 24−96 0 12 0

72 12 −12 −624 0 −6 0

,

A2 :=

192 80 −48 −880 32 −20 −4−48 −20 11 3−8 −4 3 −1

, A3 :=

0 16 0 016 0 0 00 0 0 00 0 0 0

,and

A′0 :=

−256 576 144 16

576 5616 1404 −36144 1404 −81 −916 −36 −9 −1

,

A′1 :=

3072 −10368 −2592 96

−10368 3456 2592 0−2592 2592 1944 0

96 0 0 −24

,

A′2 :=

256 1728 1008 −16

1728 −12528 −5724 3241008 −5724 −2511 153−16 324 153 1

,

34


A′3 :=

0 2304 0 0

2304 0 0 00 0 0 00 0 0 0

.If for the projection from p1 we try to define F11 as L2 which does not containthe image of a node and the two reduced lines, we discover that V(F11,∆)is set-theoretically seven points. More precisely, five points with multiplicity 2and two points with multiplicity 1. This is not two times a scheme of length6 and Algorithm 1.1.4 fails. However, if we take L1 which does contain theimage of a node instead of L2, we find the matrix

B(x) := B0x0 +B1x1 +B2x2 +B3x3,

where

B0 :=

0 97344 0 0

97344 0 0 00 0 0 00 0 0 0

,

B1 :=

−32448 −405600 624 −4368−405600 −324480 −3432 −12480

624 −3432 420 −132−4368 −12480 −132 −480

,

B2 :=

−67600 75712 1300 1040

75712 −400192 −1456 −60321300 −1456 −25 −201040 −6032 −20 −16

,

B3 :=

54080 189280 −2912 4160

189280 −432640 9464 −13520−2912 9464 −196 280

4160 −13520 280 −400

.For each of the matrices A, A′ and B, the variety of singular loci associated tothe web of quadrics is the union of a plane and a 2-nodal cubic. The base locusof the web is four points, where one of the points has multiplicity 2.

The details of the rank 2 loci and quadrics which are singular at points inthe base locus, are summarised in Table 4.3. ♥

F11 F22 rank 2 [QP ] for P ∈ Bl(W )

L1 + C L1 + 2L2 L1, p1, p3, q2, q′2 L2, p2, q1, q

′1

L2 + C 2L1 + L2 L2, p2, p3, q1, q′1 L1, p1, q2, q

′2

L1 + l1 + l2 L1 + 2L2 L1, p1, p3, q2, q′2 L2, p2, q1, q

′1

Table 4.3: Division of the ramification locus into different cubics F11 and F22in Algorithm 1.1.4, with the resulting rank 2 loci and that quadrics that aresingular at points in the base loci.

35


L1

L2

p2

p3

(a) Ramification locus when projectedfrom P1.

(b) Ramification locus when projectedfrom P3.

Figure 4.3: The ramification loci of the surface X in Example 4.2.5 projectedfrom a node. The points marked with • are the images of the embedded pointsand indicates the image of a node.

Consider now symmetroids with only two isolated singularities. Before wecan apply our projection method, we need to construct a del Pezzo surface withtwo nodes:

Example 4.2.6 (2-nodal quartic del Pezzo surface). Blow up P2 in five pointsP1,P2,P3,P4,P5 that are such that the triples of points P1, P2, P3 and P1, P4, P5are collinear. Then l− e1− e2− e3 and l− e1− e4− e5 are contracted to nodes.The line e1 between the two singularities is contained in the resulting del Pezzosurface. Readers who are familiar with Segre symbols2 will recognise this ashaving Segre symbol [221]. We compute the equation of a del Pezzo surface likethis in Appendix B.6.

Note that there is an inequivalent linear system of cubics passing throughfive points that also defines a 2-nodal del Pezzo surface. It has Segre symbol[(11)111]. Indeed, take a tangent direction P2 through a point P1, and say thatthis corresponds to a line L through P1. Let P3, P4 and P5 be three pointslying on a line L′ 6= L. In this case, the line between the nodes e1 − e2 andl − e3 − e4 − e5 is not contained in the del Pezzo surface. ♥

We use the 2-nodal del Pezzo surface from Example 4.2.6 to find a sym-metroid with only two isolated singularities.

Example 4.2.7. We begin with the del Pezzo surface S found in Appendix B.6.Curves in the linear equivalence classes l − e1 and 2l − e2 − e3 − e4 − e5 areconics that avoid both singularities on S and meet curves from the other classin two points, which span a line L. The point p := [2 : −2 : 1 : −2 : 2] lies onsuch a line L and is not contained in S. We choose p as the projection centreand V(x1) as the target hyperplane.

The image of S under the projection is the symmetroid X in Figure A.5con page 70. Its singular locus is comprised of two double lines L1, L2, two

2We will not need Segre symbols for anything than a shorthand notation for differentiatingbetween the linear systems in this example. Interested readers may consult [Jes16, Article 48]for an introduction to Segre symbols.

36


nodes p1, p2, and four embedded points: The intersection L1 ∩ L2, two pointsq1, q

′1 ∈ L1 and one other point q2 ∈ L2.The ramification locus Rp1 is the two double lines L1 and L2 coming from

L1 and L2 and a smooth, reduced conic section C. The line L2 is tangentto C. The point p2 is mapped onto L1, as shown in Figure 4.4.

L2L1

q2q1

q′1p2

Figure 4.4: The ramification locus of the surface X in Example 4.2.7 projectedfrom a node. It consists of two double lines and a reduced conic. One of thelines is tangent to the conic. The points marked with • are the images of theembedded points and indicates the image of a node.

If we let F11 := L2 + C, then we find that V(F11,∆) is the union of atriple point, a single point and four double points. Consequently, there is nosubscheme Z∆ of length 6 such that V(F11,∆) = 2Z∆. Thus Algorithm 1.1.4does not work. If we on the other hand let F11 := L1 + C, then the algorithmproduces the matrix

A(x) := A0x0 +A1x1 +A2x2 +A3x3,

where

A0 :=

0 100 0 0

100 0 0 00 0 0 00 0 0 0

, A1 :=

−9 48 −12 348 −1056 264 −16−12 264 −16 4

3 −16 4 −1

,

A2 :=

24 222 −48 2

222 216 −144 6−48 −144 96 −4

2 6 −4 −4

, A3 :=

24 2 12 22 −504 276 46

12 276 −144 −242 46 −24 −4

.The matrix A has rank 2 in L1, p1, p2 and q2. The variety of singular points ofthe quadrics aTAa is the union of a plane and a 1-nodal cubic. The base locusof the web is three points, one of which has multiplicity 2. The quadrics thatare singular at points in the base locus are the quadrics along L2, as well asthe quadrics in q1 and q′1. ♥

There are patterns emerging from the above examples, some of which maybe general. In the ramification locus, the intersection between the double andthe reduced conic was always the image of embedded points in the singularlocus of the surface in P3. In each example, nodes were mapped either to thedouble conic or the reduced conic, not to the intersection of the conics.

37


In the cases where the singular locus consisted of two double lines and nadditional nodes, the variety of singular points of the quadrics was the unionof a plane and an (n− 1)-nodal cubic. The base locus was a scheme of lengthn+ 2, with support in n+ 1 points.

When we considered a ramification locus 2L1 + 2L2 + C where L1, L2are lines and C a conic and we defined F11 as L1 + C, then Algorithm 1.1.4worked precisely when L1 contained the projected image of a node.

4.3 Quadrics Through Four General Points

We now turn to the task of expanding some of what we learnt from the examplesinto generalities.

In Example 4.2.2 the base locus of the web of quadrics was four pointsspanning P3. For this reason, we study the quadrics in P3 passing throughfour given points. We can also motivate this less heuristically. Consider a netN of quadrics containing a smooth conic C of rank-2-quadrics. Let V(l1l2),V(l′1l′2) be two points on C. These quadrics intersect generically in four lines,say L1, L

′1 ⊂ V(l1) and L2, L

′2 ⊂ V(l2). Then L1, L′1, L2, L′2 are fixed in the

pencil spanned by V(l1l2) and V(l′1l′2). Letting V(l′1l′2) run through all the pointsin C, we obtain a pencil P of line-pairs Li +L′i in each of the V(li). Since P onlycontains line-pairs and no smooth conic sections, the base locus must containa line; see Figure 2.4. Suppose that the line is Li. Then every quadric in Npasses through L1 and L2. Let W ⊃ N be a web of quadrics, generated by Nand a quadric Q. Generically, Q intersects each line in two points, so Bl(W ) isgenerically four general points.

The set of quadrics through four points form a P5 in the P9 of all quadrics.Taking the four points to have coordinates [1 : 0 : 0 : 0], [0 : 1 : 0 : 0], [0 : 0 : 1 : 0]and [0 : 0 : 0 : 1], the P5 is parametrised by the matrix

M :=

0 y01 y02 y03y01 0 y12 y13y02 y12 0 y23y03 y13 y23 0

.As discussed in Chapter 3, intersecting the rank 3 locus of M with a 3-spaceyields a symmetroid. Let X be the rank 2 locus of M . It is a surface of degree10. See Appendix B.7 for this computation. Recall Theorem 2.3.3: Since thedimensions add up to the dimension of the ambient space, a P3 ⊂ P5 willalways intersect X, and generically the intersection consists of 10 points. Soeven though we have restricted to quadrics passing through four points, thesituation is not so special that a general 3-space yields a symmetroid with anon-generic number of rank-2-points.

Furthermore, X breaks up into three smooth quadric surfaces Q1, Q2 andQ3, each spanning a P3, and four planes Π1, Π2, Π3 and Π4. The componentsare configured in the following manner: The Πi intersect each other in a singlepoint, resulting in six unique points Pk spanning P5. The Qi intersect eachother in two points, giving precisely the same six points as the planes. Theintersection between Πi and Qj is a line, totalling 12 distinct lines. Four ofthese lines pass through each Pk. The lines between the points in Qi ∩Qj do

38

4.3. Quadrics Through Four General Points

not occur as the intersection between either Qi or Qj with one of the planesΠk. Figure 4.5 illustrates the intersection between the components in X.

The singular locus of V(det(M)) consists of four additional planes. Each ofthese planes intersects three of the Πi along the same lines as the Πi intersectthe Qj . It avoids the fourth Πi completely.

Figure 4.5: Intersection of the components in the rank 2 locus X. The pointsmarked with • are the intersection points between two planes Πi and Πj , orequivalently between two quadrics Qi and Qj . The solid lines are the inter-sections between the planes and the quadrics. The dashed lines are spanned byQi ∩Qj .

Smooth Conic and Four Isolated Nodes

A symmetroid which is double along a smooth conic section and four additionalpoints inspired us to look at quadrics through four points. The first fruit wewant to harvest from this observation is a family of symmetroids having thisproperty. Let Π be a plane in the P3 spanned by Q1. Then Π intersects Q1 ina conic section C. Assume that C is smooth. Extend Π to a linear 3-space H.If H meets Q1 outside of C, then Q1 is completely contained in H, so supposethis is not the case. Generically H meets the planes Πi in a single point each.Since each Πi intersects Q1 along a line, they meet H in a point3 each on C.Similarly, H meets Q2 and Q3 generically in two points each, which we assumeto be outside of C. Therefore, H contains a conic section and four additionalpoints with rank 2, thus giving rise to a symmetroid with the desired property.

Smooth Conic and Three Isolated Nodes

We can reduce the number of rank-2-points in H, outside of C, by choosing Πsuch that some of the points in Q1 ∩ Q2 or Q1 ∩ Q3 are contained in Π. Forinstance, if one of these intersection points are in Π, we generically obtain asymmetroid which has rank 2 in a smooth conic section and three additional

3In the concrete examples we computed, these points corresponded to embedded pointsin the singular locus of the symmetroid.

39


points. For example, the matrix0 x0 3x1 − 2x2 − 3x0 x1x0 0 x3 x2

3x1 − 2x2 − 3x0 x3 0 x0x1 x2 x0 0

was produced in this fashion. The symmetroid it defines is showed in Figure A.6bon page 71. It has one point embedded in the conic in the rank 2 locus. Thisis opposed to previous examples, where we have had embedded points in thesingular locus, but not in the rank 2 locus. It has two further embedded pointsin the singular locus. Projecting from one of the isolated nodes, the branchcurve is the union of a double conic and two lines meeting on the conic, asshown in Figure 4.6.

Figure 4.6: The ramification locus of a symmetroid with a double conic andthree nodes, projected from an isolated node. It consists of a double smoothconic and two reduced lines. The point marked with N is the image of the pointembedded in the rank 2 locus, the points marked with • are the images of theembedded points in the singular locus and indicates the image of a node.Smooth Conic and Two Isolated Nodes

For a symmetroid which has rank 2 along a smooth conic section and two points,let Π be such that it contains Q1 ∩ Q2. Then the 3-space H will genericallyonly meet X in two other points. An example of this is the matrix

0 2x0 x0 + x1 x22x0 0 x3 x1

x0 + x1 x3 0 x0x2 x1 x0 0

,which defines the symmetroid in Figure A.6c on page 71. It has two pointsembedded in the conic in the rank 2 locus. The projection from one of theisolated nodes is ramified along a double conic and two lines that are bothtangent to the conic, as in Figure 4.7.

If we instead let Π contain one point p1 from Q1 ∩ Q2 and another pointp2 from Q1 ∩ Q3, then C breaks down into two lines, since the line p1p2 iscontained in Q1.

Two Lines

We can go one step further and let Π be spanned by Q1 ∩ Q2 and a point inQ1 ∩Q3. Then H will generically meet X in two lines and a single point. This

40

4.3. Quadrics Through Four General Points

Figure 4.7: The ramification locus of a symmetroid with a double conic and twonodes, projected from an isolated node. It consists of a double smooth conic andtwo tangential lines. The points marked with N are the images of the pointsembedded in the rank 2 locus and indicates the image of the node.is a type of symmetroid that was missing from our collection. An example ofsuch a symmetroid is displayed in Figure A.5d on page 70. It is fgiven by thematrix

0 x0 x1 x2x0 0 x3 x0x1 x3 0 x0x2 x0 x0 0

.It has three embedded points in the rank 2 locus; one on each line and theirintersection. The projection from the isolated node is ramified along two triplelines, as displayed in Figure 4.8.

Figure 4.8: The ramification locus of a symmetroid with two double lines anda node, projected from the isolated node. It consists of two triple lines. Thepoints marked with N are the images of the points embedded in the rank 2locus.

We can also recover symmetroids whose singular loci consist of two doublelines and four additional nodes in this picture. One way to go about is to chooseΠ such that the conic C breaks up into two lines that avoid Q2 and Q3. Bythe same argument as above, a generic P3 containing Π intersects X in fourmore points. Another approach is to let Π be a plane that cuts both Π1 and Π2along a line each, say l1 and l2. Number the points and the quadrics such thatP1 = Π1 ∩Π2 ⊂ Q1 ∩Q2. Extend Π to a linear 3-space H. Then H genericallymeets Π3 and Π4 in a single point each, and the Qi in two points each. SinceP1 is contained in Π, H only meets each of Q1 and Q2 in one other point. Wededuce that there is no further contribution from Q3, since Π1 intersects Q3 ina line that meets l1, and similarly for Π2.

41


It is possible to reduce the number of rank-2-points outside the two lines inthe last case. For instance, if l1 is spanned by P1 and P2 := Π1 ∩Π3 then thenumber of additional rank-2-points drops to two, since either Q1 or Q2 passthrough P2, say Q1. If, in addition, l2 is spanned by P3 := Π2 ∩Π4, the numberof rank-2-points only drops to one, since Q1 also passes through P3, which canbe seen from the computations in Appendix B.7.

Degenerate Cases

In the above account we utilised that a P3 ⊂ P5 generically meets the quadricQi in two points. It can well happen that these two points coincide, so the P3

intersects Qi in one point with multiplicity 2.For instance, we can start with a plane containing Q1 ∩Q2, and extend this

to a 3-space meeting Q3 in a single point. Thus the resulting symmetroid hasrank 2 along a conic section and in that point. The matrix

0 x0 −x0 − x1 x2x0 0 x3 −x0 + x1

−x0 − x1 x3 0 x0x2 −x0 + x1 x0 0

is an example of this, defining the symmetroid in Figure A.6d on page 71. Wecan unearth this geometric description by excavating algebra. The saturatedideal of the rank 2 locus is

I :=⟨x2

1 + x2x3, x0x2x3, x0x1x3, x20x3, x0x1x2, x

20x2⟩.

The associated prime ideals are 〈x0, x21 + x2x3〉, corresponding to the conic

section; 〈x1, x2, x3〉, corresponding to where the P3 meets Q3; the ideals〈x0, x1, x2〉, 〈x0, x1, x3〉, corresponding to the embedded points Q1∩Q2, whichis also the points where the planes Πi meet the conic section. One possibleprimary decomposition of I is⟨

x0, x21 + x2x3

⟩∩⟨x2

1, x2, x3⟩∩⟨x2

0, x21, x0x1, x2

⟩∩⟨x2

0, x21, x0x1, x3

⟩.

Notice that the point corresponding to the intersection with Q3 is representedby the ideal

⟨x2

1, x2, x3⟩of degree 2 in the decomposition.

Another degenerated example is

M :=

0 x0 x1 7x1 − 3x2x0 0 x2 x3x1 x2 0 x0

7x1 − 3x2 x3 x0 0

.The matrix M was found by letting a point p ∈ Q1 ∩Q2 be contained in Π andthen extend Π to a 3-space H that intersects Q1 ∩Q2 twice in p, rather thanin two different points. Hence the rank 2 locus consists of a conic section andtwo points, instead of three.

4.4 Conjectures

Having a accumulated a sizeable collection of symmetroids, we have an ideaabout what is possible and what is not. We propose two conjectures. The

42

4.4. Conjectures

attempted proofs are missing the same detail. Namely, we have not been ableto exclude the possibility that the ramification locus consists of a double conicand two lines meeting on the conic.

Lemma 4.4.1. All lines on a projected quartic del Pezzo surface X in P3 meetthe double conic.

Proof. In the proof of Theorem 4.1.5, Dolgachev gives the following descriptionof the curve C ⊂ S ⊂ P4 which is mapped onto the double conic: Let H be thetangent hyperplane of Qp at p, then C = S ∩H.

Let l be a line on X and Π the plane in P4 spanned by l and p. Then Hcuts Π along a line L. And Π meets S in some curve K, since l ⊂ X. The claimfollows because ∅ 6= K ∩ L ⊂ S ∩H = C.

Corollary 4.4.2. All singularities on the branch curve in P2 of the projectionfrom an isolated node on a projected quartic del Pezzo surface X ⊂ P3 are theimages of singularities on X.

Proof. In general, a singularity on the branch curve of the projection from anode on a quartic Q in P3, is either the image of a singularity on Q or the imageof a line on Q passing through the projection centre. Lemma 4.4.1 states that ifsuch a line exits on X, its image coincides with the image of a singularity.

Conjecture 4.4.3. Let X be an irreducible quartic surface in P3 which isdouble along a smooth conic section and has a single isolated node p. If X is asymmetroid, then p is a rank-3-point.

Partial proof. If p has rank 2 or less, then Theorem 1.1.2 implies that theprojection P3 99K P2 from p is ramified along two cubics. The ramification locuscontains a double smooth conic section C, which is the image of the doubleconic on X, and another conic section C ′. For this sextic to split into two cubics,C ′ must be a pair of lines, hence have a singular point. By Theorem 4.1.5, X isa projected 1-nodal quartic del Pezzo surface. Corollary 4.4.2, combined withthe fact that X has no further nodes, implies that C ′ is either nonsingular ortwo lines meeting on C. We are done if can show the latter is not possible.

Recall from Section 4.3 that we saw the conic break up into lines when weattempted to reduce the number of isolated nodes to one.

Conjecture 4.4.4. Let X ⊂ P3 be an irreducible quartic surface which isdouble along a smooth conic section and has two additional nodes, p1 and p2.Suppose that X is the projection of a 2-nodal del Pezzo surface S ⊂ P4 withSegre symbol [221]. If X is a symmetroid, then p1 and p2 are rank-3-points.

Partial proof. Consult Example 4.2.6 for details about del Pezzo surfaces withSegre symbol [221].

Just as for Conjecture 4.4.3, the ramification locus for the projection fromeither p1 or p2 is the union of a smooth double conic C and another conic C ′.For pi to be a rank-2-point, C ′ must be singular.

The line p1p2 is contained in X, since the line between the two nodes on Sis contained in S. By Lemma 4.4.1, p1p2 intersects the double conic. Thus p2 ismapped to C in the projection from p1. Since there are no other singularitieson X, C ′ is either smooth or two lines meeting on C.

43


One could imagine that one way to complete the proofs is to start with asextic curve R in the plane, consisting of a double conic and two lines meetingon that conic. Then one constructs a surface S having R as ramification locuswhen projected from a node. Finally, one shows that S is not one of the surfacesin the conjectures. Alas, this gives no conclusion; several surfaces can have thesame ramification locus. For instance, if l, l1, l2 ∈ C[x1, x2, x3]1 are linear forms,q ∈ C[x1, x2, x3]2 is a quadratic form and k1, k2 ∈ C are such that k1k2 = 1

4 ,then

V(k1l1l2x

20 − k2q

2) V(k1l1l2x

20 + l1l2lx0 + k2

(l1l2l

2 − q2))V(q(k1x

20 − k2l1l2

))V(q(k1x

20 + lx0 + k2

(l2 − l1l2

)))V(k1(l2 − l1l2

)x2

0 + qlx0 + k2q2)

are quartic surfaces in P3 that all have VP2(l1l2q

2) as their ramification locus,when projected to V(x0) with projection centre [1 : 0 : 0 : 0].

44

CHAPTER 5

Symmetroids with a Double Line

Art, like morality, consists ofdrawing the line somewhere.

G. K. Chesterton

GENERICALLY, a quartic symmetroid has only finitely many singularities.We have found several symmetroids that are singular along a curve ofdegree 2. There ought to be intermediate specimens, so we direct our

focus at symmetroids that have nodes along a single line, apart from someisolated singularities.

Pictures of some of the surfaces in this chapter can be found in Appendix A.

A First Estimate

Consider a quartic symmetroid S ⊂ P3. Let p ∈ S be an isolated node suchthat the projection πp : S \ p → P2 from p is ramified along two cubics, F11and F22. Suppose that S has a determinantal representation obtained fromAlgorithm 1.1.4. In this representation, the rank 2 locus of S is mapped ontoF11 ∩ F22.

Now assume that S has rank 2 along a line. Then F11 and F22 share acommon line. The remaining components of F11 and F22 are two conics thatgenerically intersect in four points. Taking the projection centre into account,we expect, a priori, that a symmetroid which has rank 2 along a line to havegenerically five other rank-2-points.

Note that the same type of analysis yields that a generic symmetroid whichhas rank 2 along a double conic, will have two isolated rank-2-points. This iscontrary to our experience with four nodes being the least special in Section 4.3.When we computed the ramification curves of symmetroids with more thantwo isolated nodes, we saw that some of the isolated nodes were mapped to thedouble conic in the plane.

5.1 Quadrics Through Four Coplanar Points

We are primarily interested in symmetroids which have rank 2 along a line andno rank-1-points. Recall from Section 3.3 that if a pencil of rank-2-quadricscontains no rank-1-quadric, then it fixes a plane and a line not contained inthat plane.

5. Symmetroids with a Double Line

We thus start with two quadrics, Q1 and Q2, having a plane Π and a line lin common. A symmetroid gives rise to a whole web of quadrics, so we extendthis to four independent quadrics, Q1, Q2, Q3, Q4. Both Q3 and Q4 intersect Πin a conic and l in two points each. Generically, the points on l do not coincide,but the two conics intersect in four points. Hence the base locus of the web isfour coplanar points.

First, observe that if three of the points are collinear, then the symmetroidis a double quadric. Indeed, let Π := V(x3) and choose the points [1 : 0 : 0 : 0],[0 : 1 : 0 : 0], [1 : 1 : 0 : 0] on the line V(x2, x3), as well as the point [0 : 0 : 1 : 0].Then the P5 of quadrics passing through these four points is parametrised by

0 0 y02 y030 0 y12 y13y02 y12 0 y23y03 y13 y23 y33

,which has determinant (y03y12 − y02y13)2. Therefore the intersection of a 3-space in P5 with the rank 3 locus is a double quadric.

Assume instead that the four coplanar points are in general position. Let thepoints have coordinates [1 : 0 : 0 : 0], [0 : 1 : 0 : 0], [0 : 0 : 1 : 0] and [1 : 1 : 1 : 0].Then the P5 of quadrics passing through these points is parametrised by

M :=

0 y01 y02 y03y01 0 −y01 − y02 y13y02 −y01 − y02 0 y23y03 y13 y23 y33

.We calculate similarly as in Appendix B.7 and find that rankM = 1 only inthe point P := [0 : 0 : 0 : 0 : 0 : 1], which corresponds to the quadric 2Π. Therank 2 locus X consists of a linear 3-space T and three quadric surfaces K1, K2and K3. All the Ki meet in P . Each Ki intersects T along two lines, li and Li.The lines are concurrent, meeting in P . It is possible to label the lines in such away that the all the li are contained in a plane Πl, and the Li in another planeΠL. The singular locus Y of V(det(M)) has the same description, but it alsoincludes four embedded planes in T that intersect each other along the lines liand Li.

We will now look at the symmetroids we can obtain by choosing differentlinear 3-spaces in this P5.

Six Isolated Rank-2-Points

A generic P3 ⊂ P5 will cut T along a line and meet each Ki in two points. Theresulting symmetroid thus has a rank 2 locus consisting of a line and six otherpoints. How does this fit in with our estimate of the number of isolated nodes?Let us look at an example, such as the symmetroid given by

0 −x0 −x1 − 2x2 + x3 x1−x0 0 x0 + x1 + 2x2 − x3 x2

−x1 − 2x2 + x3 x0 + x1 + 2x2 − x3 0 x3x1 x2 x3 x0

.

46

5.1. Quadrics Through Four Coplanar Points

It is drawn in Figure A.2a on page 68. There are four embedded points in thesingular locus and the rank 2 locus is contained in a quadric. The ramificationlocus is as predicted two conics and a double line although the conclusionabout the number of rank-2-points is wrong. This is because one of the nodesis mapped onto the double line. The situation is completely analogous to Ex-ample 4.2.2, where two nodes were mapped onto the double conic. The branchcurve is displayed in Figure 5.1 on the following page.

It can happen that a 3-space H in P5 intersects the rank 3 locus of M insuch a way that the intersection has singularities that are not in Y . For instance,the symmetroid in Figure A.1b on page 67, given by

0 x0 x1 2x2 + x3x0 0 −x0 − x1 2x3x1 −x0 − x1 0 x2

2x2 + x3 2x3 x2 x3

is singular along a line and has seven isolated nodes, of which only six arerank-2-points. The rank 2 locus is contained in a quadric surface, but the rank-3-node is not on the quadric. In addition, there are four points embedded inthe singular locus.

Projection from one of the rank-2-points gives a ramification locus thatconsists of a double line, two reduced lines and a smooth conic section. Three ofthe six points are collinear. If the projection centre is one of these, then the twoother nodes on that line are mapped to the same point in P2. This causes one ofthe reduced lines to be tangent to the conic. This story is depicted in Figure 5.2on the following page. Considering the projection from the rank-3-node instead,we get five general lines, with one of them being double, as in Figure 5.3 on thenext page.

There is one lesson to take away from these ramification loci: It is impossibleto divide them into cubics such that the cubics have all the images of the nodesin common. Thus there exists no matrix representation of this symmetroid suchthat all the isolated nodes have rank 2.

We can also find symmetroids like this in the context of Section 4.3. Usingthe notation of Section 4.3, start with a line l in Π1 and extend it to a 3-spaceH containing l. Generically, H meets the other Πi in one point each and Qi intwo points each. But since Π1 cuts each Qi along a line, one of these points areon l. Hence H contains six isolated rank-2-points. Since there is a plane in thesingular locus of V(det(M)) not meeting Π1, the symmetroid will have sevenisolated nodes.

Likewise, extend a line l on Q1 to a 3-space H. A plane through l cuts Q1in another line l′. As noted earlier, the Πi all meet the conic l + l′. It is notpossible to find a point pi ∈ Πi ∩ Q1 for each i such that three of them arecollinear. One consequence is that exactly two of the Πi meet l. The same istrue of the four extra planes in the singular locus of V(det(M)). Therefore Hmeets the rank 2 locus generically in l and six isolated rank-2-points and tworank-3-points. An example is the symmetroid given by

0 x0 − x1 + x2 x0 + x3 − x1 x0x0 − x1 + x2 0 x3 x1x0 + x3 − x1 x3 0 x2

x0 x1 x2 0

.

47


A similar symmetroid with a less pretty matrix, but with only real nodes isshown in Figure A.1a on page 67.

Figure 5.1: The ramification locus of a symmetroid with a double line and sixnodes, projected from an isolated node. It consists of two conics and a doubleline. The points marked with • are the images of the embedded points in thesingular locus and indicates the image of a node.

(a) No two nodes lie on the same linethrough the projection centre.

(b) Two nodes lie on the same linethrough the projection centre.

Figure 5.2: The ramification locus of a symmetroid with a double line andseven nodes, projected from an isolated rank-2-point. It consists of a conic, tworeduced lines and a double line. The point marked with is the image of therank-3-node, the points marked with • are the images of the embedded pointsin the singular locus and indicates the image of a node.

Figure 5.3: The ramification locus of a symmetroid with a double line and sevennodes, projected from the rank-3-node. It consists of five general lines, one ofwhich is double. The points marked with • are the images of the embeddedpoints in the singular locus and indicates the image of a node.

48


Five Isolated Rank-2-Points

Let p be a point on one of the lines li or Lj . A P3 that contains p will genericallyintersect X in a line and five points. The symmetroid defined by

0 −x0 + x1 x2 x3−x0 + x1 0 x0 − x1 − x2 x0

x2 x0 − x1 − x2 0 x3x3 x0 x3 x1

was found this way. A similar symmetroid with real nodes is plotted in Fig-ure A.3a on page 68. The rank 2 locus is contained in a quadric surface andit contains an embedded point. Two additional points are embedded in thesingular locus. The projection from one of the isolated nodes is ramified alongtwo conics and a double line that is tangent to one of the conics, as shown inFigure 5.4.

Figure 5.4: The ramification locus of a symmetroid with a double line and fivenodes, projected from an isolated node. It consists of two conics and a doubleline, which is tangent to one of them. The point marked with N is the imageof the point embedded in the rank 2 locus, the points marked with • are theimages of the embedded points in the singular locus and indicates the imageof a node.

Again, we can choose the 3-space such that the symmetroid has singularitiesnot contained in Y . The determinant of the matrix

0 x0 x1 x2x0 0 −x0 − x1 x3x1 −x0 − x1 0 x2x2 x3 x2 x3

defines a symmetroid that is double along a line and has six isolated nodes, ofwhich five are rank-2-points. The rank 2 locus is contained in a quadric whichalso contains the rank-3-node. One point is embedded in the rank 2 locus andtwo other points are embedded in the singular locus.

The five isolated rank-2-points are coplanar, and the conic passing throughthem is a line-pair. The lines intersect in one of the rank-2-points, say q, andthere are two additional rank-2-points on each line. Hence, no matter which ofthe nodes we project from, two nodes are mapped to the same point. Projectingfrom q, the other rank-2-points are mapped to two points. The ramification

49


locus is a conic, two reduced lines and a double line. Projecting from the rank-3-point, the branch curve consists of five lines, one of which is double. Thedouble line and two of the other lines are concurrent. The ramification loci aredisplayed in Figures 5.5 and 5.6. It is impossible to divide the branch curvesinto cubics such that they have the image of all the nodes in common. Hencethis symmetroid is different from the symmetroid with a double line and sixrank-2-nodes.

(a) Two nodes lie on the same linethrough the projection centre.

(b) Two pairs of two nodes lie onthe same lines through the projectioncentre.

Figure 5.5: The ramification locus of a symmetroid with a double line, fiverank-2-nodes and one rank-3-node, projected from a rank-2-node. It consists ofa conic, two reduced lines and a double line. The point marked with is theimage of the rank-3-node, the point N is embedded in the rank 2 locus, thepoints marked with • are the images of the embedded points in the singularlocus and indicates the image of a node.

Figure 5.6: The ramification locus of a symmetroid with a double line, fiverank-2-nodes and one rank-3-node, projected from the rank-3-node. It consistsof five lines, one of which is double. Three of the lines are concurrent. The pointmarked with N is embedded in the rank 2 locus, the points marked with • arethe images of the embedded points in the singular locus and indicates theimage of a node.

Four Isolated Rank-2-Points

Let p1 be a point on one of the li, and p2 a point on one of the Lj . A 3-spacecontaining both p1 and p2 will generically intersect the rank 2 locus X in theline p1p2 and in four points. The matrix

0 x0 −x1 + x2 x3x0 0 −x0 + x1 − x2 x1

−x1 + x2 −x0 + x1 − x2 0 x1x3 x1 x1 x2

50


was obtained in this procedure. A symmetroid like this with real singularities isdrawn in Figure A.3b on page 68. It has one point embedded with multiplicity2 in the rank 2 locus and two points embedded in the singular locus. The rank2 locus lies on a quadric surface. The projection from one of the isolated nodesis branched along two smooth conics that are tangent in a point q and a doubleline passing through q. This is illustrated in Figure 5.7.

Figure 5.7: The ramification locus of a symmetroid with a double line and fournodes, projected from an isolated node. It consists of two tangential conics anda double line passing through the point of tangency. The point marked with Nis the image of the point embedded in the rank 2 locus, the points marked with• are the images of the embedded points in the singular locus and indicatesthe image of a node.

Three Isolated Rank-2-Points

Symmetroids that are singular along a line and have three isolated rank-2-pointscome in two flavours.

For the first flavour, let p1 ∈ l1 and p2 ∈ l2. Then the line p1p2 meets l3,since the li are coplanar. Hence a 3-space containing p1 and p2 genericallyintersects X in p1p2 and three other points. The matrix

0 x0 x1 x2x0 0 −x0 − x1 x3x1 −x0 − x1 0 x3x2 x3 x3 x1

is an example of a symmetroid like this. Another symmetroid with real nodes isshown in Figure A.4a on page 69. It has no rank-1-points. The rank 2 locus iscontained in a quadric and there are three points embedded in it. The branchcurve has the same description as for the symmetroid with a double line andfour nodes. The images of the embedded points are marked in Figure 5.8.

The second flavour comes about by taking a 3-space H that contains therank-1-point P . Since P ∈ K1 ∩K2 ∩K3, generically H intersects each Ki inone further point. Hence the resulting symmetroid has a rank-1-point on a lineof rank-2-points and three additional rank-2-points. One such symmetroid isgiven by

0 x0 + x1 x0 x2x0 + x1 0 −2x0 − x1 x1x0 −2x0 − x1 0 x2x2 x1 x2 x3

.

51


It is drawn in Figure A.4b on page 69. The rank-1-point is embedded in therank 2 locus and there is one other point embedded in the singular locus. Therank 2 locus is not contained in a quadric. The branch curve is a conic, a tripleline that is tangent to the conic and another reduced line. This is drawn inFigure 5.9.

Figure 5.8: The ramification locus of a symmetroid with a double line, threenodes and no rank-1-point, projected from an isolated node. It consists of twotangential conics and a double line passing through the point of tangency. Thepoints marked with N are the images of the points embedded in the rank 2locus and indicates the image of a node.

Figure 5.9: Ramification locus for a symmetroid with a double line, three nodesand a rank-1-point, projected from an isolated node. It consists of a triple line,a reduced line and a conic that is tangent to the triple line. The point markedwith N is the image of the rank-1-point, the point marked with • is the imageof the embedded point in the singular locus and indicates the image of anode.

Two Isolated Rank-2-Points

If H contains the rank-1-point P and a point p ∈ l1, then l1 is contained in H.Also, H will generically not intersect K1 further and only meet K2 and K3 inone more point each. In summary, the symmetroid will have one rank-1-point,a line of rank-2-points and two isolated rank-2-points. An example of such asymmetroid is shown in Figure A.4c on page 69. It is given by the matrix

0 2x0 + x1 x1 x02x0 + x1 0 −2x0 − 2x1 x1

x1 −2x0 − 2x1 0 x2x0 x1 x2 x3

.

52

5.2. Linear System

The rank-1-point is embedded in the rank 2 locus, and there are two furtherembedded points in the singular locus. There is a quadric surface containingthe rank 2 locus. The projection from one of the isolated nodes has a branchcurve consisting of a quadruple line and two reduced lines. See Figure 5.10.

Figure 5.10: The ramification locus of a symmetroid with a double line and twonodes, projected from an isolated node. It consists of a quadruple line and tworeduced lines. The point marked with N is the image of the rank-1-point, thepoints marked with • are the images of the embedded points in the singularlocus and indicates the image of the node.

If we attempt to reduce the number of nodes further, by letting H containa point on one of the other lines li or Lj , then H will cut T along a plane. Thesymmetroid will thus be reducible.Remark 5.1.1. Discount the symmetroids that either have a rank-1-point or asingularity that was not originally a singular point on the discriminant hyper-surface in the P5 of quadrics through four coplanar points. Then the number ofisolated rank-2-points and points embedded in the rank 2 locus is six, countedwith multiplicity for all of the above examples.

In Chapter 4, the corresponding statement is that the number of isolatednodes and points embedded in the rank 2 locus is four. This is not true forExamples 4.2.5 and 4.2.7, but the rank 2 locus is not equal to the singular locusin the matrix representations we have for those symmetroids. ♣

5.2 Linear System

Consider the linear system of quartic curves passing through nine given pointsP1, . . . , P9 ∈ P2, going twice through P1. By [Jes16, Article 79], any quarticsurface S ⊆ P3 with a double line is the image of the map ϕ : P2 → P3 induced bythe linear system. Consequently, all the symmetroids found above are rational.

Let us acquaint ourselves with this linear system. It is the plane cubicpassing through the Pi that is contracted to the double line on S. Indeed,the plane quartics in the linear system correspond to the hyperplane sectionsH := 4l− 2e1 −

∑9i=2 ei on S. Since these are singular in one point, S must be

singular along a curve of degree 1, that is, a line L. Intersect S with a planecontaining L. By Bézout’s theorem, this intersection is 2L+ C for some conicC. The pencil of such planes gives a pencil of conics, which in P2 correspondsto the pencil l− e1 of lines through P1. Hence B := H − (l− e1) = 3l−

∑9i=1 ei

is common for all the hyperplanes containing L, which is precisely the linearequivalence class of the cubic passing through Pi.

If the Pi constitute the complete intersection of two cubic curves, then ϕ isa two-to-one map and the image is a quadric [Jes16, Article 80].

53


A quartic surface with a double line can have up to eight isolated nodes, inwhich case it is called Plücker’s surface. To see this, let Pi+4 be a point on theexceptional line Ei for i = 2, 3, 4, 5 such that P1, Pi, Pi+4 are collinear. Thenthe classes

l − e1 − e2 − e6 e2 − e6

l − e1 − e3 − e7 e3 − e7

l − e1 − e4 − e8 e4 − e8

l − e1 − e5 − e9 e5 − e9

are effective and contracted to eight distinct nodes on S. One could imaginethat it is possible to push this to nine nodes by letting P2, P3, P4, P5 be collinearso that l − e2 − e3 − e4 − e5 is another node. But observe that

(l − e2 − e3 − e4 − e5) ·B = −1.

Hence it is contained in the double line. It should be clear how one can let thePi have a more general position to obtain fewer nodes.

Ideally, we would like to give necessary conditions on the configuration ofthe nine points for the linear system to give rise to a symmetroid. Recall fromSection 1.1 that if S is a symmetroid, then there exist cubic surfaces that aretangent to S along a sextic curve of arithmetic genus 3. We would like todescribe these sextics via the linear system. The intersection between S and acubic surface has linear equivalence class 3H. The aim is therefore to write 3Has 2C +A, where C has degree 6 and genus 3, and A has degree 0. A speciouscandidate is C := 5l − 3e1 −

∑9i=2 ei and A := 2l −

∑9i=2 ei. We check that C

is as required:

deg(C) = C ·H = 5 · 4− 3 · 2− 8 · 1 = 6,

ρa(C) = 12((5− 1)(5− 2)− 3 · (3− 1)− 8 · 1 · (1− 1)

)= 3.

The obvious problem is that if A is effective, then the Pi constitute the completeintersection of two cubics and S is a quadric. We therefore settle for describingA and C in examples.

Example 5.2.1. In Appendix B.8 we let the Pi be such that P3 P2, P5 P4,P7 P6 and the triples P1, P2, P3; P1, P4, P5 and P1, P6, P7 are collinear. Lastly,we let P8 be P9 be general points. The resulting quartic S is then singular alongthe double line B = 3l −

∑9i=1 ei and in the six nodes

A1 := l − e1 − e2 − e3, A4 := e2 − e3,

A2 := l − e1 − e4 − e5, A5 := e4 − e5,

A3 := l − e1 − e6 − e7, A6 := e6 − e7.

We project from A4 and find a symmetric matrix M representing S, usingAlgorithm 1.1.4. The rank 2 locus is all of the Ai and B.

We consider the curve C defined by the (3 × 3)-minors of the submatrixof M obtained by removing the first column. Having defined the ring mapcorresponding to ϕ in Macaulay2, we can investigate how C is represented inthe linear system. The sextic decomposes into the quartic curve

3l − e1 − 2e2 − e4 − e6 − e8 − e9,

54

5.2. Linear System

the exceptional line e2 and the associated reduced line of B. Hence

2C = 2(3l − e1 − 2e2 − e4 − e6 − e8 − e9) + 2e2 +B

= 9l − 3e1 − 3e2 − e3 − 3e4 − e5 − 3e6 − e7 − 3e8 − 3e9.

Note that 3H − 2C = A1 +A2 +A3 +A4 +A5 +A6.Lastly, removing different columns ofM gives four independent sextic curves

of genus 3. The system

|(3l − e1 − 2e2 − e4 − e6 − e8 − e9) + e2| = |3l − e1 − e2 − e4 − e6 − e8 − e9|

has the correct dimension to account for this, since there is a web of planecubics passing through the six points. ♥

Let S be a symmetroid with matrix M . The (3 × 3)-minors of a (3 × 4)-submatrix of M define a sextic curve with arithmetic genus 3. The rank 2 locusis given by all the (3 × 3)-minors of M and hence is in the base locus of theweb generated by these four sextics.

In all the examples we have seen of symmetroids with a double line, thedouble line has been contained in the rank 2 locus. Therefore it is tempting toconjecture that B must be contained in 2C, just as in the previous example.We will now present a counterexample to this conjecture.

Example 5.2.2. In Appendix B.9 we start with a smooth conic K and let P2,P4 and P6 be three points on K. Then for i = 2, 4, 6, let Pi+1 be the point onthe exceptional divisor over Pi corresponding to the tangent direction of K atPi. Define P1 as the intersection of the tangent lines of K at P2 and P4. Finally,let P8 and P9 be two points not on K, on a line through P1.

Then the image S of ϕ has, in addition to the double line, the nodes

A1 := l − e1 − e2 − e3,

A3 := e2 − e3,

A5 := e6 − e7.

A2 := l − e1 − e4 − e5,

A4 := e4 − e5,

The cubic passing through the Pi is the union of K and the line through P1,P8 and P9. Because of this, the singularity A := l − e1 − e8 − e9 is a point onthe double line B.

The projection from A1 is ramified along a singular, irreducible cubic curve, areduced line and a double line. The two lines meet on the cubic, as in Figure 5.11on the next page. Hence it is not possible to divide the branch curve into twocubics having a common line, so B is not contained in the rank 2 locus. UsingAlgorithm 1.1.4 we find a matrix M for S. The rank 2 locus is a scheme oflength 10. It consists of A, with multiplicity 3, and two other points on B, withmultiplicity 2 each. These three points are embedded in the singular locus. Inaddition, A1, A4 and A5 are rank-2-points. The rank 2 locus is not containedin a quadric.

Let C be the curve on S determined by the (3× 3)-minors of the submatrixobtained by removing the first column ofM . We have not been able to describeC via the linear system, but it is a simple task to verify that the reduced lineassociated to B is not a component of C. ♥

55


This does not mean that no choice of curve C contains B or its associatedreduced line:

Example 5.2.3. If we in Example 5.2.2 instead define C by removing thesecond column of M , rather than the first, then C consists of two double linesand a conic. These correspond to B, two times the exceptional line e3 andl − e4 − e6, respectively. Since

C = B + 2e3 + (l − e4 − e6)= 4l − e1 − e2 + e3 − 2e4 − e5 − 2e6 − e7 − e8 − e9,

then

3H − 2C = 4l − 4e1 − e2 − 5e3 + e4 − e5 + e6 − e7 − e8 − e9.

This can be written as A+ 3A1 + 2A3 +A4 +A5. ♥

Figure 5.11: Ramification locus for a symmetroid with a double line that isnot contained in the rank 2 locus. It consists of a singular cubic, as well as areduced line and a double line meeting on the cubic. The points marked withare the images of isolated rank-3-nodes. Both • and indicate the images ofrank-2-points, but • means that the points are embedded in the singular locus.

5.3 Double Line Not Contained in the Rank 2 Locus

How rare is it that a symmetroid is double along a line, but that line is notcontained in the rank 2 locus? It turns out that we are able to find at least asmany of these as of symmetroids where the double line is contained in the rank2 locus.

Consider the web W of quadrics defined by the matrix in Example 5.2.2. Itsbase locus is a scheme of length 4, consisting of a scheme of length 2 with supportin a point p1, as well as two other points p2 and p3. Choosing coordinates suchthat the points are p1 := [1 : 0 : 0 : 0], p2 := [0 : 1 : 0 : 0] and p3 := [0 : 0 : 1 : 0],we may take V(xy + xw), V(xz), V(yz), V(yw), V(zw) and V

(w2) as a basis

for the quadrics passing through Bl(W ). The matrix parametrising this P5 ofquadrics is

M :=

0 y01 y02 y01y01 0 y12 y13y02 y12 0 y23y01 y13 y23 y33

.

56

5.4. Dimensions of Families of Rational Symmetroids

With a computation similar to the one in Appendix B.7, we find thatSingV(det(M)) is the union of a linear 3-space T , a smooth quadratic sur-face Q and four planes Πi. The quadric Q and two of the planes, Π1 and Π2,are in the rank 2 locus. In addition, there is a double quadric surface and adouble plane in the rank 2 locus which are completely contained in T . A general3-space H ⊂ P5 will intersect the singular locus in a line and six points. Also,H meets the rank 2 locus in seven points, three of which are double. One suchsymmetroid is

0 x0 + x1 x2 + x3 x0 + x1x0 + x1 0 x2 x0x2 + x3 x2 0 x1x0 + x1 x0 x1 x3

,which is displayed in Figure A.2b on page 68.

The 3-space T is the set of quadrics passing through Bl(W ) that are sin-gular at p1. That the discriminant hypersurface in P5 is singular along T isimmediate from Lemma 3.2.2. The quadratic surface of rank 2 quadrics insideT corresponds to surfaces Π1 ∪ Π2, where Π1 is a plane containing p1p2 andΠ2 is a plane containing p1p3. The plane of rank 2 quadrics in T is the set ofquadrics that are the unions of the plane p1p2p3 and another plane throughp1. We see that both these surfaces contain the rank-1-point corresponding totwice p1p2p3.

The components Q and Πi are cut along a line each by T . The planes Π1and Π2 of rank-2-quadrics meet in a single point. So do the planes Π3 and Π4;they intersect in the rank-1-point. The two planes Π1 and Π3 meet in a line onQ. Likewise for Π2 and Π4. The intersections Π1 ∩Π4 and Π2 ∩Π3 are empty.

Example 5.3.1. Let H be a 3-space containing the line L := Π3 ∩ T . Then Lintersects Π4 in the rank-1-point. The line Π1 ∩ Q ⊂ Π3 cuts L in a point P .Generically, H will meet Q in another point and intersect Π2 in a single pointoutside L. However,H can in degenerate cases intersect Q with multiplicity 2 inP . Thus the resulting symmetroid is double along a line and has one additionalrank-2-point. An example of such a symmetroid is defined by the matrix

0 x0 x1 x0x0 0 x0 x1x1 x0 0 x2x0 x1 x2 x3

.It is shown in Figure A.4d on page 69. ♥

5.4 Dimensions of Families of Rational Symmetroids

Recall from Section 3.1 that there is a correspondence between the quarticsymmetroids in P3 and linear 3-spaces in P9. By [Har92, Lecture 11, p. 138],the dimension of the Grassmannian of k-dimensional, linear subspaces of Pn is

dimG(k, n) = (k + 1)(n− k). (5.1)

Therefore the symmetroids form a 24-dimensional variety in the P(3+44 )−1 = P34

of all quartic surfaces in P3.We have looked at webs of quadrics with three different base loci:

57


(i) four general points in Section 4.3,

(ii) four coplanar points in Section 5.1,

(iii) a scheme of length 4 with support in three points in Section 5.3.

We will now compute the dimensions of the families of rational symmetroidsthese induce.

Four General Points

We want to compute the dimension of the family of rational symmetroids thatare found by considering webs of quadrics with a base locus containing fourgeneral points. While we have mostly focused on symmetroids with a doubleconic, for ease of computation, we restrict to the case where the P3 cuts one ofthe Πi or the extra planes in SingV(det(M)) in a line. These symmetroidsare briefly discussed in Section 5.1.

To give four points in P3 is the same as giving a 4-tuple in P3×P3×P3×P3.Naturally, there are 4-tuples that do not correspond to points spanning P3, butremoving them do not affect the dimension. Therefore, the set of all choices offour general points has dimension 4 · 3 = 12.

We have from [Har92, Example 11.42] that the subvariety of G(k, n) consist-ing of k-planesK, which intersects a fixedm-planeM such that dim(K∩M) ≥ l,has dimension

(l + 1)(m− l) + (k − l)(n− k). (5.2)

Hence, given four general points we get in this way a 6-dimensional family ofsymmetroids with a double line. In total, varying the four points yields a familyof rational quartic symmetroids of dimension 12 + 6 = 18.

Four Coplanar Points

The first three points can be chosen freely, and the last point must be in theplane spanned by the other three. Hence giving four coplanar points is the sameas giving a 4-tuple in P3 × P3 × P3 × P2, which is 11-dimensional.

We have seen that a general 3-space in the P5 of quadrics through fourcoplanar points gives a rational symmetroid. Equation (5.1) gives that thedimension of G(3, 5) is 8. So all together with the different choices of fourcoplanar points, we get a family of rational quartic symmetroids of dimension11 + 8 = 19.

Scheme of Length 4 with Support in Three Points

The three points can be chosen freely. For the scheme to be of length 4, one of thepoints must be doubled in one direction. In P3 there is a P2 of directions througha point. Hence the scheme can be represented by a 4-tuple in P3×P3×P3×P2.

Again, we know that a generic P3 in the space of quadrics through a schemeof length 4 with support in three points gives rise to a rational symmetroid.Just as before, we get in entirety a family of dimension 11 + 8 = 19 of rationalquartic symmetroids.

58

5.4. Dimensions of Families of Rational Symmetroids

A Larger Family

While we explained why a symmetroid with a line of rank-2-points induces aweb of quadrics with a base locus that is generically four coplanar points, wehad no such justification for base loci of length 4 with support in three points.We merely studied those after stumbling upon an example with that property.It is in fact a rather special situation.

For a surface to be double along a line it must contain a line. Suppose thatS := V(F ) is a quartic symmetroid containing the line L := V(x0, x1). Then Fcannot contain any x4

2, x32x3, x2

2x23, x2x

23 or x4

3 terms. This is five conditions,but since dimG(1, 3) = 4, it is only one condition for a quartic to contain anarbitrary line.

Because S is a symmetroid, then L is a pencil of rank-3-quadrics. A pencil notcontaining a single rank-4-quadric must contain at least three rank-2-quadrics.Hence S has at least three singularities on L. We claim that if S has foursingularities on L, then it must be singular along the whole of L. Let again Lbe V(x0, x1). Then F can be written as

F = x20F00 + x0x1F01 + x2

1F11 + x0F0 + x1F1,

where F00, F01, F11 ∈ C[x0, x1, x2, x3] and F0, F1 ∈ C[x2, x3]. Note that

SingS ∩ L = V(F0, F1) ∩ L.

Since V(F0) and V(F1) are cubics, they contain L if L intersects them in morethan three points, so the claim follows.

Thus to require a quartic symmetroid to be double along a line is the sameas requiring it to contain a line and to have an additional singularity on thatline. As we saw, it is one condition to contain a line. Demanding that the quarticis singular at a given point imposes four conditions, but since the symmetroidalready passes through the point, it reduces to three extra conditions. Moreover,the extra node can be anywhere on the line, so it amounts to just two conditions.We conclude that the quartic symmetroids with a double line form a family ofdimension 24− 1− 2 = 21.

We let Macaulay2 compute the singular locus and the rank 2 locus of thediscriminant hypersurface in the P8 of quadrics through a single given pointp ∈ P3. The rank 2 locus is a fivefold of degree 10. The singular locus consistsin addition of a linear fivefold X. By Equation (5.2), there is an 18-dimensionalfamily of linear 3-spaces intersecting X in a line. Since p can be any point in P3,this totals to a 21-dimensional family of quartic symmetroids that are doublealong a line.

59

CHAPTER 6

Degenerated Symmetroids

He who hears the rippling of rivers inthese degenerate days will not utterlydespair.

Henry David Thoreau

ALL the summetroids we have considered thus far are of course degeneratein the sense that they are not generic. However, here we will take a brieflook at surfaces that are even more special. This includes symmetroids

that are double along a cubic curve, have a triple point or are cones. Lastly, wediscuss our work.

6.1 Symmetroids with No Isolated Singularities

We present here three symmetroids that were discovered in the context ofSection 5.3. Two of them are singular along reducible cubic curves. We startout with a surface that is singular along two intersecting lines and has a rank-1-point where they meet. It is a degeneration of the surfaces in Chapter 4, havingno isolated singularities.

In the notation of Section 5.3, let L1 be a line in Π3 passing through therank-1-point. The line Π1 ∩Q ⊂ Π3 meets L1 in a point. Let L2 be a line in Tpassing through the rank-1-point and meeting Q. Then a 3-space H containingL1 and L2 will generically not meet SingV(det(M)) in any further points. Forinstance, the symmetroid X given by

0 x0 x1 x0x0 0 x0 x2x1 x0 0 x1x0 x2 x1 x3

is double along two lines and has no isolated singularities. It is plotted inFigure A.5e on page 70.

Next, let L1 ⊂ Π3, L2 ⊂ Π4, L3 ⊂ T be lines passing through the rank-1-point. The P3 spanned by the Li gives rise to a symmetroid with three double

6. Degenerated Symmetroids

lines. One such symmetroid is given by0 x0 x1 x0x0 0 2x0 − x2 x2x1 2x0 − x2 0 x1x0 x2 x1 x3

.By a change of coordinates we obtain

A :=

0 x0 x1 x0x0 0 −2x2 4x0 + 2x2x1 −2x2 0 2x1x0 4x0 + 2x2 2x1 6x0 + 2x1 + 2x2 − x3

,which has determinant

det(A) = 4(x0x1x2x3 + x2

0x21 + x2

0x22 + x2

1x22).

We recognise this from Equation (2.1) as the Steiner surface.Finally, let Π be a plane intersecting Q in a conic. IfH is a 3-space containing

Π, then it will generically meet T in a line, so the resulting symmetroid is doublealong a conic and a line. The symmetroid defined by

0 x0 x1 x0x0 0 2x2 − x3 x2x1 2x2 − x3 0 x0x0 x2 x0 x3

is an example of this. It is shown in Figure A.7 on page 72.

Base Loci

Both the Steiner surface andX are uniquely determined by the base loci of theirwebs of quadrics. For each symmetroid, the base locus is a scheme of length 6.Hence there is a unique web of quadrics passing through it.

The base locus of the Steiner surface consists of three points and a directionthrough each. The directions correspond to mutually skew lines, none of whichare contained in the plane spanned by the three points.

For X the base locus also consists of three points and a direction througheach. Again, the directions correspond to mutually skew lines, but one of themis contained in the plane spanned by the three points.

Cones

As mentioned in Remark 3.1.4, the space of quadrics aTMa degenerates to a netwhen the symmetroid is a cone. The arguments in Section 4.3 and Section 5.1usually involve a plane Π, in some subset of the P9 of quadrics, which is extendedto a P3. The discriminant of this P3 is then the sought-after symmetroid. Thesame type of arguments can be carried out for cones, except that Π is notextended to a P3.

62

6.2. Quartic Symmetroids with a Triple Point

If Π cuts the rank 2 locus in some points, then the resulting cone has rank 2in the lines through those points and the apex. This symmetroid has no othersingularities. For instance, the matrix

0 x0 −x1 + x2 x0x0 0 −x0 + x1 − x2 x1

−x1 + x2 −x0 + x1 − x2 0 x1x0 x1 x1 x2

defines a symmetroid that is only singular along a line through the rank-0-point[0 : 0 : 0 : 1]. It is drawn in Figure A.4e on page 69.

Equation (5.1) gives that the Grassmannian G(2, 9) of planes in P9 hasdimension 21.

6.2 Quartic Symmetroids with a Triple Point

A quartic surface S := V(F ) in P3 with a triple point p is automatically rational.Indeed, the projection πp : S \ p 99K P2 from p is a birational map.

If we let p := [1 : 0 : 0 : 0], then p is a triple point for S if F contains nox4

0, x30f1 or x2

0f2 terms, where f1, f2 ∈ C[x1, x2, x3] are polynomials of degree 1and 2, respectively. Then F can be written as

F = x0F3 + F4,

where F3, F4 ∈ C[x1, x2, x3] are polynomials of degree 3 and 4, respectively.The surfaces X3 := V(F3) and X4 := V(F4) meet in a curve of degree 12 on S.Since both X3 and X4 are cones with apex p, the curve breaks up into twelveconcurrent lines.

The lines through p are blown down to points pi via πp. These twelve pointsare the intersection of a cubic and a quartic curve, namely

p1, . . . , p12 = VP2(F3) ∩ VP2(F4).

Now S can be represented as the image of the map induced by the linearsystem of quartics through the pi. The triple point p has linear equivalenceclass 3l −

∑12i=1 ei.

It is a simple task to find quartic symmetroids with triple points. Assumethat lij ∈ C[x1, x2, x3]1 are linear forms for 0 ≤ i ≤ j ≤ 3. Then the matrix

l00 + x0 l01 l02 l03l01 l11 l12 l13l02 l12 l22 l23l03 l13 l23 l33

defines a symmetroid which is triple in [1 : 0 : 0 : 0]. Clearly p is a rank-1-point.

If S is a symmetroid with p := [1 : 0 : 0 : 0], then

F3 =

∣∣∣∣∣∣l11 l12 l13l12 l22 l23l13 l23 l33

∣∣∣∣∣∣.63

6. Degenerated Symmetroids

As explained in Section 1.1, X3 touches S along a sextic curve of genus 3. SinceX3 intersects S in the twelve lines through p, these must coincide in such a waythat the lines through p have even multiplicity.

Consequently, VP2(F3) is a contact curve for VP2(F4); their intersection isa scheme which is two times a scheme of length 6. Consider the special casewhere this scheme has support in six points, pi for i = 1, . . . , 6. This occurswhen two and two lines through p coincide. Then pi+6 is a tangent directionthrough pi and S have the six nodes ei− ei+6. It is proved in [Jes16, Article 93]that a quartic surface with a triple point and six nodes is always a symmetroid.

Recall from Section 5.4 that the quartics in P3 form a P34. To require atriple point at a given point imposes ten conditions on the quartic, whereas adouble point imposes only four conditions. Allowing the nodes to be any pointsin P3, only imposes 4 − 3 = 1 condition for each. Hence the set of quarticswith a given triple point and six nodes has dimension 34 − 10 − 6 = 18. Thedimension is 21 if we allow the triple point to be arbitrary.

6.3 Final Remarks

There are still numerous questions we would like to answer. The long-term goalis a complete classification of rational quartic symmetroids.

How far away are we from achieving that aim? Are there many symmetroidsthat we have not described? We argued in Section 5.4 that the family of quarticsymmetroids that are singular along a line has dimension 21. Under the assump-tion that the base locus is non-empty, we found a component with the correctdimension. We have not proved that a double line implies a non-empty baselocus, so it is unclear if we have missed some symmetroids using this procedure.

We demonstrated that symmetroids with rank 2 along a line have genericallyfour coplanar base points. In addition, symmetroids with rank 2 along a conicsection have generically base loci consisting of four general points. We haveno results on symmetroids with a curve contained in the rank 2 loci, but withnon-generic base loci. Furthermore, we have not examined the possibility ofsymmetroids with rank-3-nodes along a conic section.

One important step towards a classification is to determine whether allquartic surfaces with the same singular locus, say a double line and six isolatednodes, are symmetroids, or if only a subset of these surfaces has a symmetric,determinantal representation.

There are issues concerning the number of isolated nodes which remainsto be addressed. Firstly is the verification or disproving of Conjectures 4.4.3and 4.4.4. Furthermore, does there exist quartic symmetroids that are singularalong a smooth conic section and have only one node in a nondegenerate way or no isolated nodes? Similarly, can a symmetroid be singular along a singleline and have only one node or no further singularities, in a nondegenerate way?

In our experience, it is more common that a symmetroid with a double linehas six nodes than any other number of isolated nodes. Likewise for symmetroidswith a double conic and four isolated nodes. Are these heuristic results true?

We have not conducted much work on symmetroids that are double alonga cubic curve. The only ones we have seen are the Toeplitz symmetroid inExample 3.3.2, the Steiner surface and the symmetroid with the double lineand conic. The Toeplitz symmetroid is singular along a twisted cubic curve and

64

6.3. Final Remarks

a line. Its singular locus is the complete intersection of two quadrics, so thesurface is reducible. It is not true in general that a quartic surface which isdouble along a twisted cubic is reducible. Indeed, [CW05, Theorem 2.1] provides

256x30x2 − 256x2

0x21 − 288x0x1x2x3 + 256x3

1x3 + 27x22x

23 = 0

as a counterexample. But it is worth investigating if all symmetroids which aresingular along a twisted cubic curve are reducible.

We also desire suitable conditions for the rank 2 locus to be contained in aquadric surface. This is not the case for the generic symmetroids, but it is truefor the majority of the examples we have given.

65

APPENDIX A

Gallery of Symmetroids

Drawing is the honesty of the art.There is no possibility of cheating.It is either good or bad.

Salvador Dalí

HERE we showcase some of the symmetroids referred to in the text. Thepictures are drawn using the visualising tool surf, developed by StephanEndraß [End10]. All the singularities are real and contained in the affine

chart used. For some of the surfaces, in particular Figure A.3a, you may needto squint in order to spot every singular point.

A.1 Symmetroids Singular along a Line

(a) Eight additional nodes. (b) Seven additional nodes.

Figure A.1: Rational quartic symmetroids singular along a line.

A. Gallery of Symmetroids

(a) The double line is contained in therank 2 locus.

(b) The double line is not contained inthe rank 2 locus.

Figure A.2: Rational quartic symmetroids singular along a line, with six isolatednodes.

(a) Five additional nodes. (b) Four additional nodes.


68

A.1. Symmetroids Singular along a Line

(a) Three additional nodes,no rank-1-points.

(b) Three additional nodesand a rank-1-point.

(c) Two additional nodes. (d) One additional node.

(e) No additional nodes.


69


A.2 Symmetroids Singular along a Conic

(a) Four additional nodes. The lines meetat infinity.

(b) Three additional nodes.


(e) No additional nodes.

Figure A.5: Rational quartic symmetroids singular along two intersecting lines.

70

A.2. Symmetroids Singular along a Conic

(a) Four additional nodes. (b) Three additional nodes.


Figure A.6: Rational quartic symmetroids singular along a smooth conic section.

71


A.3 Symmetroid Singular along a Reducible Cubic

Figure A.7: Quartic symmetroid singular along a smooth conic section and aline.

72

APPENDIX B

Code and Computations

Worst-case estimates suggested that trying to computeGröbner bases might be a hopeless approach to solvingproblems. But from the first prototype, Macaulay wassuccessful surprisingly often.

Computations in Algebraic Geometry with Macaulay2

THE purpose of this appendix is to collect all the code necessary for thecomputations mentioned earlier and the unsightly long polynomial outputthat otherwise makes for a boring read. The calculations have been per-

formed using Macaulay2, which is a computer algebra system for research inalgebraic geometry, developed by Daniel Grayson and Michael Stillman [M2].

Due to the limitations of Macaulay2, the calculations have to be done overQ, so some of the output is reducible.

B.1 Computations for Example 3.3.1

Consider the following snippet:

k = QQ;R = k[x_0..x_3];

P = matrixx_3, x_0, 0, x_0,x_0, x_3, x_1, 0 , 0, x_1, x_3, x_2,x_0, 0, x_2, x_3;

pillow = ideal det P

rank2 = saturate minors(3, P)rank1 = saturate minors(2, P)sing = saturate ideal singularLocus pillow

degree rank2codim rank2

B. Code and Computations

minimalPrimes rank2minimalPrimes singassociatedPrimes sing

The equation defining the pillow is

det(P ) = x20x

21 − 2x2

0x1x2 + x20x

22 − 2x2

0x23 − x2

1x23 − x2

2x23 + x4

3 = 0.

There are no rank-1-points on the surface. Its rank 2 locus is the one-dimensionalvariety of degree 2 given by the ideal

I := < (x1 + x2)x3,(2x2

2 − x23)x3,(

2x20 − x2

3)x3,

x0(x1 + x2)(x1 − x2),x0(x1x2 − x2

2 + x23),

x20x1 − x2

0x2 + x2x23 >,

where the important observation is that the first generator is a quadric. We candescribe how the quadric relates to the rank 2 locus: The ideal I decomposesas the intersection of the ideals

〈x0, x3〉, 〈x1 − x2, x3〉,⟨x0 − x2, x1 + x2, 2x2

2 − x23⟩,

⟨x0 + x2, x1 + x2, 2x2

2 − x23⟩.

In other words, the rank 2 locus is the union of the two lines L1 := V(x0, x3)and L2 := V(x1 − x2, x3) lying in the plane x3 = 0 and the four points[

1 : 1 : −1 : ±√

2] [

1 : −1 : 1 : ±√

2]

in the plane x1 + x2 = 0.The singular locus consists of the rank 2 locus and five embedded points: The

intersection L1∩L2, the two points [0 : 1 : ±i : 0] ∈ L1 and [1 : ±i : ±i : 0] ∈ L2.Let us now turn to the variety of singular points on quadrics stemming from

the pillow. The related claims can be verified easily by hand. Take for instancethe corner

[1 : 1 : −1 :

√2]. Then Pa = 0 reduces toa1 + a3 +

√2a0

a0 + a2 +√

2a1a1 − a3 +

√2a2

a0 − a2 +√

2a3

=

0000

.Adding the first and the third row, and the second and the last row, we findthat the solutions to these equations are described by the ideal⟨

2a0 +√

2(a1 + a3), 2a1 +√

2(a0 + a2)⟩.

We can use Macaulay2 to confirm that the line defined by this ideal does indeedlie on Q:

74

B.2. Computations for Example 3.3.2

R1 = k[a_0..a_3];

A = matrixa_1 + a_3, 0, 0, a_0, a_0, a_2, 0, a_1, 0, a_1, a_3, a_2, a_0, 0, a_2, a_3;

f = factor det AQ = ideal f#2#0

R2 = R1[s] / (s^2 - 2);-- s = sqrt(2)-- Polynomial factorisation is only-- implemented over rational numbers.

i = map(R2, R1, gens R1); -- InclusionQ = i(Q);

L = ideal(2*a_0 + s*(a_1 + a_3),2*a_1 + s*(a_0 + a_2));

L + Q == L


The next lines of code are used to examine the stratification of the Toeplitzsymmetroid.

k = QQ;R = k[x_0..x_3];

T = matrixx_0, x_1, x_2, x_3,x_1, x_0, x_1, x_2,x_2, x_1, x_0, x_1,x_3, x_2, x_1, x_0;

f = factor det T

C1 = ideal f#0#0C2 = ideal f#1#0

S1 = saturate ideal singularLocus C1S2 = saturate ideal singularLocus C2

rank2 = saturate minors(3, T)rank1 = saturate minors(2, T)

MP = minimalPrimes rank2

75


rank1 == MP_0 + MP_1rank1 == intersect(S1, S2)

The determinant detT factors as the product of the quadratic forms

− x20 − x0x1 + x2

1 + 2x1x2 + x22 − x0x3 − x1x3

and

− x20 + x0x1 + x2

1 − 2x1x2 + x22 + x0x3 − x1x3.

Denote the cones these forms define by C1 and C2. The Ci have apices in thepoints [1 : −1 : 1 : −1] and [1 : 1 : 1 : 1], respectively. These two pointsconstitute the rank 1 locus of the Toeplitz symmetroid.

The rank 2 locus is not contained in any quadric; its ideal is generated bysix cubics. It decomposes as the intersection of the ideals⟨

x20 − 2x2

1 + x0x2, x0x1 − 2x1x2 + x0x3, x20 − x0x2 − 2x2

2 + 2x1x3⟩

and

〈x0 − x2, x1 − x3〉.

The twisted cubic and the line defined by these intersect precisely in the tworank-1-points.


Put P1 := [1 : 0 : 0], P3 := [0 : 1 : 0] and P5 := [0 : 0 : 1]. The requirementthat a cubic V(f) passes through these three points amounts to there being nox3

0, x31 or x3

2 terms in f . Furthermore, the lines P1P5 and P3P5 are V(x1) andV(x0), respectively. Thus, for V(f) to have these as tangent lines at P1 and P3,we must have

∂f

∂x0(P1) = ∂f

∂x2(P1) = ∂f

∂x1(P2) = ∂f

∂x2(P2) = 0.

Therefore, there cannot be any x20x2 or x2

1x2 terms in f either. Hence a cubicin this linear system is on the form

f(x0, x1, x2) = a210x20x1 + a120x0x

21 + a111x0x1x2 + a102x0x

22 + a012x1x

22.

We use the obvious basis x20x1, x0x

21, x0x1x2, x0x

22, x1x

22. The rest we leave

to Macaulay2:k = QQ;P2 = k[x_0..x_2];P4 = k[y_0..y_4];

f = map(P2, P4, x_0^2*x_1, x_0*x_1^2, x_0*x_1*x_2,x_0*x_2^2, x_1*x_2^2);

DelPezzo = ker f

minimalPrimes ideal singularLocus DelPezzo

76



We start with the 4-nodal del Pezzo surface from Appendix B.3.

k = QQ;P2 = k[x_0..x_2];P3 = k[y_0..y_3];P4 = k[z_0..z_4];

f = map(P2, P4, x_0^2*x_1, x_0*x_1^2, x_0*x_1*x_2,x_0*x_2^2, x_1*x_2^2);

S = ker f -- Del Pezzo Surface.

minimalPrimes ideal singularLocus S

Then we compute the strict transform of a line passing through P5, and the stricttransform of a conic that meets all the Pi except P5. These strict transformsare two conics, C1 and C2, that do not meet any of the nodes on the surface.Also, C1 and C2 intersect in two points, which span a line L. In this case,L = V(x3 +x4, x1 +x4, x0−x4) and we let the projection centre p be the point[1 : −1 : 0 : −1 : 1], which we verify is not on the surface.

-- Conics that avoid the singularities:C1 = preimage(f, ideal(x_0 + x_1));C2 = preimage(f, ideal(x_0*x_1 + x_2^2));

-- Find the line spanned by their intersection points.L = trim(C1 + C2);L = ideal(L_0, L_1, L_2)

-- Take the projection centre p to be a point on this line.-- p = [1 : -1 : 0 : -1 : 1]Ip = ideal(z_1 + z_0, z_2, z_3 + z_0, z_4 - z_0);

Ip + L == Ip -- p is on LIp + S == Ip -- p is not on the del Pezzo

Under the projection, a point q on the del Pezzo surface S is mapped to theunique intersection point between the line pq and the P3. Consider the unionof all lines from p to a point on S. This is a cone C over S with apex in p. Theimage of the projection is C ∩ P3. Because the del Pezzo surface is a quartic,p is a quadruple point on C. Thus, the equation of C is given by the uniquequartic contained in the ideal

I := I4p ∩ IDelPezzo.

This explains why we did not use [0 : 0 : 1 : 0 : 0] for p; one of the quadrics inthe pencil has apex in [0 : 0 : 1 : 0 : 0], in which case I contains ten quartics,rather than just one. Not to mention that [0 : 0 : 1 : 0 : 0] is not contained inthe P3 spanned by the nodes. However, it is of minor importance whether theirprojected images are coplanar or not.

77


We use V(x4) as the linear hyperspace.

C = trim intersect (Ip^4, S); -- Cone with apex in p.degrees C -- Check number of quartics.

-- Intersect with the hyperplane V(z_4):i = map(P3, P4, gens P3 | 0);

X = i(ideal(C_0)) -- Projected del Pezzo Surface.

First, we make sure that the singular locus of the projected del Pezzo surfacedoes indeed consist of two lines, L1 and L2, and four points.

Next, there are embedded points in the singular locus: The intersectionL1 ∩ L2 and two other points on each Li. We pull the latter back to P4 andfind that the preimages of the two embedded points on Li span a line L′i in theplane spanned by the conic Ci. The line L′i is tangent to Ci.

sing = ideal singularLocus X;minimalPrimes sing

-- Confirm that the embedded points are-- ramification points for the projection:AP = associatedPrimes singL1 = preimage(i, AP_3)L2 = preimage(i, AP_2)L1 = ideal(L1_0, L1_1, L1_2)L2 = ideal(L2_0, L2_1, L2_2)minimalPrimes (L1 + C1)minimalPrimes (L2 + C2)

Since X has a node in [1 : 0 : 0 : 0], its defining equation is of the formf = ax2

0+bx0+c, where a, b, c ∈ Q[x1, x2, x3]. When projecting from [1 : 0 : 0 : 0]to V(x0), the branch curve is given by the discriminant b2 − 4ac.

-- Trick in order to use the function ’coefficients’:R1 = k[y_1..y_3][y_0];

X = sub(X, R1);

(Mon, Coeff) = coefficients(X_0,Monomials => y_0^2, y_0, 1);

X = sub(X, P3);

a = sub(Coeff_(0, 0), P3);b = sub(Coeff_(1, 0), P3);c = sub(Coeff_(2, 0), P3);

use P3;

branch = factor(b^2 - 4*a*c)

78


The built-in method adjoint does not give us what we want, so we copy thecorrect definition from [Eis+02].

classicalAdjoint = (G) -> (n := degree target G;m := degree source G;matrix table(n, n, (i, j) -> (-1)^(i + j) * det(

submatrix(G, 0..j - 1, j + 1..n - 1,0..i - 1, i + 1..m - 1))));

Then we just follow Algorithm 1.1.4 and check that matrix we end up with hasthe desired determinant.

F11 = branch#0#0 * branch#2#0 / 4;F22 = branch#0#0 * branch#1#0^2;

q = -a;g = b/2;delta = c;

I = ideal(F11, delta);RI = radical I;

-- Check that the space of cubics is 4-dimensional:degrees RI

F11*F22 - g^2 == q*delta

F12 = g;F13 = RI_0;F14 = RI_3;

-- Check that the Fij are independent:degrees trim ideal(F11, F12, F13, F14)

F23 = (F12*F13 // gens I)_(0,0);F24 = (F12*F14 // gens I)_(0,0);F33 = (F13*F13 // gens I)_(0,0);F34 = (F13*F14 // gens I)_(0,0);F44 = (F14*F14 // gens I)_(0,0);

F = matrixF11, F12, F13, F14,F12, F22, F23, F24,F13, F23, F33, F34,F14, F24, F34, F44;

M = classicalAdjoint(F) // delta^2;

cc = det F // delta^3;

79


N = matrix 0, cc*y_0, 0, 0,cc*y_0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

A = M + N

det A / X_0

Next, we find the variety of the singular loci of the quadrics aTAa. This isstraightforward: Look at the product Aa and create the corresponding matrixM called B in the code, since M is already in use of the coefficients to thevariables ai, such that Aa = Mx. Then det(M) = 0 is the desired variety. Itturns out to be the union of a plane and a 3-nodal cubic.

-- Trick in order to use the function ’coefficients’:R2 = k[w_0..w_3][y_0..y_3];

A = sub(A, R2);

W = matrixw_0, w_1, w_2, w_3;

P = A*W;

(M0, C0) = coefficients(P^0,Monomials => y_0, y_1, y_2, y_3);




B = matrixC0_(0,0), C0_(1, 0), C0_(2, 0), C0_(3,0),C1_(0,0), C1_(1, 0), C1_(2, 0), C1_(3,0),C2_(0,0), C2_(1, 0), C2_(2, 0), C2_(3,0),C3_(0,0), C3_(1, 0), C3_(2, 0), C3_(3,0)

R3 = k[w_0..w_3];

g = factor sub(det B, R3)

minimalPrimes ideal singularLocus ideal g#1#0

The base locus of the web of quadrics aTAa, equals the common intersectionof four generating quadrics. To find four such quadrics, take aTA(Pi)a for fourpoints Pi that span P3. In this case, the base locus consists of six points, countedwith multiplicity. More precisely, it contains five points where one of the pointsis counted twice.

80


-- General quadratic form:Q = (transpose(W)*P)_(0, 0)

R4 = k[w_0..w_3,y_0..y_3];

Q = sub(Q, R4)

-- Generators for the base locus:Q0 = sub(Q, y_0 => 1, y_1 => 0, y_2 => 0, y_3 => 0);Q1 = sub(Q, y_0 => 0, y_1 => 1, y_2 => 0, y_3 => 0);Q2 = sub(Q, y_0 => 0, y_1 => 0, y_2 => 1, y_3 => 0);Q3 = sub(Q, y_0 => 0, y_1 => 0, y_2 => 0, y_3 => 1);

base = ideal(Q0, Q1, Q2, Q3);degree baseminimalPrimes baseprimaryDecomposition base

For the points P in the base locus, we find the set of points [QP ] such that thequadrics QP are singular at P .

-- The function ’singularLocus’ differentiates with respect-- to y_i as well, hence we have to do it manually.d0 = diff(w_0, Q);d1 = diff(w_1, Q);d2 = diff(w_2, Q);d3 = diff(w_3, Q);

dQ = ideal(d0, d1, d2, d3);

saturate sub(dQ, w_0 => 1, w_1 => 0, w_2 => 0, w_3 => 0)saturate sub(dQ, w_0 => 1, w_1 => 0, w_2 => 2, w_3 => 0)saturate sub(dQ, w_0 => 0, w_1 => 1, w_2 => 1, w_3 => 0)saturate sub(dQ, w_0 => 0, w_2 => 0, w_3 => 2)


We use the same points as in Appendix B.3, albeit we must swap the names ofP1 and P3 in order to be consistent with Example 4.2.4. Again, we can have noterms of the forms x3

0, x31, x3

2 or x21x2, but x2

0x2 terms are allowed.The cubics must have the same tangent at P3, and the tangent line must

be different from P3P5. We choose the line V(x1 − x2) and letx2

0x1 + x20x2, x

20x1 + x2

0x2 + x1x22, x

20x1 + x2

0x2 + x0x1x2,

x20x1 + x2

0x2 + x0x22, x

20x1 + x2

0x2 + x0x21

be the basis for cubics passing through the Pi. Using this, we find the del Pezzo

81


surface with ideal

〈x0x2 + x1x2 − 3x22 − x1x3 + 2x2x3 + x2x4 − x3x4,

x21 − 2x1x2 + x2x3 + x2x4 − x3x4〉.

It has nodes in [1 : 0 : 0 : 0 : 0], [0 : 0 : 0 : 0 : 1] and [1 : 1 : 1 : 1 : 1].

k = QQ;P2 = k[x_0..x_2];P4 = k[y_0..y_4];

f = map(P2, P4, x_0^2*x_1 + x_0^2*x_2 + x_1*x_2^2,x_0^2*x_1 + x_0^2*x_2 + x_0*x_1*x_2,x_0^2*x_1 + x_0^2*x_2,x_0^2*x_1 + x_0^2*x_2 + x_0*x_2^2,x_0^2*x_1 + x_0^2*x_2 + x_0*x_1^2);

DelPezzo = ker f



We choose the points P1 := [0 : 0 : 1], P2 := [0 : 1 : 0] and P3 := [0 : 1 : −1]on the line V(x0). On the line V(x1) through P1, we pick P4 := [1 : 0 : 0] andP5 := [1 : 0 : −1]. Macaulay2 produces the basis

x20x1, x0x

21, x0x1x2, x

21x2 + x1x

22, x

20x2 + x0x

22

of cubics through the Pi. The resulting del Pezzo surface in P4 has ideal⟨x1x3 + x2

3 − x2x4, x0x1 − x2x3 − x23⟩.

It has nodes in [1 : 0 : 0 : 0 : 0] and [0 : 0 : 0 : 0 : 1].

k = QQ;P2 = k[x_0..x_2];P4 = k[y_0..y_4];

I1 = ideal(x_0, x_1); -- [0 : 0 : 1]I2 = ideal(x_0, x_2); -- [0 : 1 : 0]I3 = ideal(x_0, x_1 + x_2); -- [0 : 1 : -1]I4 = ideal(x_1, x_2); -- [1 : 0 : 0]I5 = ideal(x_1, x_0 + x_2); -- [1 : 0 : -1]

I = intersect(I1, I2, I3, I4, I5)

-- The generators are listed in this order such that-- one node gets coodinates [1 : 0 : 0 : 0 : 0].f = map(P2, P4, I_1, I_0*x_0, I_0*x_1, I_0*x_2, I_2);

82

B.7. Computations for Section 4.3

DelPezzo = saturate ker f


B.7 Computations for Section 4.3

This is a straightforward factorisation into components and then examinationof their intersections.

k = QQ;R = k[x_0..x_5];

M = matrix 0, x_0, x_1, x_2,x_0, 0, x_3, x_4,x_1, x_3, 0, x_5,x_2, x_4, x_5, 0

rank2 = saturate minors(3, M);sing = saturate ideal singularLocus ideal det M;

dim rank2degree rank2

dim singdegree sing

MP = minimalPrimes rank2mp = minimalPrimes sing

Pi1 = MP_0;Pi2 = MP_2;Pi3 = MP_3;Pi4 = MP_5;Pi5 = mp_0;Pi6 = mp_4;Pi7 = mp_6;Pi8 = mp_7;

Q1 = MP_1;Q2 = MP_4;Q3 = MP_6;

saturate(Pi1 + Pi2)saturate(Pi1 + Pi3)saturate(Pi1 + Pi4)saturate(Pi2 + Pi3)saturate(Pi2 + Pi4)saturate(Pi3 + Pi4)

83


minimalPrimes(Q1 + Q2)minimalPrimes(Q1 + Q3)minimalPrimes(Q2 + Q3)

saturate(Pi1 + Q1)saturate(Pi1 + Q2)saturate(Pi1 + Q3)saturate(Pi2 + Q1)saturate(Pi2 + Q2)saturate(Pi2 + Q3)saturate(Pi3 + Q1)saturate(Pi3 + Q2)saturate(Pi3 + Q3)saturate(Pi4 + Q1)saturate(Pi4 + Q2)saturate(Pi4 + Q3)

saturate(Pi5 + Pi1)saturate(Pi5 + Pi2)saturate(Pi5 + Pi3)saturate(Pi5 + Pi4)saturate(Pi6 + Pi1)saturate(Pi6 + Pi2)saturate(Pi6 + Pi3)saturate(Pi6 + Pi4)saturate(Pi7 + Pi1)saturate(Pi7 + Pi2)saturate(Pi7 + Pi3)saturate(Pi7 + Pi4)saturate(Pi8 + Pi1)saturate(Pi8 + Pi2)saturate(Pi8 + Pi3)saturate(Pi8 + Pi4)


We let P1 := [0 : 0 : 1], P2 := [1 : 0 : 0] P4 := [0 : 1 : 0] and P6 := [1 : 1 : 1].So we want the quartics to have V(x1), V(x0) and V(x0 − x1) as their tangentlines at P2, P4 and P6, respectively. Lastly, we put P8 := [1 : −1 : 2] andP9 := [2 : 1 : −1]. We used Matlab to find a basis for the quartics that passthrough these points with the correct tangent lines and a singularity at P1:

6x0x21x2 − 5x2

0x22 + 3x0x1x

22 + 2x2

1x22 − 6x2

0x1x2,

48x0x21x2 + 55x2

0x22 + 3x0x1x

22 − 82x2

1x22 − 24x0x

31,

48x0x21x2 + 73x2

0x22 − 3x0x1x

22 − 94x2

1x22 − 24x2

0x21,

48x0x21x2 + 103x2

0x22 + 3x0x1x

22 − 130x2

1x22 − 24x3

0x1.

We use this basis to define the ring map f associated to the map ϕ induced bythe linear system:

84


k = QQ;P2 = k[x_0..x_2];P3 = k[y_0..y_3];

q1 = 6*x_0*x_1^2*x_2 - 5*x_0^2*x_2^2 + 3*x_0*x_1*x_2^2 +2*x_1^2*x_2^2 - 6*x_0^2*x_1*x_2;

q2 = 48*x_0*x_1^2*x_2 + 55*x_0^2*x_2^2 + 3*x_0*x_1*x_2^2 -82*x_1^2*x_2^2 - 24*x_0*x_1^3;

q3 = 48*x_0*x_1^2*x_2 + 73*x_0^2*x_2^2 - 3*x_0*x_1*x_2^2 -94*x_1^2*x_2^2 - 24*x_0^2*x_1^2;

q4 = 48*x_0*x_1^2*x_2 + 103*x_0^2*x_2^2 + 3*x_0*x_1*x_2^2 -130*x_1^2*x_2^2 - 24*x_0^3*x_1;

f = map(P2, P3, q1, q2, q3, q4);

S = ker f

The result is a surface S with the equation

41472x40 − 2016x3

0x1 − 34848x30x2 + 31680x3

0x3 − 13444x20x

21 + 18550x2

0x1x2

+ 2110x20x1x3 − 1252x2

0x22 − 13310x2

0x2x3 + 6050x20x

23 − 1752x0x

21x2

+ 368x0x21x3 + 4456x0x1x

22 − 3292x0x1x2x3 + 1100x0x1x

23

+ 220x0x22x3 − 1100x0x

32 − 219x1x

32 + 219x2

1x2x3 − 169x21x

23

− 46x1x22x3 + 165x1x2x

23 + 215x4

2 − 165x32x3.

We are able to use f to relate three of the nodes to the linear system right away:The ideal f(node) is contained in the ideal of one the lines P1P2, P1P4, P1P6.MP = minimalPrimes ideal singularLocus S;

-- See which singularities we can identify:minimalPrimes f(MP_1)minimalPrimes f(MP_2)minimalPrimes f(MP_3) -- l - e_1 - e_6 - e_7minimalPrimes f(MP_4) -- l - e_1 - e_2 - e_3minimalPrimes f(MP_5)minimalPrimes f(MP_6) -- l - e_1 - e_4 - e_5

We then apply Algorithm 1.1.4 to obtain a matrix representation of S. Animplementation of the algorithm is given in Appendix B.4. The output is asymmetric matrix with so large entries that we will not present it here. Weremove the first column and consider the curve C determined by the (3 × 3)-minors of the remaining submatrix.C = minors(3, M_1, 2, 3);

mp = minimalPrimes C;apply(mp, i -> degree i) -- 4, 1, 1

It consists of a quartic and two lines. One of the lines is immediately identifiedas the reduced line associated to B:

85


-- Check that the reduced line-- associated to B is a component:mp_1 == MP_0 -- True

The quartic is identified with a plane cubic. It is singular at P2. We check whichpoints it passes through.

cubic = (minimalPrimes f(mp_0))_0

saturate ideal singularLocus cubic -- p2

p1 = ideal(x_0, x_1); -- [0 : 0 : 1]p2 = ideal(x_1, x_2); -- [1 : 0 : 0]p4 = ideal(x_0, x_2); -- [0 : 1 : 0]p6 = ideal(x_0 - x_1, x_0 - x_2); -- [1 : 1 : 1]p8 = ideal(x_0 + x_1, 2*x_0 - x_2); -- [1 : -1 : 2]p9 = ideal(x_0 - 2*x_1, x_1 + x_2) -- [2 : 1 : -1]

cubic + p1 == p1 -- Truecubic + p2 == p2 -- Truecubic + p4 == p4 -- Truecubic + p6 == p6 -- Truecubic + p8 == p8 -- Truecubic + p9 == p9 -- True

It passes through all of them. We then check if it has some of the tangent linescorresponding to P3, P5 or P7.

dx0 = diff(x_0, cubic_0);dx1 = diff(x_1, cubic_0);dx2 = diff(x_2, cubic_0);

sub(dx0, x_0 => 1, x_1 => 0, x_2 => 0)sub(dx1, x_0 => 1, x_1 => 0, x_2 => 0)sub(dx2, x_0 => 1, x_1 => 0, x_2 => 0)



It does not, so we conclude that the quartic is 3l− e1 − 2e2 − e4 − e6 − e8 − e9.This curve is singular at e2 − e3, so we can relate another node to the linearsystem.

86


saturate ideal singularLocus mp_0 -- e_2 - e_3

saturate(mp_2 + mp_0) -- e_2 - e_3mp_2 + MP_4 == MP_4 -- True

The last component in C passes through both e2 − e3 and l− e1 − e2 − e3. Wededuce that the line is e2.


We let the conic K be V(x0x1 − x22). Let P2 := [1 : 0 : 0], P4 := [0 : 1 : 0] and

P6 := [1 : 1 : 1]. The tangent lines to K at P2, P4 and P6 are V(x1), V(x0) andV(x0 + x1 − 2x2), respectively. That means that P1 is [0 : 0 : 1]. Finally, putP8 := [1 : −1 : 2] and P9 := [1 : −1 : 3]. We found the following basis of quarticspassing through the Pi using Matlab:

19x30x1 + 5x0x

31 − 20x2

0x1x2 − 4x20x

22,

7x30x1 − 5x2

0x21 − 5x2

0x1x2 + 5x0x21x2 − 2x2

0x22,

x30x1 + x2

0x21 − x2

0x22 − x0x1x

22,

24x30x1 − 20x2

0x1x2 − 9x20x

22 + 5x2

1x22.

We use this to define the ring map induced by the linear system:

k = QQ;P2 = k[x_0..x_2];P3 = k[y_0..y_3];

q1 = 19*x_0^3*x_1 - 20*x_0^2*x_1*x_2 - 4*x_0^2*x_2^2 +5*x_0*x_1^3;

q2 = 7*x_0^3*x_1 - 5*x_0^2*x_1^2 - 5*x_0^2*x_1*x_2 -2*x_0^2*x_2^2 + 5*x_0*x_1^2*x_2;

q3 = x_0^3*x_1 + x_0^2*x_1^2 - x_0^2*x_2^2 -x_0*x_1*x_2^2;

q4 = 24*x_0^3*x_1 - 20*x_0^2*x_1*x_2 - 9*x_0^2*x_2^2 +5*x_1^2*x_2^2;

f = map(P2, P3, q1, q2, q3, q4);

S = ker f

We obtain the surface S with defining polynomial

x20x

21 + 10x2

0x1x2 + 25x20x

22 − 5x2

0x2x3 + 20x0x21x2 − 2x0x

21x3 − 20x0x1x

22

− 10x0x1x2x3 − 25x0x22x3 + 5x0x2x

23 + 300x2

1x22 − 20x2

1x2x3 + x21x

23

+ 100x1x32 − 80x1x

22x3 − 400x4

2 + 100x32x3.

Applying Algorithm 1.1.4 as in Appendix B.4, we obtain the matrix

M(x) := M0x0 +M1x1 +M2x2 +M3x3,

87


where

M0 :=

−16 800 200 100800 −60000 −15000 −5000200 −15000 −2500 −1250100 −5000 −1250 −625

,

M1 :=

−32 1600 400 01600 −120000 −20000 0400 −20000 −5000 0

0 0 0 1250

,

M2 :=

−240 3200 1000 5003200 120000 10000 50001000 10000 12500 −6250500 5000 −6250 3125

,

M3 :=

0 400 0 0

400 0 0 00 0 0 00 0 0 0

.We remove the first column ofM . Then we verify that the reduced line associatedto B is not a component in the curve C defined by the (3× 3)-minors of this(4× 3)-matrix.

C = saturate minors(3, M_1, 2, 3);

MP = minimalPrimes ideal singularLocus SMP_2 + C == MP_2 -- False

88

APPENDIX C

Rational Double Points

Put thyself into the trick of singularity.

William Shakespeare

DYNKIN diagrams crop up in many areas of mathematics. This appendixis dedicated to explaining the connection between simply laced Dynkindiagrams those with no multiple edges and rational double points,

and thereby making sense of the symbols An, Dn and En used in the thesis.

C.1 ADE Singularities

The collection of simply laced Dynkin diagrams is called the ADE classification.The complete list, comprised of two families and three exceptional diagrams, isgiven in Table C.1.

An

Dn (n ≥ 4)

E6

E7

E8

Table C.1: ADE classification. The requirement n ≥ 4 for Dn is put in to avoidredundancy, as D3 ∼= A3.

Definition C.1.1. An ADE curve is an exceptional divisor C :=∑Ci, where

each irreducible component Ci is a (−2)-curve. Moreover, Ci · Cj ≤ 1 forall i 6= j, that is, two components intersect in at most one point and thentransversally. ♠

The ADE curves are named so because their dual graphs the graphsconstructed by representing Ci with a node, and the intersection between Ci

C. Rational Double Points

and Cj with an edge connecting their corresponding nodes are the Dynkindiagrams above. The An curves also go under the alias Hirzebruch-Jung strings.

Definition C.1.2. Let X and Y be normal surfaces and assume that X isnonsingular. A curve C on X is said to be exceptional if there is a birationalmap π : X → Y such that C is exceptional for π. This means that there is aneighbourhood U ⊂ X of C, a point y ∈ Y and a neighbourhood V ⊂ Y ofy, such that π(C) = y and π gives an isomorphism of X \ U onto Y \ V . Thissituation is expressed by saying C contracts to y. ♠

Definition C.1.3. Let X and Y be normal and irreducible surfaces. Assumefurther that X is smooth. Suppose that C ⊂ X is the exceptional curve con-tracting to the singularity y ∈ Y under the proper, birational map π : X → Y .Then y is called a rational singularity if the first higher direct image R1π∗OX

vanish. ♠

Proposition C.1.4 ([Bar+04, III, Proposition 3.4]). The ADE curves contractto rational singularities.

Suppose that C :=∑Ci is an exceptional curve, and that U ⊂ X is a

neighbourhood of C with ωU = OU . In particular, such a neighbourhood alwaysexists for an ADE curve [Bar+04, III, Proposition 3.5]. An effective divisor Don U is the principal divisor (f) of a rational function f on U , if and onlyif D · Ci = 0 for all i. Any effective divisor D can be written on the formD = Z +

∑Uj , where Z :=

∑riCi has support on C and each Uj intersects C

in at most finitely many points.

Proposition C.1.5 ([Bar+04, III, Proposition 3.7]). The divisors Z :=∑riCi,

where ri ≥ 0, satisfying

Z · Ci ≤ 0 (C.1)

for all i, are exactly the parts contained in C of divisors of rational functionsdefined on some neighbourhood of C.

Let (f), (f ′) be two principal divisors with corresponding parts Z :=∑riCi

and Z ′ :=∑r′iCi in C. Then for α, α′ ∈ C \ 0, the part of (αf + α′f ′) in

C is∑

minri, r′iCi. Thus there is a minimal effective divisor satisfying (C.1)

for all i, which is called the fundamental cycle of C or of the singularity Ccontracts to.

Table C.2 on the next page indicates the coefficients for the fundamentalcycles of the ADE curves.

An immediate consequence is:

Proposition C.1.6 ([Bar+04, III, Proposition 3.9]). For the fundamental cycleZ of an exceptional ADE curve, we have

Z2 = −2.

Definition C.1.7. The multiplicity of the maximal ideal m in a local ring Rof dimension d, is (d− 1)! times the leading coefficient of the Hilbert–Samuelpolynomial of R. The multiplicity of a variety X at a point p ∈ X is defined tobe the multiplicity of the maximal ideal m in the local ring OX,p. ♠

90

C.1. ADE Singularities

An1 1 1 1 1

Dn (n ≥ 4)

1

1 12 2 2

E6

2

1 2 3 2 1

E7

2

2 3 4 123

E8

3

234562 4

Table C.2: Fundamental cycles for ADE curves.

Artin showed in [Art66] that if X is the spectrum of a two-dimensionallocal ring and p ∈ X is a rational singularity with fundamental cycle Z, thenthe multiplicity at p equals −(Z2). Hence Proposition C.1.6 implies that ADEcurves contract to rational double points. This provides a converse to du Val’sresult in [Val34], that the resolution of an isolated double point on an embeddedsurface is an ADE curve.

In light of this, we denominate rational double points according to the ADEclassification of their exceptional curves. Rational double points are known inthe literature as du Val singularities, simple surface singularities and Kleiniansingularities.

Example C.1.8 (E6 singularity). We shall construct an ADE curve by se-quentially blowing up P2 in six points. Set X0 := P2 and let Pi be a point inXi−1. Denote the blow-up of Xi−1 with centre Pi by πi : Xi → Xi−1, and takeEi to be the exceptional line π−1

i (Pi). Consider these steps:

(i) Let L be a line in P2 and P1 any point on L.

(ii) Let P2 be the intersection of E1 and L− E1, where L− E1 is the stricttransform of L under π1. Put C1 := E1 − E2.

(iii) Let P3 be the intersection of E2 and L−E1−E2. Put C2 := E2−E3 andC3 := L− E1 − E2 − E3.

(iv) Let P4 be any point on E3 that is not on C2 or C3. Put C4 := E3 − E4.

(v) Let P5 be any point on E4 that is not on C4. Put C5 := E4 − E5.

(vi) Let P6 be any point on E5 that is not on C5. Put C6 := E5 − E6.

91

C. Rational Double Points

EachCi is irreducible with self-intersection C2i = −2. Moreover, they fit together

in the following dual graph:

C3

C1 C2 C4 C5 C6

This means that

C := C1 + C2 + C3 + C4 + C5 + C6 = L− E2 − E3 − E6

is an E6 curve.We know from Proposition C.1.4 that C contracts to a single point. We

can see this explicitly. Let π : X6 → P2 be the composition π1 · · · π6 andd the linear system of plane cubic curves with assigned base points P1, . . . , P6.Suppose D ∈ d, then we have a corresponding linear system d′ on X6 given by

d′ := |π∗(D)− E1 − · · · − E6|.

Note that d′ is base-point free, so it gives rise to a morphism X6 → X, whereX is a cubic surface in P3. We claim that this morphism maps C to a point.

Let l denote the linear equivalence class of π∗(L) and ei the classes of Ei.By abuse of notation, identify divisors on X6 with their image in X. Recall thata hyperplane section h of X is in the linear equivalence class 3l−

∑6i=1 ei. The

intersection number of C and h is then

C · h = (l − e2 − e3 − e6) ·(

3l −6∑

i=1ei

)= 3l2 + e2

2 + e23 + e2

6

= 3− 1− 1− 1= 0.

Since the image of C in X does not meet a general hyperplane section, it mustbe a point. In other words, it is an E6 singularity. ♥

92

Bibliography

[Art66] Michael Artin. ‘On isolated rational singularities of surfaces’. In:Amer. J. Math. 88 (1966), pp. 129–136. issn: 0002-9327.

[Bar+04] Wolf P. Barth, Klaus Hulek, Chris A. M. Peters and Antonius Vande Ven. Compact complex surfaces. Second. Vol. 4. Ergebnisse derMathematik und ihrer Grenzgebiete. 3. Folge. A Series of ModernSurveys in Mathematics [Results in Mathematics and Related Areas.3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2004, pp. xii+436. isbn: 3-540-00832-2. doi: 10 .1007/978-3-642-57739-0. url: http://dx.doi.org/10.1007/978-3-642-57739-0.

[Ble+12] Grigoriy Blekherman, Jonathan Hauenstein, John Christian Ottem,Kristian Ranestad and Bernd Sturmfels. ‘Algebraic boundaries ofHilbert’s SOS cones’. In: Compos. Math. 148.6 (2012), pp. 1717–1735. issn: 0010-437X. doi: 10.1112/S0010437X12000437. url:http://dx.doi.org/10.1112/S0010437X12000437.

[Cas41] C. Ronald Cassity. ‘On the Quartic Del Pezzo Surface’. In: AmericanJournal of Mathematics 63.2 (1941), pp. 256–262. issn: 00029327,10806377. url: http://www.jstor.org/stable/2371521.

[Cay69] Arthur Cayley. ‘A Memoir on Quartic Surfaces’. In: Proc. LondonMath. Soc. S1-3.1 (1869), p. 19. issn: 0024-6115. doi: 10.1112/plms/s1-3.1.19. url: http://dx.doi.org/10.1112/plms/s1-3.1.19.

[Cob29] Arthur B. Coble. Algebraic geometry and theta functions. Vol. 10.American Mathematical Society Colloquium Publications. Reprintof the 1929 edition. American Mathematical Society, Providence,R.I., 1982, pp. vii+282. isbn: 0-8218-1010-3.

[Cos83] François R. Cossec. ‘Reye congruences’. In: Trans. Amer. Math. Soc.280.2 (1983), pp. 737–751. issn: 0002-9947. doi: 10.2307/1999644.url: http://dx.doi.org/10.2307/1999644.

[CW05] Fabrizio Catanese and Bronislaw Wajnryb. ‘The 3-cuspidal quarticand braid monodromy of degree 4 coverings’. In: Projective varietieswith unexpected properties. Walter de Gruyter GmbH & Co. KG,Berlin, 2005, pp. 113–129.

http://dx.doi.org/10.1007/978-3-642-57739-0

http://dx.doi.org/10.1007/978-3-642-57739-0

http://dx.doi.org/10.1007/978-3-642-57739-0

http://dx.doi.org/10.1007/978-3-642-57739-0

http://dx.doi.org/10.1112/S0010437X12000437

http://dx.doi.org/10.1112/S0010437X12000437

http://www.jstor.org/stable/2371521

http://dx.doi.org/10.1112/plms/s1-3.1.19



http://dx.doi.org/10.2307/1999644

http://dx.doi.org/10.2307/1999644

Bibliography

[Dol12] Igor V. Dolgachev. Classical Algebraic Geometry. A modern view.Cambridge University Press, Cambridge, 2012, pp. xii+639. isbn:978-1-107-01765-8. doi: 10.1017/CBO9781139084437. url: http://dx.doi.org/10.1017/CBO9781139084437.

[Edge38] William L. Edge. ‘Determinantal representations of x4 + y4 + z4’.In: Proceedings of the Cambridge Philosophical Society 34 (1938),p. 6. doi: 10.1017/S0305004100019848.

[Eis+02] David Eisenbud, Daniel R. Grayson, Michael Stillman and BerndSturmfels, eds. Computations in Algebraic Geometry with Macaulay2. London, UK, UK: Springer-Verlag, 2002. isbn: 3-540-42230-7.

[Eis95] David Eisenbud. Commutative algebra. Vol. 150. Graduate Textsin Mathematics. With a view toward algebraic geometry. Springer-Verlag, New York, 1995, pp. xvi+785. isbn: 0-387-94268-8; 0-387-94269-6. doi: 10.1007/978-1-4612-5350-1. url: http://dx.doi.org/10.1007/978-1-4612-5350-1.

[End10] Stephan Endraß. surf 1.0.6. 2010. url: http://surf.sourceforge.net.[FH79] William Fulton and Johan Hansen. ‘A connectedness theorem for

projective varieties, with applications to intersections and singular-ities of mappings’. In: Ann. of Math. (2) 110.1 (1979), pp. 159–166.issn: 0003-486X. doi: 10.2307/1971249. url: http://dx.doi.org/10.2307/1971249.

[Har77] Robin Hartshorne. Algebraic geometry. Graduate Texts in Mathem-atics, No. 52. Springer-Verlag, New York-Heidelberg, 1977. isbn:0-387-90244-9.

[Har92] Joe Harris. Algebraic geometry. Vol. 133. Graduate Texts in Mathem-atics. A first course. Springer-Verlag, New York, 1992, pp. xx+328.isbn: 0-387-97716-3. doi: 10.1007/978- 1- 4757- 2189- 8. url:http://dx.doi.org/10.1007/978-1-4757-2189-8.

[Jes16] Charles Minshall Jessop. Quartic surfaces with singular points. Uni-versity Press, 1916.

[M2] Daniel R. Grayson and Michael E. Stillman. Macaulay2, a softwaresystem for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/.

[Ott+14] John Christian Ottem, Kristian Ranestad, Bernd Sturmfels andCynthia Vinzant. ‘Quartic spectrahedra’. English. In: MathematicalProgramming (2014), pp. 1–28. issn: 0025-5610. doi: 10.1007/s10107-014-0844-3. url: http://dx.doi.org/10.1007/s10107-014-0844-3.

[PSV12] Daniel Plaumann, Bernd Sturmfels and Cynthia Vinzant. ‘Com-puting linear matrix representations of Helton-Vinnikov curves’.In: Mathematical methods in systems, optimization, and control.Vol. 222. Oper. Theory Adv. Appl. Birkhäuser/Springer Basel AG,Basel, 2012, pp. 259–277. doi: 10.1007/978-3-0348-0411-0_19.url: http://dx.doi.org/10.1007/978-3-0348-0411-0_19.

94

http://dx.doi.org/10.1017/CBO9781139084437



http://dx.doi.org/10.1017/S0305004100019848

http://dx.doi.org/10.1007/978-1-4612-5350-1

http://dx.doi.org/10.1007/978-1-4612-5350-1

http://dx.doi.org/10.1007/978-1-4612-5350-1

http://surf.sourceforge.net

http://dx.doi.org/10.2307/1971249

http://dx.doi.org/10.2307/1971249

http://dx.doi.org/10.2307/1971249

http://dx.doi.org/10.1007/978-1-4757-2189-8

http://dx.doi.org/10.1007/978-1-4757-2189-8

http://www.math.uiuc.edu/Macaulay2/

http://www.math.uiuc.edu/Macaulay2/

http://dx.doi.org/10.1007/s10107-014-0844-3

http://dx.doi.org/10.1007/s10107-014-0844-3

http://dx.doi.org/10.1007/s10107-014-0844-3

http://dx.doi.org/10.1007/s10107-014-0844-3

http://dx.doi.org/10.1007/978-3-0348-0411-0_19

http://dx.doi.org/10.1007/978-3-0348-0411-0_19

Bibliography

[Seg84] Corrado Segre. ‘Etude des différentes surfaces du 4e ordre à coniquedouble ou cuspidale (générale ou décomposée) considérées commedes projections de l’intersection de deux variétés quadratiques del’espace à quatre dimensions’. In: Mathematische Annalen 24.3(1884), pp. 313–444. issn: 1432-1807. doi: 10.1007/BF01443412.url: http://dx.doi.org/10.1007/BF01443412.

[Tev03] Evgueni A. Tevelev. ‘Projectively dual varieties’. In: J. Math. Sci.(N. Y.) 117.6 (2003). Algebraic geometry, pp. 4585–4732. issn:1072-3374. doi: 10.1023/A:1025366207448. url: http://dx.doi.org/10.1023/A:1025366207448.

[Val34] Patrick du Val. ‘On isolated singularities of surfaces which do notaffect the conditions of adjunction (Part I.)’ In: Mathematical Pro-ceedings of the Cambridge Philosophical Society 30 (04 Oct. 1934),pp. 453–459. issn: 1469-8064. doi: 10.1017/S030500410001269X.url: http://journals.cambridge.org/article_S030500410001269X.

95

http://dx.doi.org/10.1007/BF01443412

http://dx.doi.org/10.1007/BF01443412

http://dx.doi.org/10.1023/A:1025366207448

http://dx.doi.org/10.1023/A:1025366207448

http://dx.doi.org/10.1023/A:1025366207448

http://dx.doi.org/10.1017/S030500410001269X

http://journals.cambridge.org/article_S030500410001269X

Glossary of Symbols

Rp Ramification locus for projection from the node p, page 3

F11 Cubic in Rp, page 4

F22 Cubic in Rp, page 4

∆ Ternary quartic such that f = −qx20 + 2gx0 + ∆, page 4

π∗ Pullback, page 94

l Linear equivalence class of π∗ of a line in P2, page 27

Ei Exceptional divisor over point Pi, page 27

ei Linear equivalence class of exceptional divisor Ei, page 27

Pi Pj Pi is a point on the exceptional divisor over Pj , page 27

D ·D′ Intersection number of divisors D and D′, page 28

ρa(D) Arithmetic genus of divisor D, page 54

[f ] Point corresponding to the quadric f , page 18

d Linear system, page 19

Bl(d) Base locus of d, page 19

G(k, n) Grassmannian of k-dimensional, linear subspaces of Pn, page 11

νd Veronese map of degree d, page 9

ΓΦ Span of subsetes Γ,Φ ⊆ Pn, page 11

(Pn)∨ Dual projective space, page 14

X∗ Dual variety of X, page 14

X⊥ Orthogonal subspace to X, page 15

deg(X) Degree of the variety X, page 10

Sec(X) Secant variety of the variety X, page 12

Tan(X) Tangential variety of the variety X, page 18

Fn The n-th Hirzebruch surface, page 25

Index

ADEclassification, 89curve, 89

Bertini’s theorem, 10Bézout’s theorem, 10Blow-up, 2, 3, 27, 33, 36, 76, 81, 82,

91

Contact curve, 5Contract a curve, 28, 33, 36, 90

Deficient, see secantDel Pezzo surface, 25–44, 76, 81,

82Discriminant hypersurface, 18Du Val singularities, 91Dual

graph, 89projective space, 14variety, 14

Dynkin diagram, 89

Exceptional curve, 90

Fundamental cycle, 90

Generically transversal intersection,10

Grassmannian, 11

Hirzebruch surfaces, 25Hirzebruch-Jung strings, 90

Kleinian singularities, 91

Multiplicity, 90

Nodal surface, 2

Nondegenerate variety, 25

Orthogonal subspace, 15

Pillow, 20, 25, 73Plücker’s surface, 54

Ramification locus, 3Rank k locus, 2Rank-k-point, 2Rational, see also symmetroid

normal curve, 9singularity, 90

Secantdeficient variety, 12, 18line, 12line map, 11variety, 12

Segresurface, 25symbol, 36, 43

Simple surface singularities, 91Span, 11Spectrahedron, 1Steiner surface, 11, 62Symmetroid, 1

quartic, 1rational, 2, 26, 53transversal, 2with a double conic

1 node, 422 nodes, 40, 423 nodes, 394 nodes, 31, 32, 39

with a double line and conic,62

99

Index

with one double line0 nodes, 631 node, 572 nodes, 523 nodes, 514 nodes, 505 nodes, 49, 556 nodes, 46, 49, 547 nodes, 478 nodes, 47

with three double lines, 62with two double lines0 nodes, 611 node, 40

2 nodes, 36, 40, 423 nodes, 344 nodes, 28, 30, 41

Tangent hyperplane, 14Tangential variety, 18Toeplitz, 21, 75Transversal intersection, 10

Variety of singular loci, 19–22, 30,32, 35, 37, 73, 80

Veronesemap, 9, 17surface, 9variety, 9, 17

100

Documents

Rational Quartic Symmetroids · 2017. 12. 7. · 1.1. Basic Properties isomorphic,sotheseareallsymmetroids. ♥ Genus 3 Sextics The(3×3)-minorsofa(3×4)-matrixoflinearformsinC[x0,x