Direct computation of the degree 4 Gopakumar–Vafa invariant on a Calabi–Yau 3-fold

Journal of Geometry and Physics 62 (2012) 935–952

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Journal of Geometry and Physics

journal homepage: www.elsevier.com/locate/jgp

Direct computation of the degree 4 Gopakumar–Vafa invariant on aCalabi–Yau 3-foldMehmet SahinUniversity of Illinois, Department of Mathematics, United States

a r t i c l e i n f o

Article history:Received 14 February 2011Received in revised form 14 November2011Accepted 6 January 2012Available online 30 January 2012

Keywords:Moduli of sheavesGopakumar–Vafa invariantGromow–Witten invariantHilbert schemeCalabi–YauDonaldson Thomas invariant

a b s t r a c t

In this work we compute the topological Euler characteristic of the moduli space of stablesheaves of Hilbert polynomial 4n + 1 on P2 to be 192, using tools of algebraic geometry.This Euler characteristic is equal up to sign to the degree 4 BPS (Gopakumar–Vafa) invariantof local P2, a (noncompact) Calabi–Yau 3-fold. This is a new result verifying an instance ofconjecture motivated by physics.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

1.1. Background

A generic Calabi–Yau 3-fold X is expected to have finitelymany curves of any given genus g , representing a given integralhomology class β ∈ H2(X). There were numbers predicted by physics (see [1]), called instanton numbers on a specificCalabi Yau 3-fold (the blowup of a weighted hypersurface in the weighted projective space P(1, 1, 1, 6, 9)) which werebelieved to be equal to the expected number of rational curves of a given degree. Later a rigorous mathematical theory ofGromov–Witten theory was built based on Kontsevich’s notion of stable maps allowing people to look at such objects in amathematically rigorous way. The rigorous definition of Gromov–Witten theory required a homology class or an elementin the Chow group of the moduli of stable maps, called the virtual fundamental class, constructed by Li and Tian in theanalytic category and Behrend and Fantechi in the algebraic category. Gromov–Witten invariants are integrals of pullbacksof cohomology classes on a variety by evaluationmaps (which send a point representing a stablemap to images of itsmarkedpoints), over the virtual fundamental class. Roughly speaking this should correspond to the class in themoduli space of stablemaps, corresponding to the set of stable maps whose images of marked points lie on some dual cycles of given cohomologyclasses (because of excess intersection, this is not quite true, and this is why one needs a virtual fundamental class instead ofjust the fundamental class). Let’s denote the moduli space of stable maps whose domains are genus g curves with nmarkedpoints representing a class β ∈ H2(X) by Mg,0(X, β) and the corresponding virtual class by [Mg,0(X, β)]vir whose expecteddimension is c1(X) · β + (dim X − 3) · (1 − g) + n. If n = 0 and X is a Calabi–Yau 3-fold, we get 0 as the virtual dimension.The degree of this virtual fundamental class is a Gromov–Witten invariant which we call Ng,0(X, β). These are not numbers

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936 M. Sahin / Journal of Geometry and Physics 62 (2012) 935–952

of genus g curves in class β in X , because in a stable map, the genus of the domain and the image can be different, stablemaps can have automorphisms, etc.

We now consider the (noncompact) Calabi–Yau 3-fold X containing a P2, namely the total space of the canonical bundleof P2, which is called local P2, and a class β = d · [line] ∈ H2(P2) (here P2 is embedded as the 0-section). It is known thatthat Mg,0(P2, β) is a component of Mg,0(X, β). Let’s take g = 0 and denote our Gromov–Witten invariant by Nd. There is arecursive formula for computing Nd. The Nd are rational numbers. Another sequence of rational numbers nd is recursivelygiven by:

Nd =

k|d

nd/k/k3. (1)

According to the integrality conjecture nd’s defined this way must be integers. This doesn’t follow from their definitions.See [2] for a different formulation of the same claim. We call these BPS invariants or Gopakumar–Vafa invariants.

There are other approaches to counting curves. Curves in X look like their structure sheaves, as well as their inclusionmaps into X . The numbers nd defined above are conjecturally up to sign equal to the topological Euler characteristics ofM(dn + k), the moduli space of semistable sheaves of a fixed linear Hilbert polynomial of the form dn + k on P2, when thismoduli space is smooth and all semistable sheaves are stable. See [2]. Here we should mention a result by Sheldon Katzstating that any stable sheaf of Hilbert polynomial 4n + 1 on local P2 is scheme theoretically supported on the P2 (there isa natural O(1) on the local P2 restricting to OP2(1)).

In this work we study M(4n + 1) by describing individual sheaves and putting them into various families to describesome locally closed subsets of M(4n + 1), which enables us to compute its Euler characteristic; verifying a conjecture for aspecific case of interest.

Moduli spaces are important spaces which parametrize ‘‘objects’’ satisfying some conditions. Some examples of modulispaces that appear often in algebraic geometry are Hilbert/Quot Schemes, Picard schemes/Jacobians, moduli of curves/stablemaps, moduli of semistable sheaves.

To get a nice moduli space of sheaves on a space, one needs to fix some numerical data (such as Hilbert Polynomial,or Chern classes and rank), and after this still some ‘‘bad’’ or degenerate cases need to be eliminated and this is done byenforcing some semistability condition. A general construction of moduli of sheaves was done by Simpson (see [3]). Thegeneral construction is done by taking a good part of a Quot scheme and dividing out by a reductive group by GIT (geometricinvariant theory).

The spaceM(dn + k) is projective, smooth if (d, k) = 1, of dimension d2 + 1 (see [4]). (How do we get d2 + 1? Here is aheuristic argument. It is equal to

d+22

− 1

+

d−12

. The first term is the dimension of P(d), the space of hypersurfaces

of degree d, and most of these are smooth curves of genus

d−12

, which is the second term, the dimension of Jacobians,

spaces of fixed degree line bundles on such curves.)It is not known whether M(dn + k) ≃ M(dn − k), although there is a bijection between these, matching points

corresponding to dual sheaves, but this has been proven for d = 4 in [5].The space M(4n + 1) is a fine moduli space. This means any family is a pullback of a universal family, up to a twist by

pullback of a line bundle on the parametrizing space (see [4, Theorem 3.19]).

1.2. Strategy of the computation

In this work we study the moduli space of stable sheaves on P2 of Hilbert polynomial 4n + 1, denoted by M(4n + 1) orsimply byM; and compute its Euler characteristic. The main result (Theorem 4.23) says that χ(M) = 192. We compute thisnumber by computing Euler characteristics of three disjoint locally closed subsets of M (Propositions 4.11, 4.16 and 4.22)and adding them up (by quoted result: Proposition 4.2).

In Section 2 we go through the basics of 1 dimensional sheaves on P2. Hypersurfaces and finite schemes play an essentialrole in our treatment of the subject. We define pureness (torsion-freeness) for coherent sheaves in the usual way.We definestability and semistability using reduced Hilbert polynomials. We define intrinsic duals, and prove that taking duals is aninvolution on the set of pure one dimensional sheaves taking stable (resp. semistable) sheaves to stable (resp. semistable)sheaves. We create two pure sheaves from a hypersurface and a finite subscheme of it and call these removing and addingpoints.

In Section 3 we build some universal families of sheaves. We define the space of hypersurfaces of degree d, and denote itby P(d), together with the universal family over it, which we call the universal hypersurface and denote it by UHd. The spaceP(d) is defined as a Hilbert scheme andUHd as the universal subscheme of P2

×P(d) over it. Wemake an alternative definitionfor P(d) which allows us to view it as a projective space in a concrete way. The universal hypersurface sheaf is defined asOUHd .

Then we define the universal family of hypersurface unions. We build a scheme (resp. a sheaf) by putting together allunions of pairs of hypersurfaces of a fixed bidegree.

Then at the end of Section 3, we create universal families of hypersurfaces with removed points. Flattening stratificationis the key to this construction, and many others in this work. (We need a similar construction in the next section, but since

M. Sahin / Journal of Geometry and Physics 62 (2012) 935–952 937

we don’t have a good notation for it, we don’t devote a section to that construction. Many similar constructions can be done,but we don’t make definitions just for the sake of making definitions.)

Section 4 is the main section. First we quote some results about Euler characteristics of bundles and varieties. Then weanalyze stable sheaves of Hilbert polynomial 4n + 1 (members of M) based on global sections they have. We prove thatthe image of a section generates a hypersurface sheaf of degree 3 or 4, in which case we say there is an H3 section, or an H4section, respectively. Using Serre duality (among other things) we prove that that amember ofM can have atmost 2 linearlyindependent sections, and explicitly write down all possibilities. We analyze how having 2 independent sections relates tohaving an H3 or H4 section.

For each member ofM , there is either a unique section (up to scalar) which is an H4 section or an H3 section, or there aretwo independent sections. We denote the corresponding subsets ofM by A, B, and C respectively.

We put all members of C into a family over the universal quartic, (more precisely P(4,1), rather than UH4), which inducesan injectivemorphism intoM whose image is exactly C . This shows that C is a closed subset ofM , and its Euler characteristicis 42 (Proposition 4.11).

Next using semicontinuity theorems (and more) we show that A is open, and B is closed in A ∪ B (Section 4.4). We showthat global sections come in families, and these families induce morphisms from A and B to the space of quartics and to thespace of cubics respectively. We also look at cokernels to extract more information about A and B.

Then we compute the Euler characteristic of A as 120 (Proposition 4.16), by proving that it is a topological P11 bundleover an open subset of Hilbert scheme of 3 points on P2, whose Euler characteristic is 10. We use smoothness of M and theHilbert scheme of 3 points on P2 in proving the existence of local trivializations.

Toward the end we describe B as the complement of an ‘‘almost’’ P1 bundle inside an ‘‘almost’’ P2 bundle over a productof projective spaces. We don’t prove existence of local trivializations (to say that we can remove ‘‘almost’’), but we proveenough to be able to use deep theorems (Propositions 4.2 and 4.3) related to the topology of varieties.

At the end we add up the Euler characteristics of A, B, C and get 192 as the Euler characteristic ofM , as predicted.

1.3. Notations and conventions

Everything is over C, the field of complex numbers. We use Zariski and Euclidean topologies.We are considering sheaves or families of sheaves on P2. A (flat) family of coherent sheaves on P2 is a sheaf on P2

× X(flat over X). We always put this P2 on the left in a such product. Also we try to avoid using P2 for any other space, unless weare saying something is isomorphic to P2. For example we denote the space of lines on P2 by P(1) although it is isomorphicto P2.

When we talk about maps between sheaves, we always mean homomorphisms (respecting O module structures).Hom(·, ·) denotes global Hom, while hom(·, ·) denotes local hom of sheaves.

If we talk about a variety/scheme as a set, we mean the set of its complex (closed) points. If we prove that a subset of aparameter space is closed, we also consider that subset as a schemewith the reduced induced scheme structure. This usuallyhappens when we use flattening stratification.

Some notations expire quickly, some don’t. Here are some that are permanent.

P2: Projective plane where all individual sheaves live.Hd: A hypersurface of degree d on P2.Hd-section: A section of a sheaf, whose image generates a sheaf isomorphic to OHd . (Here Hd may represent a specifichypersurface, or an arbitrary hypersurface. It will be clear what is meant in each case.)Qm: A 0-dimensional subsceheme of P2 of lengthm. This means that the structure sheaf of Qm has Hilbert polynomialm.Hd(−Qm) (resp.Hd(+Qm)): The sheaf obtained by ‘‘removing from’’ (resp. ‘‘adding to’’) a hypersurface sheafOHd

of degreed, a zero dimensional scheme Qm of lengthm.P(d): Space of hypersurfaces of degree d (a Hilbert scheme, and a projective space).UHd: Universal hypersurface.OUHd : Universal hypersurface sheaf.Hilb(m) = HilbP2(m): Hilbert scheme ofm points on P2.HilbP1(m): Hilbert scheme ofm points on P1.P(d,m) Space parametrizing pairs (Hd,Qm), where Qm is a subscheme of Hd.UHd(−Qm): Sheaf on P2

× P(d,m) which restricts to Hd(−Qm) over points of P(d,m) (read this as ‘‘Universal Hd(−Qm)’’).PF (n): Hilbert polynomial of F (see Section 2.3).pF (n): Reduced Hilbert polynomial of F (see Section 2.3).Fpure: Purification of F (see Section 2.3).M(dn + k) Moduli space of semistable sheaves on P2, with Hilbert polynomial dn + k (here n is the variable).M := M(4n + 1).A: The subset ofM corresponding to sheaves with an H4 (degree 4) section, and no other sections, (up to scalar).B: The subset ofM corresponding to sheaves with an H3 (degree 3) section, and no other sections, (up to scalar).C: The subset ofM corresponding to sheaves whose space of sections is 2-dimensional.


C+: Subset of C corresponding to sheaves with at least one H3 section (there can be up to 4 pairwise independent H3sections).T := (the set of triples)M×P(3)×P(1). T also appears (togetherwith S) as a variable in proofs using flattening stratification,but there can’t be any confusion between these two different uses of the same symbol (Section 4.6.3).E: The set of exact triples in T (Section 4.6.3).U, V : Some subsets of T (Section 4.6.3).V ′: A space with a surjective birational map to C+ and a bijective map to V (Section 4.6.3).

2. Some basic facts about 1-dimensional sheaves on P2

2.1. Hd: hypersurfaces on P2

A hypersurface on P2 is a subscheme of P2 defined as a zero scheme of a homogenous polynomial in 3 homogenouscoordinates of P2. A hypersurface sheaf is structure sheaf of a hypersurface as a coherent OP2 module. The degree of ahypersurface or hypersurface sheaf is the degree of defining polynomial.

We usually denote a hypersurface of degree d by Hd. Any hypersurface sheaf OHd of degree d fits into an exact sequence:

0 −→ OP2(−d) −→ OP2 −→ OHd −→ 0. (2)

Conversely any sheaf fitting into a such sequence (as 4th term) is a hypersurface sheaf of degree d. The inclusion maphere is multiplication by the defining polynomial of the hypersurface.

Let’s write P2 as Proj(C[x, y, z]). Let f and g be homogenous polynomials of degree d and e respectively in x, y, z; andHd,He,Hd+e be hypersurfaces defined as zero schemes of f , g and f · g respectively. We call Hd+e the union of Hd and He;Hd and He subhypersurfaces of Hd+e; and Hd and He complementary hypersurfaces in Hd+e. There is a canonical surjectivequotient map OHd+e −→ OHd and its kernel can be computed using fundamental theorems of module theory, and we getthe exact sequence:

0 −→ OHe(−d) −→ OHd+e −→ OHd −→ 0. (3)

2.2. Pure sheaves

Definition 2.1. A coherent sheaf F on scheme X is called pure of dimension d if any coherent subsheaf of F has d dimensionalsupport.

If F is a d dimensional coherent sheaf (on a Noetherian scheme), there is a maximal coherent subsheaf, say G, of F whichis at most d− 1-dimensional. F/G is pure of dimension d. We call it the purification of F and denote it by Fpure. If the schemewhere F is defined has a given very ample line bundle (for example if it is a projective space), highest order terms in Hilbertpolynomials of F and Fpure are the same. If F is a structure sheaf of a scheme, so is Fpure.

Proposition 2.1. Let F be a structure sheaf of a closed subscheme of P2 of pure dimension 1. Then F is a hypersurface sheaf.

Proof. This is a standard result (also true for any smooth projective surface). �

2.3. (Semi)stability

Definition 2.2. Let F be a sheaf of pure dimension d on a projective schemewith a very ample line bundle O(1). The integervalued function PF (n) := χ(F(n)) =

(−1)idim(H i(F(n))) is a polynomial in n, and is called the Hilbert polynomial of F .

The reduced Hilbert polynomial pF (n) of F is the scalar multiple of the Hilbert polynomial which is monic. The sheaf F issemistable (resp. stable) if for any coherent subsheaf G of F , pG(n) ≤ pF (n) (resp. pG(n) < pF (n)).

Remark 2.1. (Semi)stability can equivalently be defined by replacing subsheaves with d-dimensional quotient sheaves andreversing inequalities in the definition above (easy exercise).

Proposition 2.2. Let F be a hypersurface sheaf of degree d on P2. Then F is stable.

Proof. F is a cokernel of an inclusion OP2(−d) ↩→ OP2 . From this we find a Hilbert polynomial of F as

n+22

−

n+2−d

2

=

nd − d(d − 3)/2 and pF (n) = n − (d − 3)/2. Let G be a 1-dimensional (coherent) quotient sheaf of F . Let G0 be the torsionsubsheaf of G. Let H := G/G0 (=Gpure). H is pure of dimension-1 and a quotient of G, so it is also a quotient of F , whichimplies that it is a structure sheaf of a subcheme of P2 (since F is). By the Proposition 2.1 H is a hypersurface sheaf of degreed′ for some d′ strictly smaller than d. By calculation above pH(n) = n − (d′

− 3)/2 > n − (d − 3)/2 = pF (n). Hilbertpolynomials of G and H have the same main term (namely d′

· n) and so pG(n) ≥ pH(n). Combining these inequalities wefind pF (n) < pG(n). By the Remark 2.1, F is stable. �


Remark 2.2. Above, we actually provedmore than the stability of hypersurface sheaves on P2. We found upper (resp. lower)bounds for Hilbert polynomials of subsheaves (resp. quotient sheaves) of such sheaves, which are better than the onesprovided by the definition of stability.

2.4. FD: duals

Definition 2.3. Let F be a coherent sheaf of codimension c on a smooth projective variety X . The dual of F is defined asFD

= ExtcX (F , ωX ).

This definition, when compared to hom definition of dual, has the significant advantage of being intrinsic: It doesn’tdepend on the ambient space where F lives. See [6].

Proposition 2.3. Let G be a 0-dimensional coherent sheaf on P2. Then Hilbert polynomials of G and GD are the same.

Proof. Induction on the Hilbert polynomial = length of G = k. True for k = 0, because then G = GD= 0. Also true for

k = 1, because the unique nonzero (up to scalar) section has to generate a skyscraper sheaf at a closed point, and dual ofthat sheaf is isomorphic to itself (because its dual can be determined inside its support as hom dual of trivial sheaf). Nowlet k > 1. Then there is a nozero section s of G. If O · s = G, then G is an extension of a proper subsheaf O · s, and thecokernel of the inclusion O · s → G. If O · s = G then it is structure sheaf of a scheme and maps surjectively onto some Op, astructure sheaf of a closed point p on its support, and G is an extension of Op and a kernel of G → Op. In either case we getan exact sequence: 0 → G1 → G → G2 → 0 where lengths of G1 and G2 are smaller than that of G. Taking duals (applyinghom(·, ω ≃ O(−3))) we get the exact sequence: 0 → GD

2 → GD→ GD

1 → 0. Using these two exact sequences, additivityof Hilbert polynomials in exact sequences, and the induction hypothesis we get the result for k. �

Proposition 2.4. For any coherent sheaf F , there is a natural map F −→ FDD. If F is pure of dimension 1 or 0-dimensional sheafon P2 then this map is an isomorphism. If F is pure of dimension 1 on P2, so is FD.

Proof. Follows from [6]. �

By this proposition we see that taking duals is an involution on the set of pure 1-dimensional coherent sheaves on P2.We’ll see that it takes (semi)stable elements to (semi)stable elements.

Proposition 2.5. Let Hd be a hypersurface of degree d on P2. Then (OHd)D

≃ OHd(d − 3).

Proof. Since duals are intrinsic, (OHd)D is the canonical sheaf of Hd, which is isomorphic to OHd(d − 3) by adjunction. �

Proposition 2.6. Let F be a coherent sheaf on P2 which is pure of dimension 1 and with Hilbert polynomial dn + k. Then theHilbert polynomial of FD is dn − k.

Proof. True for hypersurfaces by previous results: POHd(n) = dn − d(d − 3)/2 and P(OHd )D(n) = POHd (d−3)(n) =

d(n + d − 3) − d(d − 3)/2 = dn + d(d − 3)/2. Also true for twists of hypersurfaces: POHd (m)(n) = POHd(n) + dm and

P(OHd (m))D(n) = P(OHd )D(−m)(n) = P(OHd )D(n) − dm. Now assume that the statement above is not true. Pick an F violating this

claimwith the smallest possible d. FD is pure and FD(m) has a global section for somem, which generates a pure sheaf whichis a structure sheaf of a subscheme of P2, and by a proposition above, a hypersurface sheaf H of degree, say, r . We have anexact sequence 0 −→ H(−m) −→ FD

−→ Q −→ 0 for some Q . Apply hom(·, O(−3)) to get the exact sequence

0 −→ Ext1(Q , O(−3)) −→ F −→ H(r + m − 3) −→ Ext2(Q , O(−3)) −→ 0. (4)

(Zeros on both ends come from pureness of H and F , see [6].)If Q is zero dimensional, we have Ext1(Q , O(−3)) = 0 and the sequence (4) above becomes

0 −→ F −→ H(r + m − 3) −→ Q D−→ 0.

By additivity of Hilbert polynomials in short exact sequences we get PFD(n) = PH(−m)(n) + PQ (n) and PF (n) =

PH(r+m−3)(n) − PQD(n) = P(H(−m))D(n) − PQ (n).If Q is 1-dimensional, from (4) we get the exact sequence:

0 −→ Q D−→ F −→ H(r + m − 3) −→ Ext2(Q , O(−3)) −→ 0.

The image of F in this sequence is subsheaf of a pure sheaf. So we get F as an extension of two pure 1-dimensionalsheaves, which both have main coefficient less than d. Let’s write this extension as 0 −→ F1 −→ F −→ F2 −→ 0 and letPF1(n) = d1n + k1, PF2(n) = d2n + k2, by induction hypothesis we get PFD1 (n) = d1n − k1, PFD2 (n) = d2n − k2. Dualizing

the sequence (applying hom(·, ω ≃ O(−3))) we get the exact sequence: 0 −→ FD2 −→ FD

−→ FD1 −→ 0. From these two

sequences,we see PF (n) = PF1(n)+PF2(n) = (d1+d2)n+(k1+k2) and PFD(n) = PFD2 (n)+PFD1 (n) = (d2+d1)n−(k2+k1). �


Proposition 2.7. Let F be a (semi)stable sheaf of pure dimension 1 on P2. Then so is FD.

Proof. Follows from the proposition above (Proposition 2.6) and two equivalent definitions of (semi)stability(2.2 and 2.1). �

2.5. Hd(−Qm) and Hd(+Qm): removing and adding points

Definition 2.4. Let H be a hypersurface of degree d on P2, and Q be a 0-dimensional subscheme of H . We call the kernelof the canonical quotient map OH −→ OQ the sheaf obtained by removing Q from H , and denote it by H(−Q ). We defineH(+Q ) := (H(−Q ))D(3 − d) and call it the sheaf obtained by adding Q to H .

Remark 2.3. Clearly these operations produce pure sheaves.

Lemma 2.8. Let H be a hypersurface of degree d and L a line on P2. Then either OH ⊗ OL is isomorphic to OL or it is a length-dsheaf.

Proof. By tensoring the defining sequence of H by OL, we find that OH ⊗ OL is isomorphic to cokernel of a map OL(−d)−→ OL. We can consider these as sheaves on L ≃ P1. OP1(−d) −→ OP1 is either zero; in which case we get OL as cokernel;or injective, in which case we get a sheaf of length d. �

Proposition 2.9. Let H be a hypersurface of degree d, and L a line which is not a component of H (on P2), and Q be their schemetheoretic intersection. Then H(−Q ) ≃ OH(−1).

Proof. Start with the defining sequence of L, tensor it with OH to get 0 99K OH(−1) −→ OH −→ OQ −→ 0, which is exacton the right. We know that OQ = OH ⊗OL has length (Hilbert polynomial) d. By comparing Hilbert polynomials we see thatthe sequence is exact on the left also, which implies that OH(−1) ≃ H(−Q ). �

3. Some families of 1-dimensional sheaves on P2

3.1. Families of hypersurfaces

3.1.1. UHd over P(d): universal hypersurface

Let Hd be a hypersurface of degree d on P2. The Hilbert polynomial of OHd is

n+22

−

n+2−d

2

= dn − d(d − 3)/2.

Conversely any scheme on P2 whose structure sheaf has this Hilbert polynomial is a hypersurface of degree d. This is becausepurification (Section 2.2) of the structure sheaf of a such scheme is a pure 1-dimensional sheaf which is a structure sheaf of ascheme and somust be a hypersurface sheaf of degree d by Proposition 2.1. Therefore there is aHilbert schemeparametrizingall hypersurfaces of degree d (and nothing else). This space is known to be isomorphic to Pd(d+3)/2 (see [7, page 111]). In therest of this work we will use P(d) to denote the Hilbert scheme of hypersurfaces of degree d in P2. We call the universalsubscheme in OP2×P(d)

, the universal hypersurface of degree d and denote it by UHd.Let’s sketch a direct construction of this family without reference to Hilbert schemes. For d = 2 let’s consider the zero

scheme Zs2 of s2 = ax2 + by2 + cz2 + dxy + exz + fyz on Proj(C[x, y, z]) × Proj(C[a, b, c, d, e, f ]) ≃ P2× P5 together with

the projection onto Proj(C[a, b, c, d, e, f ]) ≃ P5. Fibers of this projection are degree 2 hypersurfaces, and each degree 2hypersurface is isomorphic to exactly one fiber. By the universal property of Hilbert schemes, there is a bijective morphismfrom P5 (in this paragraph) to theHilbert scheme P(2) (≃ P5, from the previous paragraph). It iswell known that such amap isan isomorphism. Certainly the same construction directly extends to all degrees d. If onewants towrite P2 (= Proj(C[x, y, z]),the space where individual sheaves live) in a coordinate free way as P(V ), where V is a 3 -dimensional vector space, onewould write P(d) as P(SymdV ∗). There is a natural bihomogenous polynomial sd of degree (d, 1) on P(V ) × P(SymdV ∗) (asection of a line bundle, not a regular function) analogous to s2 above and the zero set Zsd of sd as a family over P(SymdV ∗)

gives rise to an isomorphism P(SymdV ∗) → P(d).Let IUHd denote the ideal sheaf of UHd. Locally w.r.t. to the parametrizing space P(d), it is isomorphic to the pullback of

OP2(−d) (Exercise 3.12.4 in [8]). We have the exact sequence of OP2×P(d)modules:

0 −→ IUHd −→ OP2×P(d)−→ OUHd −→ 0. (5)

Each sheaf in the sequence above is flat over the parametrizing space P(d). We call OUHd the universal hypersurface sheafof degree d (viewed as a family over P(d)).


3.1.2. UHd+e over P(d) × P(e): universal union of hypersurface pairsLetHd,He be hypersurfaces on P2 of degree d and e respectively defined by homogenous polynomials f and g . We defined

the union of Hd and He to be the hypersurface defined by f · g , which is of degree d + e. We would like to put all these intoa natural flat family over P(d) × P(e).

We have subschemes UHd in P2× P(d) and UHe in P2

× P(e). We pull them back via obvious projections to P2× P(d) × P(e),

and let’s denote these pullback schemes by S and T respectively. (Here S is the fiber product of UHd and P2× P(d) × P(e) over

P2× P(d) w.r.t. inclusion and projection maps. Similarly for T .) Let IS and IT be their ideal sheaves. We have exact sequences:

0 → IS → O → OS → 0 (6)0 → IT → O → OT → 0. (7)

IS ⊗ IT is mapped to O ⊗ O via inclusions above. This map is an inclusion because it is the composition of

IS ⊗ IT → O ⊗ IT → O ⊗ O (8)

which are inclusions because IS and IT are locally free (locally (w.r.t. to the base) they are isomorphic to O(−d) and O(−e)).Via this inclusion IS ⊗ IT becomes a coherent subsheaf of O ⊗ O ≃ O = OP2×P(d)×P(e)

, that is, an ideal sheaf of a subscheme,which we will denote by UHd+e, of P2

× P(d) × P(e). (For example we can talk about UH1+3 and it is not the same as UH4.)We have the exact sequence (defining sequence of UHd+e):

0 → IS ⊗ IT → O ⊗ O ≃ O → OUHd+e → 0. (9)

By these three exact sequences (and by fundamental theorems of module theory) we get the exact sequence:

0 → OS ⊗ IT → OUHd+e → O ⊗ OT → 0. (10)

(This can be viewed as the relative version of (3).) OUHd+e is flat over P(d) × P(e) because it is an extension of sheavesflat over P(d) × P(e). Let (x, y) ∈ P(d) × P(e), and Hd,He,H be restrictions of S, T and UHd+e to the fiber of the projectionP2

× P(d) × P(e) → P(d) × P(e) over (x, y). The exact sequence above restricts to an exact sequence

0 → OHd(−e) → OH → OHe → 0. (11)

OH is structure sheaf of a scheme in P2, and it is pure of dimension 1 because it is an extension of sheaves pure ofdimension 1. So H is a hypersurface by Proposition 2.1. By additivity of Hilbert polynomials in exact sequences we seethat degree of H is d + e. We claim that H is actually the union of Hd and He. From the sequence above we see that He isa subhypersurface of H (meaning that defining polynomial of He divides that of H). We know that the structure sheaf ofthe remaining ‘‘component’’ (complementary subhypersurface) is O(e), a twist of the kernel of the canonical quotient mapOH → OHe . But the canonical quotient differs by a scalar factor from any other nonzeromapOH → OHe , and kernels of suchmaps are the same. This proves that H is the union of Hd and He.

3.2. UHd(−Qm) family version of removing points

Removing points was a simple operation of taking a kernel. We now create a (flat) family parametrizing all sheavesobtained by removing fixed length schemes from fixed degree hypersurfaces.

Let UHd be the universal hypersurface as before, which we consider as a subscheme of P2× P(d). Let Hilb(m) be the

Hilbert scheme of m points on P2, parametrizing length m subschemes of P2, and let’s denote the corresponding universalsubscheme of P2

× Hilb(m) by UHilb(m).Let’s denote the pull backs of UHd and UHilb(m) to P2

× P(d) × Hilb(m) (in the obvious way via projection maps) by Sand T respectively. Clearly S and T are flat and projective over the parameter space P(d) ×Hilb(m) which parametrizes pairs(Hd,Qm).

We’d like to restrict our attention (and families) to (the space of) pairs (Hd,Qm) where Qm is a subscheme of Hd, whichwe denote by P(d,m). This is the case if and only if OHd ⊗ OQm has the maximum possible length, that is m. Sheaves of theform OHd ⊗ OQm are restrictions of OS ⊗ OT to the fibers of the projection P2

× P(d) × Hilb(m) → P(d) × Hilb(m) (copies ofP2). By flattening stratification (see [7, 5.4.2 on page 123]), P(d,m) is closed.

Let UHd(−Qm) be the kernel of OS |P2×P(d,m)→ (OS ⊗ OT )|P2×P(d,m)

, which is a sheaf on P2× P(d,m), viewed as a family

of sheaves over P(d,m). Since UHd(−Qm) is the kernel of a surjective homomorphism of sheaves that are flat over P(d,m), thesheaf UHd(−Qm) is flat over P(d,m). By construction this family parametrizes all sheaves of the form Hd(−Qm), where Hd isa degree d hypersurface on P2 and Qm is a subscheme of Hd of length m. (It is possible for this family to have isomorphicmembers, and this would be the case whenm is big enough compared to d.)


4. On the geometry and topology ofM = M(4n + 1)

In this chapter we study the geometry of M , the moduli space of stable sheaves of Hilbert polynomial 4n + 1 on P2, toprove themain Theorem 4.23. First we quote a few general results about Euler characteristics of bundles and varieties. Thenwe classify ‘‘points’’ of M using global sections of individual sheaves. Then we divide M mainly into three locally closedsubsets based on this classification, and compute the Euler characteristic of each piece, and finally prove the main theoremby adding these numbers.

4.1. A few general results about Euler characteristics of bundles and varieties

Proposition 4.1. The (topological) Euler characteristic of the total space of a fiber bundle is the product of those of the base andthe fiber.

Proof. This is a well known result in algebraic topology. �

Proposition 4.2. Let X be a complex algebraic variety, Z a Zariski closed subset.Then χ(X) = χ(Z) + χ(X \ Z).

Proof. See [9], it is sketched on page 2. �

Proposition 4.3. Let f : X → Y be a surjective projective morphism of projective varieties such that topological Eulercharacteristic of fibers is constant. Then χ(X) = χ(F) · χ(Y ), where F is a fiber.

Proof. A trivial consequence of the result above and the first paragraph of [10]. �

4.2. About (closed) points of M

In this sectionwe prove results about coherent sheaves of Hilbert polynomial 4n+1 on P2, based on their global sections.

Theorem 4.4. Let F be a stable sheaf of Hilbert polynomial4n+1 on P2. Let s be a (nonzero) global section of F . ThenOP2 ·s := OHdis a hypersurface sheaf of degree d = 3 or 4.

If d = 3 we have an exact sequence:

0 −→ OH3 −→ F −→ OH1 −→ 0 (12)

for some line H1. A coherent sheaf fitting into such an exact sequence as the middle term is stable if and only if this sequence is notsplit.

If d = 4 we have

F ≃ H4(+Q3) (13)

for some length 3 subscheme Q3 of H4. A sheaf of the form H4(+Q3) is stable if and only Q3 is not a subscheme of a line that is acomponent (subhypersurface) of H4.

Proof. Pureness of F forces that O · s is isomorphic to a hypersurface sheaf. The condition of stability for F forces that thedegree of this hypersurface is d = 3 or d = 4.

d = 3 case:Let K be the cokernel of OH3 ↩→ F . K has Hilbert polynomial n+ 1. K is pure because Kpure is a quotient of F , and stability

of F forces that PKpure(n) can’t be less than n + 1. K has a section, which has to generate a hypersurface sheaf (by 2.1) withHilbert polynomial n + 1 (at most n + 1 because it is a subsheaf of K , and at least n + 1 because it is a hypersurface sheaf),and this implies that K is a line sheaf.

Now we prove that F fitting into a such sequence (12) is stable if and only if it is not split. One direction is trivial, if it issplit OH1 becomes a subsheaf of F with a higher reduced Hilbert polynomial than that of F .

Now let’s prove the if part. Let G be a proper subsheaf of F violating stability of F . There is an exact sequence

0 −→ G ∩ OH3 −→ G ⊕ OH3 −→ G + OH3 −→ 0. (14)

Alternating sum of Hilbert polynomials is 0. The second (out of five) terms has hilb = 0, n − 1 − a, 2n − 1 − a or 3n − afor some nonnegative a, because it is a subsheaf of hypersurface sheaf OH3 .

The third term has a Hilbert polynomial with positive constant term.The fourth term has Hilbert polynomial 3n+ 1+ b or 4n+ 1− b for some nonnegative b, because of obvious inclusions.

If G + OH3 has Hilbert polynomial 3n + 1 + b then G + OH3/OH3 which is a subsheaf of F/OH3 ≃ OH1 must have Hilbertpolynomial b + 1, contradicting the pureness of OH1 .


But under these conditions the only ways to get a 0 constant term in the alternating sum are to have G ∩ OH3 = OH3and G + OH3 = F implying G = F , which contradicts the choice of G; or G ∩ OH3 = 0 and G + OH3 = F , equivalently thatF ≃ G ⊕ OH3 and G ≃ OH1 .

d = 4 case:Clearly we have an exact sequence:

0 −→ OH4 −→ F −→ K3 −→ 0 (15)

where K3 is a length 3 sheaf. Apply hom(·, ωP2). We get:

0 −→ Ext1(F , ωP2) −→ Ext1(OH4 , ωP2) −→ Ext2(K3, ωP2) −→ 0. (16)

(See [6] for zero terms.) Rewrite this in terms of duals:

0 −→ FD−→ HD

4 −→ KD3 −→ 0 (17)

ODH4

≃ OH4(4 − 3) = OH4(1). By twisting the sequence above by O(−1) and, and noting that twisting doesn’t affect finitelength sheaves we see that KD

3 is actually isomorphic to a quotient of OH4 . KD3 is isomorphic to the structure sheaf of a

0-dimensional subscheme of H4, let’s call this Q3.By replacing KD

3 with OQ3 above, we get FD≃ H4(−Q3)(1). Take duals of both sides to get F ≃ H4(+Q3).

F is stable if and only if N := FD(−1) is stable. Now let’s check when N := FD(−1) = H4(−Q3) fails to be stable. Let’sstart with the exact sequence:

0 −→ N −→ OH4 −→ OQ3 −→ 0. (18)

The Hilbert polynomial ofN is (4n−2)−3 = 4n−5.N is pure, because quartic sheafOH4 is pure. AssumeN is not stable. LetG be a subsheaf of N violating the stability requirement. The Hilbert polynomial of G is of the form n+ a or 2n+ b or 3n+ c(a constant Hilbert polynomial would violate the pureness ofN , while 4n+dwouldn’t be violating the stability requirementfor N). G is a subsheaf of OH4 , and so it can have a maximum (4n−2)− (n+1) = 3n−3 or (4n−2)− (2n+1) = 2n−3 or(4n− 2) − (3n) = n− 2 as a Hilbert polynomial (because (OH4/G)pure must be a hypersurface sheaf). In other words a, b, ccan be −2, −3, −3 maximum, respectively. We also have that pG(n) > n − 5/4, which means a, b, c have to be −1, −2,−3 at least, respectively. We have eliminated all possibilities except PG(n) = 3n − 3. This can only be possible if OH4/G is aline sheaf ; call this OH1 . OH1 is a quotient of OH4 and OQ3 is a quotient of OH1 , in other words OH4 −→ OQ3 factors throughthe line sheaf OH1 . It is also clear that the converse is true: If OH4 −→ OQ3 factors through a line sheaf, the kernel of thismap N , contains the kernel of OH4 −→ OH1 which has a reduced Hilbert polynomial equal to n − 1 which is greater thann − 5/4. �

Let’s say a few words about scheme theoretic supports. In general if we have a short exact sequence of coherent sheaves

0 → F1 → F2 → F3 → 0, (19)

scheme supports of F1 and F3 are contained in that of F2. Also scheme theoretic support of F2 is contained in the union ofthose of F1 and F3 (here by the union of two subschemes, we mean the scheme whose ideal sheaf is the product of idealsheaves of these two sheaves). These directly follow from definitions.

In (12) if we take H4 to be the union of H3 and H1, F is supported on H4, this means that scheme theoretic support of F iscontained in H4. In other words the natural quotient map F � F ⊗ H4 is an isomorphism.

If F is supported on a (sub)scheme Z , so is FD by intrinsicness of duals. This implies that the scheme theoretic supportof FD is contained in that of F . From this we see that, for 0 or 1 dimensional pure sheaves, the scheme support of the dualis the same as the scheme theoretic support of the original sheaf. In particular we see that the Hd is squeezed betweenthe scheme theoretic supports of Hd(−Qm) and Hd(+Qm) (by their defining sequences) which are identical (by duality andtwist), implying that scheme theoretic supports of Hd(−Qm),Hd(+Qm), and their twists are exactly Hd.

By Theorem 4.4 and the above discussion, the scheme theoretic support of any stable sheaf of Hilbert polynomial 4n+ 1on P2 is a closed subscheme of a quartic hypersurface. In principle it could be for example a cubic hypersurface containedin a reducible quartic. We haven’t checked if this is possible because it will not be needed in the following.

Proposition 4.5. For any quartic H4 on P2 and a point p on it, h0(H4(−p)(1)) = 2.

Proof. Let’s first prove that h0(OH4(1)) = 3 and OH4(1) is globally generated. We have an exact sequence:

0 → O(−3) → O(1) → OH4(1) → 0. (20)

H0(O(−3)) ≃ 0 and H1(O(−3)) ≃ 0, so from the long exact sequence of cohomology we get the exact sequence:

0 → H0(O(1)) → H0(OH4(1)) → 0. (21)

From this we see: h0(OH4(1)) = h0(O(1)) = 3, and since O(1) is globally generated (it is even very ample), andO(1) → OH4(1) is surjective, we see that OH4(1) is globally generated.


Now let’s write down the defining exact sequence for H4(−p)(1):

0 → H4(−p)(1) → OH4(1) → Op(1)(≃ Op) → 0. (22)

Apply the global section functor:

0 → H0(H4(−p)(1)) → H0(OH4(1)) → H0(Op(1))(≃ C) → · · · . (23)

Since OH4(1) is globally generated and OH4(1) → Op(1) is surjective, images of global sections of OH4(1) generate Op(1).So at least one global section of OH4(1) is sent to a nonzero element under H0(OH4(1)) → H0(Op(1)). H0(Op(1)) is1-dimensional, so H0(OH4(1)) → H0(Op(1)) is surjective. We can put 0 in place of · · · above and preserve exactness.Alternating sums of dimensions must be zero and we get: h0(H4(−p)(1)) = 3 − 1 = 2. �

Theorem 4.6. Let F be a stable sheaf on P2 of Hilbert polynomial 4n + 1. Then h0(F) = 1 or 2. If h0(F) = 2, then it can bewritten in the form: F ≃ H4(−p)(1).

Proof. We already know that h0(F) ≥ 1. We also know that F is supported on some quartic, and let’s fix one such quartic,say H4. H4 is Cohen–Macaulay (being a complete intersection) and equidimensional and ωH4 ≃ OH4(1). We apply Serreduality [8, Theorem 7.6, p. 243] to F viewed as a sheaf on H4. There is an isomorphism: Hom(F , OH4(1)) ≃ H1(F)∗.Hom(F , OH4(1)) = 0 if and only if h1(F) = 0 and equivalently h0(F) = 1. If h0(F) ≥ 2 then there is a nonzero map:F → OH4(1). It is easy to see that any such map has to be an inclusion (image of the map is a quotient of F and the subsheafof OH4(1), F is stable, and we know ‘‘more than the stability’’ (Remark 2.2) about a twist of a hypersurface, and inequalitiesforces that that image must have Hilbert polynomial 4n + 1). The cokernel of such an inclusion is a quotient of OH4 of alength 1 (by Hilbert polynomial count), which means a skyscraper at a point p on H4. In other words F ≃ H4(−p)(1). In thiscase we get h0(F) = 2 by the previous result. �

Theorem 4.7. Any H4(−p)(1) is a stable sheaf of Hilbert polynomial 4n + 1.

Proof. Clearly the Hilbert polynomial of F = H4(−p)(1) is (4(n + 1) − 2) − 1 = 4n + 1. Let G be a subsheaf of Fviolating the stability of F . Then G(−1) is a subsheaf of OH4 . We know that such a sheaf can have Hilbert polynomialn − 2 − a, 2n − 3 − a, 3n − 3 − a for nonnegative a. Then G can have Hilbert polynomial n − 1 − a, 2n − 1 − a, 3n − a.None of these violate the stability of F . We get a contradiction, proving that F is stable. �

Proposition 4.8. Let F be a stable sheaf of Hilbert Polynomial 4n + 1 on P2 s.t. h0(F) = 2. Then F can be written in the formH4(−p)(1) in a unique way (in other words, quartic H4 and a point p on it are uniquely determined).

Proof. H4 is the scheme theoretic support of F so it is uniquely determined. So it suffices to check that H4(−p)(1) ≃

H4(−q)(1) ⇒ p = q. We proved before in the proof of Theorem 4.6 that Hom(F , OH4(1)) ≃ H1(F)∗. H1(F) ≃ C sinceh0(F) = 2 and so h1(F) = 1. Hom(F , OH4(1)) must be 1-dimensional. Standard inclusions of H4(−p)(1) and H4(−q)(1) inOH4(1) are two nonzero elements of Hom(F , OH4(1)), andmust be nonzero scalar multiples of each other, and consequentlytheir images must be the same, and their cokernels, which are Op and Oq respectively, must be isomorphic. This is onlypossible if p = q. �

Proposition 4.9. Let F be a stable sheaf of the form H4(+Q3). Then h0(F) = 2 if and only if Q3 is a subscheme of a line.

Proof. Let F = H4(+Q3) be stable. Let Q3 be a subscheme of a line H1. H1 can’t be a component of H4, because otherwiseH4(+Q3)wouldn’t be stable. Let Q4 be the scheme theoretic intersection ofH4 andH1. Q4 is a 0-dimensional scheme andOQ4has length 4.Q3 is a subscheme ofQ4. Corresponding to the surjectionOQ4 � OQ3 , there is an injectionH4(−Q4) ↩→ H4(−Q3)and its cokernel is isomorphic to the kernel of OQ4 � OQ3 , which is a length 1 (skyscraper) sheaf which we call Op, for someclosed point p. Also recall that H4(−Q4) is isomorphic to OH4(−1). So we have an exact sequence:

0 −→ OH4(−1) −→ H4(−Q3) −→ Op −→ 0. (24)

By dualizing and twisting with O(−1) we get:

0 −→ H4(+Q3) −→ OH4(1) −→ Op −→ 0. (25)

This means F ≃ H4(+Q3) ≃ H4(−p)(1). So h0(F) = 2.Now conversely suppose F ≃ H4(+Q3) and h0(F) = 2. Then there is a point p s.t. F ≃ H4(−p)(1), FD

≃ H4(+p). Wehave an exact sequence:

0 −→ (FD(−1) ≃)H4(+p)(−1) −→ OH4 −→ OQ3 −→ 0. (26)

There is an inclusion OH4(−1) ↩→ H4(+p)(−1). By composing injections we get an embedding of OH4(−1) in OH4 , whosecokernel surjects ontoOQ3 and has Hilbert polynomial 4. So it suffices to check that cokernel of an injectionOH4(−1) ↩→ OH4is the quotient of a line sheaf. So, let’s pick an injection: OH4(−1) ↩→ OH4 (actually we will only use that it is nonzero,


and there can be nonzero noninjective ones). Twist by O(1) to get an injection OH4 ↩→ OH4(1). The image of OH4 isgenerated by a global section (namely the image of a global section of OH4 ), say s ∈ H0(OH4(1)). The natural restrictionmap H0(OP2(1)) → H0(OH4(1)) is an isomorphism (proved in Proposition 4.5). Let t ∈ H0(O(1)) be the section mappedto s. (By fundamental theorems of module theory.) OH4(1)/O · s is a quotient of O(1)/O · t . Since s is nonzero, t is nonzero,so O(1)/O · t ≃ OH1(1) for some line H1 (by hypersurface theory). By going back we see that OQ3 is a quotient of OH1 , inother words Q3 is a subscheme of the line H1. �

The following results describe which H3-sections a sheaf of the form H4(−p)(1) can have.

Proposition 4.10. Let F = H4(−p)(1). If there is an injection OH3 ↩→ F then H3 is a subhypersurface of H4, in other words H4is the union of H3 and a line H1; and p is a point on H1. Conversely if a quartic H4 is the union of a cubic H3 and a line H1, and pis a point on H1, then there is an injection OH3 ↩→ H4(−p)(1), and this inclusion is unique up to scalar.

Proof. Let’s suppose there is an inclusion OH3 ↩→ F = H4(−p)(1). Then there is a sequence of inclusions:

OH3(−1) ↩→ H4(−p) ↩→ OH4 . (27)

The quotient of the last term by the first term must be a line sheaf, by familiar arguments, call this OH1 . The existence of ashort exact sequence 0 → OH3(−1) → OH4 → OH1 → 0 is equivalent to the fact that H4 is the union of H3 and H1. Bylooking at cokernels we see that there is a surjection OH1 � Op, which means that p is on H1.

Conversely assume p is on a line H1, which is a component of H4. Then there is an H3 (remaining component of H4) fittinginto an exact sequence:

0 −→ OH3(−1) −→ OH4 −→ OH1 −→ 0. (28)

Since there is a surjection OH1 � Op, there is a corresponding injection OH3(−1) ↩→ H4(−p). By twisting by O(1) we getan injection OH3 ↩→ F .

Now let’s check uniqueness. We have a cubic hypersurface H3, a line H1, their union H4, a point p on H1 and F =

H4(−p)(1) ⊂ OH4(1). There is only one surjection OH4(1) � OH1(1) up to scalar (because there is only one surjectionOH4 � OH1 up to scalar, the global section that generates OH4 must go to a nonzero global section of OH1 ). The image of OH3in OH4(1) has to be the kernel of a surjection OH4(1) � OH1(1), so it is uniquely determined. If there are two such inclusionsof OH3 in OH4(1), they differ by an automorphism of their common images which is isomorphic to OH3 , and we know suchan automorphism OH3 → OH3 is multiplication by a nonzero scalar. �

4.3. C: 2 independent sections

Let C be the subset of M corresponding to sheaves with 2 linearly independent sections. Let C0 ⊂ C be the subsetparametrizing stable sheaves with no cubic sections, and C+ ⊂ C the subset parametrizing sheaves with at least one suchsection.

Proposition 4.11. There is a morphism P(4,1) → M, which is injective as a set map (as a function between sets of closed points)whose image is C. C is closed in M. The topological Euler characteristic of C is 42.

Proof. Consider the family of sheaves UH4(−Q1)(1), the trivial OP2(1) twist of ‘universal one point removed quartic sheaf’on P2

× P(4,1), that is, a family over P(4,1). By the previous section, members of this family are exactly the elements of C ,whose any two members are not isomorphic. By the coarse moduli property of M there is a morphism P(4,1) → M whoseimage is C . P(4,1) is proper, so its image C is closed. P(4,1) → C is a bijective continuous map between compact Haussdorfspaces, in the Euclidean topology, so they are homeomorphic and have the same Euler characteristic.

Let’s compute χ(P(4,1)). P(4,1) is a (Zariski-)closed subset of P(4) × HilbP2(1) ≃ P14× P2, which we consider here as

a projective space over the second factor P2. The fiber over a point p ∈ P2 is the subset of P(4) ≃ P14 corresponding tohypersurfaces passing through p, which is a linear subspace isomorphic to P13 (because p gives a single nonzero linearrelation between coefficients of the monomials of quartic polynomials). So

χ(C) = χ(P(4,1)) = χ(P13) · χ(P2) = 14 · 3 = 42. � (29)

4.4. A ∪ B: unique section

First let’s introduce some notation. Let Y = M \ C , be the subset of M corresponding to sheaves with unique nonzeroglobal section up to scalar, and let A and B be the subsets of Y corresponding to H4 and H3 sections respectively. Y is an opensubset of M by the previous section. We will show that A is open in Y (so it is also open in M), and B is closed in Y (so it islocally closed inM).

Let F be the universal family on M , which is a sheaf on P2× M . Let’s restrict this family to Y , and still use the same

notation: F for the restricted family. We have a sheaf F on X := P2× Y flat over Y . Let f : X → Y be the projection. For any


(closed) point y ∈ Y we have a sheaf Fy on the fiber Xy = f −1(y) which is canonically identified with P2 by construction.h0(Fy) = 1 and h1(Fy) = 0 (by the definition of Y ).

We are in a situation where we can apply Theorem 12.11 on page 290 in [8].Let φi(y) : Rif∗(F) ⊗ k(y) → H i(Fy) be the natural map. φ1(y) is obviously surjective (target is 0), so (by part (a) of the

Theorem 12.11 in [8]) it is an isomorphism. This means R1f∗(F) ⊗ k(y) = 0 for each y ∈ Y . R1f∗(F) is a coherent sheaf on Y(which is a reduced variety) and it restricts to 0 at each closed point. This is only possible if R1f∗(F) is the zero sheaf. Now(by part (b) of Theorem 12.11 in [8]) φ0(y) is surjective, and (again by (a) in Theorem 12.11 in [8]) it is an isomorphism. Wehave proved that the natural map:

φ0: f∗(F) ⊗ k(y) → H0(Fy) (30)

is an isomorphism and this is only possible if f∗(F) is locally free of rank 1.There is natural map f ∗f∗(F) → F . What we say above implies that f ∗f∗(F), locally w.r.t. Y , is isomorphic to OX and on

fibers Xy, f ∗f∗(F) → F restricts to a nonzero maps. Let F1 be the image of f ∗f∗(F) → F , and F2 be the cokernel. We have anexact sequence:

0 → F1 → F → F2 → 0. (31)

If we restrict to Xy we get the exact sequence:

(F1)y → Fy → (F2)y → 0. (32)

The image of (F1)y in the sequence above is generated by a nonzero global section of Fy, and must be a hypersurface ofdegree 4, or 3, and consequently (F2)y has Hilbert polynomial 3 (dual of a length 3 subscheme of the hypersurface not lyingon a line), or n + 1 (a line sheaf), corresponding to points y ∈ A or y ∈ B respectively. By flattening stratification (appliedto F2) we see that A is open and B is closed in Y .

Let’s denote restrictions of F to f −1(A) and f −1(B) by AF and BF respectively, and restrictions of f : X → Y again byf . Let’s denote restrictions of F2 to f −1(A) and f −1(B) by AK3 and BH1 respectively (notations are justified by restrictions tofibers). AK3 and BH1 are flat over A and B respectively, by the constancy of Hilbert Polynomials on fibers. Let AH4 and BH3be the kernels of AF → AK3 and BF → BH1 respectively, which are flat (over A and B respectively) by a basic property offlatness (in a short exact sequence if the last two terms are flat, then so is the first). Note that F1|f−1(A) ≃A H4, because A isopen (and a restriction to an open subset preserves exactness), but F1|f−1(B) ≃B H3 (because otherwise points x ∈ A ∪ B forwhich F1|f−1(x) ≥ 4n − 2 would be A, which is not closed). However there is a surjection F1|f−1(B) �B H3. We have exactsequences:

0→A H4 →A F →A K3 → 0 (33)0→B H3 →B F →B H1 → 0 (34)

of sheaves on f −1(A) = P2× A and f −1(B) = P2

× B respectively which are flat over A and B respectively.Let L = f ∗f∗(F). The sheaves AH4 and BH3 are images of restrictions of F1, which is a quotient of L, so AH4 ⊗ (L|f−1(A))

−1

and BH3 ⊗ (L|f−1(B))−1 are quotients of ‘‘O’’ (which restrict to the same sheaves as AH4 and BH3 on fibers over points of A and

B respectively). These quotient families induce morphisms to Hilbert schemes of quartics/cubics:

A → P(4) (35)

B → P(3). (36)

We will continue from here in the next two sections. We’d like to analyze A and B separately.

4.5. A: unique H4 section, no other sections

We defined A in the previous section. It is the open subset of M corresponding to sheaves of the form H4(+Q3) s.t. Q3 isnot the subscheme of any line. This representation of such a sheaf is unique (because there is only one embedding ofOH4 in Fup to scalar) so there is a natural one-to-one setmap from A to P(4,3) ⊂ P(4)×Hilb(3). We have shown in the previous sectionthat A → P(4) is actually a morphism. In the next subsection we will show that the map sending the point corresponding toH4(+Q3) in A to the point corresponding to Q3 in Hilb(3) is continuous in the Euclidean topology.

4.5.1. Continuity of the map A → Hilb(3)We have the coherent sheaf AK3 on P2

× A flat over A, such that on fibers we have duals of structure sheaves of length-3subschemes of P2. More precisely over the point representing H4(+Q3) in A, AK3 restricts to OD

Q3.

Let K denote the pullback of AK3 on P2× A to P2

× A × Hilb(3) via the projection, and let Q be the pullback of thesubscheme UHilb(3) of P2

× Hilb(3) to P2× A × Hilb(3) as a subscheme via the projection.

Let x ∈ A and y ∈ Hilb(3). Restrictions of K (resp. Q ) to the fiber (x, y) don’t depend on y (resp. x), so let’s denote it byKx (resp. Qy) (fibers of P2

× A × Hilb(3) → A × Hilb(3) are canonically identified with P2). K ⊗ OQ restricts to Kx ⊗ OQy


on the fiber over (x, y). Kx is the dual of structure sheaf, a subscheme of P2 of length 3, and Qy is a length 3 subscheme ofP2. Kx ⊗ OQy has length 3 or less because it is a quotient of Kx, and it has length 3 exactly when Kx and OQy are duals. LetS = {(x, y) ∈ A × Hilb(3) : Kx ≃ OD

Qy}. By flattening stratification S is closed.

Let p1 and p2 be projections from S to A and Hilb(3) respectively. p1 is one-to-one by the definition of S. Wewant to provethat p2 o p−1

1 : A → Hilb(3) is continuous in Eucliden topology. p2 is obviously continuous. It suffices to show that p−11 is

continuous in the Euclidean topology. This follows from the closedness of S in A × Hilb(3), the compactness of Hilb(3), andthe fact that A is a manifold (open subset of a smooth variety). Metrize M and Hilb(3), put induced metrics on all spaces.Let x ∈ A. Choose δ > 0 s.t. the closed ball B(x, δ) ⊂ A and B(x, δ) is compact. p−1

1 (B(x, δ)) = S ∩ (B(x, δ) × Hilb(3)) iscompact. p1|p−1

1 (B(x,δ)) : p−11 (B(x, δ)) → (B(x, δ)) is a bijective continuous map between compact metric spaces, so it is a

homeomorphism. This proves that p−11 is locally continuous so it is continuous (and a homeomorphism).

We proved:

Proposition 4.12. The set map A → Hilb(3), sending the point corresponding to H4(+Q3) to the point corresponding to Q3, iscontinuous (in the Euclidean topology).

4.5.2. Ab: ‘‘Hilbert scheme of 3 noncolinear points on P2’’In this subsection we prove that Ab := the subset of HilbP2(3) corresponding noncolinear subschemes (subschemes

which are not subschemes of any line), is an open subvariety of HilbP2(3), and compute its Euler characteristic. (Later in thissection we will prove that A is a fiber bundle over Ab.)

First we prove that the complement Z := HilbP2(3) \ Ab is closed. Z parametrizes ‘‘colinear’’ subschemes of length 3on P2.

Let S and T denote the pull backs of subschemes UHilb(3) ⊂ P2× Hilb(3) and UH1 ⊂ P2

× P(1) via projections toP2

× Hilb(3) × P(1), respectively. Let X ⊂ Hilb(3) × P(1) be the set consisting of points corresponding to pairs (Q3,H1) s.t.Q3 is a subscheme of H1. OS ⊗ OT is a coherent sheaf on P2

×Hilb(3) × P(1), which restricts to length 0, 1, 2 or 3 sheaves onfibers of the projection P2

× Hilb(3) × P(1) → Hilb(3) × P(1), and the value 3 is taken exactly on fibers over points of X . Byflattening stratification, X is closed in Hilb(3) × P(1).

Now let’s compare X and Z . Z is the image of X under the projection Hilb(3) × P(1) → Hilb(3) and X → Z is a bijection,because for any colinear Q3, there is exactly one line containing it. Z is the image of a proper scheme under a projective map,so Z is closed in Hilb(3). This proves that Ab is open.

Now let’s compute Euler characteristics. X → Z is a bijective continuous map in Euclidean topology, between compactHausdorff spaces, so X and Z are homeomorphic. In particular χ(Z) = χ(X). Let’s compute χ(X). X is a projective schemeand the projection X → P(1) is a surjective projective morphism. We will show that fibers of this projection have atopological characteristic 4. Let Y be a fiber of this map, which is a proper scheme. Y can be identified with the subsetof Hilb(3) corresponding Q3’s, which are subschemes of a fixed line H1. The family OUHilb(3)|P2×Y over Y induces a bijectivemorphism from Y to HilbH1(3) ≃ HilbP1(3) by the universal property. HilbP1(3) ≃ P3. The morphism Y → HilbP1(3) isa bijective continuous map in Euclidean topology between compact Hausdorff spaces, so Y and P3 are homeomorphic. Inparticular χ(Y ) = 4. By a theorem from the first subsection of this section χ(X) = 4 · χ(P(1)) = 4 · χ(P2) = 12. Soχ(Ab) = χ(HilbP2(3)) − χ(Z) = χ(HilbP2(3)) − 12.

It remains to compute χ(HilbP2(3)). There are various ways of computing this number. The answer is 22, and we referto [7, Thm 7.5.5. page. 174] (in the notation of the reference, put z = −1, b1 = b3 = 0, b0 = b2 = b4 = 1, find thecoefficient of t3).

We have proved:

Proposition 4.13. Ab is an open subset of HilbP2(3) and χ(Ab) = 10.

4.5.3. The space of H4’s containing a fixed noncolinear Q3

In this subsection we prove that that the subset of P(4) corresponding to quartics H4 containing a fixed noncolinearsubscheme Q3 of length 3, is a linear subset of P(4) ≃ P14 of codimension 3. Here we identify P(4) with P14 by sendingcoefficients of monomials in defining equations of quartics to standard (homogenous) coordinates (of degree 1) of P14. Inother words the hypersurface determined by

i+j+k=4 aijkx

iyjzk is represented by [a400 : a310 : · · · : a004] ∈ P14. Letg ∈ PGl(2) be a linear automorphism of P2. This bijective map on closed points naturally extends to all subschemes of P2.For a subscheme Z of P2, g(Z) is defined as the scheme theoretic pullback of Z under g−1. Clearly g preserves subschemecontainment relations and Hilbert polynomials of structure sheaves. In particular g takes a quartic to a quartic and inducesa set map P(4) → P(4). This map is actually a linear automorphism. To see this let Z := UH4 be the universal quartic, asthe subscheme of P2

× P(4). Let g(Z) be the inverse image of Z under the inverse of (g, 1) : P2× P(4) → P2

× P(4). By theuniversal property of P(4), there is a morphism h : P(4) → P(4) s.t. g(Z) is the pullback of UH4 via h. The map h is the set mapP(4) → P(4) described above, and it is a morphism. This construction respects compositions of elements in PGl(2), and wesee that h is an automorphism of P(4), and any automorphism of P(4) ≃ P14 is linear.


Our purpose in this subsection is to prove that the set of quartics containing a fixed noncolinear subscheme Q3 of P2 forma linearly embedded P11 inside P(4) ≃ P14. In light of the discussion above, it suffices to prove this claim for one elementfrom each orbit of induced (set theoretical) PGl(2) action on Ab ⊂ Hilb(3).

Let’s classify length 3 subschemes Q3 of P2 (which are parametrized by Hilb(3)), up to PGl(2) action (or equivalently upto linear change of coordinates).

If there are 3 distinct closed points inQ3 (or equivalently,Q3 has 3 components), these are colinear or not. Clearly colinearones form one orbit (or equivalence class), and noncolinear ones form another.

Now let’s analyze length 3 subschemes Q3 which are (set theoretically) supported on 2 distinct points. Q3 must be thedisjoint union of a subscheme Q1 of length 1 and another Q2 of length 2 (each supported at one point). Let’s analyze Q2.Without loss of generality assume that Q2red is the origin on an affine part spec(C[x, y]). Let f : C[x, y] → R be the quotientmap (a C[x, y]-algebra homomorphism) defining Q2. f (x) and f (y) are nilpotents (because Q2red is the origin). f (1) is clearlynot equal to 0, because f is a (nonzero) ring homomorphism. Images of linear terms under f can’t be all zero, becauseotherwise we would get a length 1 sheaf only. Images of second order terms (x2, y2, xy) are all zero. It is enough to checkthis for x2 (then by linear change of coordinates the same holds for y2 and (x + y)2). If f (x2) is not zero, then f (x) is notzero either. f (1), f (x) are linearly independent over C, (because f (x) is nilpotent, f (1) is the multiplicative identity) so theygenerate R as a C vector space and f (x2) = af (1)+bf (x) for some constants a, b. a is zero, because f (x2)−bf (x) is nilpotent.Startingwith bf (x) = f (x2) one gets bnf (x) = f (xn+1)which is 0 for big n. So bmust be zero aswell. This contradiction provesthat f (x2) = 0. Now pick a linear nonzero term, let’s say f (x) = 0 in R. f (1), f (x) form a basis for R. f (y) = af (1) + bf (x)for some constants a, b, and since f (x), f (y) are nilpotents a = 0. We have the relation f (y) = bf (x). This defines R togetherwith quadratic relations, the ideal generated by y − bx and x2 contained in the ideal of R, already define a length 2 sheaf.So (also remembering wlog’s) Q2 (set theoretically) supported at point is determined by the point and (direction of) a linethrough that point. Using this it is easy to see that the group of (linear) automorphisms of P2 acting on Q3 with 2 points ontheir support have two orbits: Q3 belongs to one or the other according to whether Q3 is subcseme of a line or not, in otherwords, whether Q1 is on the unique line supporting Q2.

We found 4 types of length 3 subschemes, in other words 4 orbits of action of PGl(2) on Hilb(3). There are 3 more types(or orbits) coming from Q3’s with one point on their support. Let’s analyze them now.

Let Q3 be a subscheme of P2 of length 3 (set theoretically) supported at a point, which wlog we assume is the origin ofan affine plane part. Let f : C[x, y] → R be the quotient map defining Q3. As before first we see that all degree 3 terms(f (x3), f (x2y), f (xy2), f (y3)) are zero, and not all linear terms are zero. Consider the 2 cases.

First case: Images of all second order terms f (x2), f (xy), f (y2) are 0. In this case the ideal of R contains x2, y2, xy whichalready define a length 3 sheaf, so R = C[x, y]/(x2, xy, y2). Such R is uniquely determined by the supporting point (here theorigin), because its ideal is the square of the ideal of the point.

Second case: Not all second order terms are zero. This implies that there is a nonzero square term. Wlog (up tolinearity) assume f (x2) is not zero. Then f (x) is not zero either. As before we see that f (1), f (x), f (x2) form a basis for R.f (y) = bf (x) + cf (x2) for some constants b, c (‘‘as before’’). Write this as f (y − bx) = cf (x2) and by linear change ofcoordinate (wlog) assume b = 0. So we have relations y− cx2 = x3 = 0 and these already define a length 3 scheme, namelyspec(C[x, y]/(x3, y − cx2)).

If c = 0, this is subcsheme of a line, and such schemes are determined by a point (here the origin) and a line through it(here the {y = 0}), and such subschemes are in the same PGl(2) orbit.

If c = 0, by linear change of coordinate c can be taken to be 1. Q3 is a subcsheme of a smooth conic in this case. If I isideal of a smooth conic and J ideal of a point on it, I + J3 defines a Q3 of this type.

We proved:

Proposition 4.14 (A Set Theoretical Statement). The automorphism group of the P2 acts on the set of closed points of Hilbertscheme of 3 points on P2. Here are the orbits:

1. Three distinct noncolinear points. Call this subset of Hilb(3), A1.2. One point on the support, nonreduced of first order in any direction. More explicitly at the origin of a standard affine part with

coordinates x, y, the ideal is generated by quadratic terms: x2, xy, y2. (Such subschemes are uniquely defined by the supportingpoint.) Call this part A2.

3. One point on the support. Nonreduced of second order along a smooth conic through this point. More explicitly it is given byan ideal generated by x3, y − x2 up to a linear transformation in PGl(2). Call this part A3.

4. Two points on the support. One of them is reduced, the other is nonreduced of first order along a line through the second point,which is not the same as the line joining these points. Up to an element in PGl(2), its ideal is generated by, x(x − 1), y2, xy.Call this part A4.

5. Three distinct colinear points. Orbit of the subscheme with ideal x(x − 1)(x + 1), y. Call this A5.6. Two points on the support. One of them is reduced, the other is nonreduced of first order along a line through the second

point, which is the same as the line joining these points. This is the orbit of the subcscheme defined by the ideal generated byy, x2(x + 1). Call this A6.

7. One point on the support. Nonreduced of second order along a line through this point. The orbit of the subscheme is defined byy, x3. Call this A7.


Now let’s see what kind of subsets of P(4) ≃ P14 we get from the subscheme containment relation OH4 � OQ3 for a fixedQ3 belonging to A1, A2, A3, or A4. (We are not interested in Q3’s belonging to A5, A6 or A7 because they are subschemes oflines.)

A hypersurface H4 of degree 4 is the zero scheme of a homogenous quartic polynomial

i+j+k=4 aijkxiyjzk. A subscheme

Q of P2 is a subscheme of a quartic if and only if the ideal sheaf of Q contains the ideal sheaf of the quartic. Since we knowgenerators of the ideal sheaves of four types of Q ’s, computing conditions on H4 to contain Q ’s in terms of coefficients of itsdefining quartic polynomial becomes a matter of substitution.

– 3 distinct noncolinear points. LetH4 pass through ([0:0:1], [1:0:1], [0:1:1]). Then a004 = 0, a400+a301+a202+a103+a004 =

0 and a040+a031+a022+a013+a004 = 0. These are 3 linearly independent linear conditions on 15 coefficients of definingpolynomial of H4, hence the solution set is a linearly embedded P11.

– On the {z = 1} part given by (x2, y2, xy). Conditions for H4 to contain Q3 are: a004 = 0, a013 = 0, a103 = 0 and againthese are 3 independent linear conditions.

– On the {z = 1} part given by (x3, y− x2). Conditions: a004 = 0, a103 = 0, a202 + a013 = 0. Again a linearly embedded P11.– On the {z = 1} part given by (x(x− 1), y2, xy). Conditions: a004 = 0, a013 = 0, a103 + a202 + a301 + a400 = 0. Again a P11

as above.

We proved:

Proposition 4.15. Let Q3 be a noncolinear length 3 subscheme of P2. The set of points corresponding to quartics in P2 containingQ3 as a subscheme, is a linear subset of codimension 3 inside the space of quartics P(4) ≃ P14 (for any isomorphism P(4) ≃ P14).

4.5.4. χ(A)

In this subsection we will prove that A is a topological fiber bundle over Ab with fibers homeomorphic to P11 which willallow us to compute the topological Euler characteristic of A.

Let q1 : A → P(4) and q2 : A → Hilb(3) be themaps sending the point corresponding toH4(+Q3) to points correspondingto H4 and Q3 respectively. Let q := (q1, q2) : A → P(4) × Hilb(3).

We know that q1 and q2 are continuous and q is injective. Let p1 : P(4)×Hilb(3) → P(4), and p2 : P(4)×Hilb(3) → Hilb(3)be projections.

Let A′:= q(A). We know that A′

= P(4,3) ∩ p−12 (Ab) and fibers of p2|A′ : A′

→ Ab are linear subsets of dimension 11 inP(4) = P14. A′ is a closed subset of P(4) × Ab = P14

× Ab. Pick a point x ∈ Ab. Let y ∈ P14 be s.t. (y, x) ∈ P14× Ab \ A′. Since A′

is closed in p−12 (Ab) = P14

× Ab, there is an open neighborhood V ⊂ Ab of x such that, {y} × V doesn’t intersect A′. By fixinga hyperplane P13 in P14

\ {y} and projecting p−12 (V ) ∩ A′ away from y × V onto P13

× V , we get a bijective morphism fromp−12 (V )∩ A′ onto a subvariety of P13

× V over V having linear P11 planes as fibers. We repeat this process twice more to findan open neighborhoodWx of x such that there is a bijective morphism from rx : p−1

2 (Wx) ∩ A′ onto P11× Wx overWx.

q is a bijective continuous map (in Euclidean topology) from A to A′, so the restriction of q to the open subsetq−1(p−1

2 (Wx)) = q−12 (Wx) is also a bijective continuous map onto p−1

2 (Wx)∩A′. Composing this with rx, we get a continuousbijection from q−1

2 (Wx) onto P11× Wx respecting natural maps onto Wx by construction. q−1

2 (Wx) is a 34 dimensionaltopological manifold (open subset of the smooth variety M), P11

× Wx is another 34 dimensional topological manifold (Wxis an open subset of the smooth variety HilbP2(3)), and by invariance of domain from algebraic topology this continuousbijection is a homeomorphism.

We have proved that q2 : A → Ab is a topological fibration whose fibers are homeomorphic to P11. So,

χ(A) = χ(P11) · χ(Ab) = 12 · 10 = 120. (37)

We record this as:

Proposition 4.16. χ(A) = 120.

4.6. B: unique H3 section, no other sections

4.6.1. Extensions of a cubic sheaf and a line sheaf

Lemma 4.17. Let H1,H3 be hypersurfaces of degree 1 and 3 on P2. Then Ext1(OH1 , OH3) ≃ C3.

Proof. We have an exact sequence.

O −→ O(−1) −→ O −→ OH1 −→ 0. (38)

Apply Hom(·, OH3) to get the long exact sequence O −→ Hom(OH1 , OH3) −→ Hom(O, OH3) −→ Hom(O(−1), OH3) −→

Ext1(OH1 , OH3) −→ Ext1(O, OH3) −→ Ext1(O(−1), OH3)Hom(OH1 , OH3) = 0 because both OH1 and OH3 are stable of


dimension 1 and pOH1(n) > pOH3

(n). Rewrite this sequence as: 0 −→ H0(OH3) −→ H0(OH3(1)) −→ Ext1(OH1 , OH3) −→

H1(OH3) −→ H1(OH3(1)).The dimension of each term can be computed easily. First of all, the last one can be computed as 0 by taking long exact

sequence associated toO(1) twist of defining sequence ofOH3 and using Serre duality. So the alternating sum of dimensionsmust be 0. 1 − 3 + dim(Ext1(OH1 , OH3)) − 1 + 0 = 0. We find dimC(Ext1(OH1 , OH3)) = 3. �

Proposition 4.18. For fixed H3 and H1, there is family over P(Ext1(OH1 , OH3))∨

≃ P2, (which means a sheaf on P2×

P(Ext1(OH1 , OH3))∨

≃ P2× P2 flat over P(Ext1(OH1 , OH3))

∨), parametrizing all stable sheaves F on P2 which fit into an exactsequence

0 −→ OH3 −→ F −→ OH1 −→ 0 (39)

whose any two members are not isomorphic.

Proof. Stable extensions are nonsplit ones. For parametrization of nonsplit extensions up to scalar, see [6, Example 2.1.12,page 37]. �

4.6.2. A morphism B → P(3) × P(1)

In a previous section we have described a morphism B → P(3) ≃ P9.Now we want to create another morphism B → P(1) ≃ P2.For any F corresponding to a point in B, we have an extension of the form 0 → OH3 → F → OH1 → 0, such that the

(H3,H1) pair is unique (maps in the sequence are unique up to scalar). So we certainly have a set map from B to the spaceof H1’s, namely P(1). We want to show that this is actually a morphism. From a previous section we already have a family ofOH1 ’s over B, but since these are not given as quotients of OP2 in a uniform way, we can’t immediately say that we have amap to the Hilbert scheme of lines.

We can do this in two ways. We can prove that BH1 is a quotient of O, locally w.r.t. B, in other words, it is a quotientof pullback of a line bundle on B. Indeed by semicontinuity theorems as before f∗(BH1) is a line bundle and there is a mapf ∗f∗(BH1) → BH1 which is surjective on global sections on fibers and must be surjective on fibers (a nontrivial subsheaf ofa line sheaf which is structure sheaf of a scheme must be the same as that line sheaf). It is easy to check that in this casef ∗f∗(BH1) → BH1 is surjective.

Or we can use the fact that we have a morphism from B toM(n + 1) (the moduli of stable sheaves of Hilbert polynomialn+1which parametrizes structure sheaves of lines on P2) and the Hilbert scheme P(1) (which parametrizes the same objectspointwise) has a bijective morphism toM(n+ 1) by universal property (both are a smooth projective varieties ≃ P2) whichmust be an isomorphism. By composing B → M(n + 1) with the inverse of P(1) → M(n + 1), we get a morphism B → P(1).This is the map described as a set map above.

4.6.3. T := M × P(3) × P(1): triplesLet’s start with some notation. By some abuse of notation, we consider M also as a set whose elements are individual

sheaves parametrized by M , namely stable sheaves of Hilbert polynomial 4n + 1, and the same applies for subset of M .Similarly for T defined below.

Let T := M × P(3) × P(1) = {(F ,H3,H1)}. We call an element t ∈ T a triple. For any triple, there is a quartic H4, whichis the union of H3 and H1 (independent of F ). Let F3 := F ⊗ OH3 , F1 := F ⊗ OH1 , F4 := F ⊗ OH4 . Let pM : T → M be theprojection.

Let E := {t = (F ,H3,H1) : There exists an exact sequence 0 → OH3 → F → OH1 → 0}. We call elements of E ‘exacttriples’.

Let U := {t ∈ E : F ∈ B}. We described a morphism B → P(3) × P(1), and combining this with the inclusion B → M weget a morphism B → T whose image is U . Since U is a graph of B, clearly U ≃ B.

Proposition 4.19. Let Z := {t ∈ T : PF3(n) ≥ 3n + 3, PF1(n) ≥ n + 1, PF4(n) ≥ 4n + 1}. Then Z is a closed set containing E.

Proof. Let first prove that Z is closed. There are sheaves on P2× T which restrict to F3, F1, F4 for points of T , which come

from the universal sheaf on P2× M, OUH3 on P(3), OUH1 on P(1), OUH3+1 on P(3) × P(1) via pull backs and tensor products in

the obvious way. By flattening stratification Z is closed (intersection of three closed subsets of T ).Now let’s prove that Z contains E. Let t = (F ,H3,H1) ∈ E, be an exact triple. We have an exact sequence

0 → OH3 → F → OH1 → 0. (40)

Let’s apply ⊗ OH1 to the sequence above. We get exact · · · F1 → OH1 ⊗ OH1 ≃ OH1 → 0. We immediately see thatPF1(n) ≥ POH1

(n) = n + 1.Again from the sequence above we see that scheme theoretic support of F is a subscheme of H4, which implies that

F4 = F ⊗ OH4 ≃ F . In particular χ(F4) = 4n + 1.


Now let’s apply ⊗ OH3 to the sequence. We get an exact sequence:

· · · Tor(OH3 , OH1) → OH3 → F3 → OH3 ⊗ OH1 → 0. (41)

From this we get PF3(n) ≥ POH3(n) + POH3⊗OH1

(n) − PTor(OH3 ,OH1 )(n) = 3n + POH1⊗OH3(n) − PTor(OH3 ,OH1 )(n).

So it suffices to check that POH3(n) ⊗ POH1

(n) − PTor(OH3 ,OH1 )(n) ≥ 3.Now let’s write the defining sequence of OH1

0 → O(−1) → O → OH1 → 0. (42)

Apply ⊗ OH3 to this to get the exact sequence:

· · · Tor(OH3 , O) → Tor(OH3 , OH1) → OH3(−1) → OH3 → OH3 ⊗ OH1 → 0. (43)

The first term is 0 because O is flat. The alternating sum of Hilbert polynomials of remaining terms must be zero, soPOH3⊗OH1

(n) − PTor(OH3 ,OH1 )(n) = POH3(n) − POH3 (−1)(n) = 3. �

Lemma 4.20. Z ∩ p−1M (C) = E ∩ p−1

M (C).

Proof. Let t = (F ,H3,H1) ∈ Z ∩p−1M (C). Then F ∈ C , which means that F is of the form H ′

4(−p)(1) for some point p in someH ′

4. It is easy to see that H ′

4 must be the same as H4, the union of H3 and H1: We know that the scheme theoretic support ofF is H ′

4, and H4 also supports F (PF4(n) ≥ 4n + 1 coming from t ∈ Z implies that F = F4).So F = H4(−p)(1) for some p on H4. By Proposition 4.10, showing t ∈ E is equivalent to showing that p lies on H1.

By contradiction assume p doesn’t lie on H1. This implies that F and OH4(1) are isomorphic in a neighborhood of H1 in P2,namely the complement of p. So F1 = F ⊗ OH1 ≃ OH4(1) ⊗ OH1 ≃ OH1(1). There is canonical exact sequence:

0 → OH1(−3) → OH4 → OH3 → 0. (44)

Apply ⊗ F to get the exact sequence:

F1(−3) ≃ OH1(−2) → F → F3 → 0. (45)

The first map above is 0 or injective (otherwise, by Hilbert polynomial restrictions either its image or kernel has to betorsion, but both F and OH1(−2) are pure). If it is injective, the alternating sum of Hilbert polynomials must be zero, so,PF3(n) = PF (n) − POH1 (−2)(n) = 4n + 1 − (n − 1) = 3n + 2 but this contradicts PF3(n) ≥ 3n + 3, because t ∈ Z . Theremaining possibility is that F3 ≃ F , which is not possible because H3 can’t support F = H4(−p)(1) (easy to check bytensoring). This contradiction proves that t ∈ E, and Z ∩ p−1

M (C) ⊂ E ∩ p−1M (C). Other inclusion is obvious by the previous

theorem. �

Let V =: p−1M (C) ∩ E. By the result above it is closed (in T ), because it is the intersection of two closed subsets of T . Let

V ′ be the space of H4(−p)(1) where H4 is a union of a cubic H3 and a line H1, and p is a point a on H1. More precisely V ′

can be defined as a closed subset of P2× P(3) × P(1) whose fibers over P(3) × P(1) are isomorphic to P1, with a family (a

coherent sheaf on P2× P2

× P(3) × P(1) flat over the product of the last there factors) parametrizing sheaves of the formmentioned above. (This is a routine construction analogous to construction of universal union of hypersuraces and familyversion of removing points.) There is a natural projection morphism V ′

→ P(3) × P(1), and by coarse moduli property of M ,a morphism V ′

→ M . These morphisms give rise to a morphism V ′→ T = M × P(3) × P(1), whose image is exactly V by

constructions and properties of sheaves in C+. The map V ′→ V is a bijective morphism with respect to projections of V ′

and V onto P(3) × P(1) by constructions. In particular fibers of V → P(3) × P(1) are images of P1 under bijective morphisms,which must be homeomorphic to P1 in the Zariski and Euclidean topologies.

Nowwe consider fibers of the projection E → P(3)×P(1). Fix anH3,H1, pair. There is a family over P(Ext(OH3 , OH1))∨

≃ P2

parametrizing all F which fit into a nonsplitting exact sequence of the form 0 → OH3 → F → OH1 → 0, whose any twomembers are not isomorphic. By the coarsemoduli property ofM , this family gives rise to amorphism toM×{x}×{y} ≃ Mwhere x ∈ P(3), y ∈ P(1) are points corresponding to H3,H1. We see that fibers of the projection E → P(3) × P(1) are imagesof one to one morphisms from P2, so they are homemorphic to P2 in Zariski and Euclidean topologies.

Proposition 4.21. Zariski closure of U (in T ) is U ∪ V = E.

Proof. First we prove that U is closed in p−1M (B) = B × P(3) × P(1). U is the graph of a morphism B → P(3) × P(1) so it is

closed. In other words U ∩ p−1M (B) = U .

U contains E, because E = U ∪ V , fibers of E → P(3) × P(1) are 2-dimensional irreducible proper varieties and fibers ofV → P(3) × P(1) are 1-dimensional irreducible subvarieties, and the remaining parts, that is fibers of U → P(3) × P(1), aredense. This means even fiberwise closure of U contains E.

Now let’s find U ∩ p−1M (C). We have inclusions E ⊂ U ⊂ Z . By intersecting each with p−1

M (C) we get E ∩ p−1M (C) ⊂

U ∩ p−1M (C) ⊂ Z ∩ p−1

M (C). The first and last terms are V , then so is the middle term. We proved U ∩ p−1M (C) = V .

U ⊂ p−1M (B ∪ C), because p−1

M (B ∪ C) is a closed set containing U . This proves that U = U ∪ V . �


E is closed in T := M ×P(3) ×P(1) and E → P(3) ×P(1) is a projective morphism (becauseM is a projective variety) whosefibers are homeomorphic to P2. So

χ(E) = χ(P(3)) · χ(P(1)) · χ(P2) = 10 · 3 · 3 = 90. (46)

V is closed in T := M × P(3) × P(1) and V → P(3) × P(1) is a projective morphism (because M is a projective variety)whose fibers are homeomorphic to P1. So

χ(V ) = χ(P(3)) · χ(P(1)) · χ(P1) = 10 · 3 · 2 = 60. (47)

Consequently

χ(B) = χ(U) = χ(E) − χ(V ) = 90 − 60 = 30. (48)

We proved:

Proposition 4.22. χ(B) = 30.

4.7. χ(M)

The spaceM is the disjoint union of A, B and C . A is open inM, B is open inM \ A.So

χ(M) = χ(A) + χ(B) + χ(C) = 120 + 30 + 42 = 192. (49)

We proved the main theorem:

Theorem 4.23. Let M be the moduli space of stable sheaves of Hilbert polynomial 4n + 1 on P2. Then χ(M) = 192.

Let X denote the local P2, the total space of the canonical bundle of P2. Let L be the pullback of OP2(1) under X → P2.We can talk about Hilbert polynomials and the stability of coherent sheaves on X with respect to L, and for sheaves (schemetheoretically) supported on P2 these notions agree with Hilbert polynomials and stability for sheaves on P2 with respect toOP2(1).

Lemma 4.24 (Katz). Let F be a coherent sheaf on X which has Hilbert polynomial 4n + 1 and is stable with respect to L. Thenthe scheme theoretic support of F is P2.

Proof (By Katz). Since the normal bundle of P2 in X is O(−3), it follows that the ideal sheaf of P2 in X is isomorphic to L3.This gives the short exact sequence

O → L3 → OX → OP2 → O. (50)

Tensoring with F gives an exact sequence

F ⊗ L3 → F → F |P2 → O. (51)

The stability assumption can then easily be used to show that F → F |P2 is an isomorphism.

Corollary 4.25. The degree 4 Gopakumar–Vafa invariant of local P2 is −192. �

Acknowledgment

The author thanks Professor Sheldon Katz from the University of Illinois in Urbana Champaign for giving this problem(finding the number in the Main Theorem) as a thesis problem.

References

[1] Philip Candelas, Anamaria Font, Sheldon Katz, David R.Morrison,Mirror symmetry for two parametermodels—II, Nuclear Phys. B 429 (1994) 626–674.[2] Sheldon Katz, Genus zero Gopakumar–Vafa invariants of contractible curves, J. Differential Geom. 79 (2008) 185–195.[3] C. Simpson, Moduli of representations of fundamental group of a smooth projective variety, Publ. Math. Inst. Hautes Études Sci. 79 (1994) 47–129.[4] J. Le Potier, Faisceaux semi-stables de dimension 1 sur le plan projectif, Rev. Roumaine Math. Pures Appl. 38 (1993) 635–678.[5] Mario Maican, A few remarks concerning the geometry of the moduli spaces of semistable sheaves supported on plane curves of multiplicity four,

2007, Preprint.[6] D. Huybrechts, M. Lehn, The Geometry of Moduli Spaces of Sheaves, F. Vieweg and Sohn, Braunschweig, 1997.[7] B. Fantechi, L. Gottsche, L. Illiusie, S.L. Kleiman, N. Nitsure, A. Vistoli, Fundamental Algebraic Geometry, American Mathematical Society, Providence,

2005.[8] Robin. Hartshorne, Algebraic Geometry, Springer, New York, 1977.[9] Sylvain E. Cappell, Laurentiu Maxim, Julius L. Shaneson, Euler characteristics of algebraic varieties, Comm. Pure Appl. Math. 61 (2008) 409–421.

[10] Sylvain E. Cappell, Julius L. Shaneson, Genera of algebraic varieties and counting of lattice points, Bull. Amer. Math. Soc. 30 (1994) 62–69.

Documents

Direct computation of the degree 4 Gopakumar–Vafa invariant on a Calabi–Yau 3-fold