On triangulations of the plane by pencils of conics. II

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On triangulations of the plane by pencils of conics. II

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Sbornik : Mathematics 204:6 869–909 c⃝ 2013 RAS(DoM) and LMS

Matematicheskiı Sbornik 204:6 93–134 DOI 10.1070/SM2013v204n06ABEH004323

On triangulations of the plane by pencils of conics. II

V. B. Lazareva and A. M. Shelekhov

Abstract. The present work continues our previous paper in which all pos-sible triangulations of the plane using three pencils of circles were listed.In the present article we find all projectively distinct triangulations ofthe plane by pencils of conics that are obtained by projecting regularthree-webs, cut out on a nondegenerate cubic surface by three pencils ofplanes, whose axes lie on this surface.

Bibliography: 6 titles.

Keywords: curvilinear three-web, regular three-web, Burau web, specialBurau three-web.

Introduction

Recall that by definition a curvilinear k-web W consists of k families of smoothcurves on the plane or, more generally, on a two-dimensional smooth manifold.The domain of definition D of a web is the domain at whose every point the weblines are transversal. The webs constitute one of the most popular objects of studyin classical differential geometry. There is an extensive literature concerned withEuclidean, affine and projective geometry of webs. In the 1920s Blaschke andthe participants of his seminar began to consider webs up to local diffeomorphisms,that is, with respect to the largest equivalence relation in local differential geometry.Such a theory of webs is now called differential-topological, even though Blaschkehimself called it topological. In the topological theory two-webs are devoid ofinterest, because by a suitable local diffeomorphism two families of lines can bemapped (locally!) onto the Cartesian net. Yet three-webs already possess a non-trivial topological invariant —curvature. Let D be an arbitrary point of the domainof definition of some three-web W (see Fig. 1). Through it there pass three lines ofthe web, one in each family. The lines of the first, second, and third families will becalled, respectively, the vertical, horizontal, and sloped lines. Consider a point Bon the horizontal line sufficiently close to the point D and draw through it thevertical and sloped lines of the web. They will intersect the existing lines of theweb at the points E and C. If we draw horizontal lines through E and C we obtainthe points F and G. The point H is constructed similarly. In general, the pointsG and H do not lie on one line of the third family of the web W . In this case, theprincipal part of the ‘gap’ between the sloped lines of W passing through G and His determined by some relative invariant b(D), which is called the curvature of theweb W . If the points G and H lie on one line of the third family, then it is said thatthe hexagonal figure GEBCFH is closed. It is easily verified that on the simplest

AMS 2010 Mathematics Subject Classification. Primary 53A60; Secondary 53A30.

870 V. B. Lazareva and A.M. Shelekhov

Figure 1

web formed by families of parallel lines on the affine plane (such a web is calledparallel and denoted PW ), all the hexagonal figures are closed. Hence, such figuresare also closed on any web that is equivalent to a parallel web. Such webs are calledregular or parallelizable (and also hexagonal) on account of the following theoremof Thomsen: if on some three-web all the hexagonal figures are closed, then thisweb is regular, see [1], [2]. This gives grounds for saying that a regular three-webtriangulates its domain of definition.

Regular webs are of special interest also because they arise quite frequently invarious applications. For example, Ferapontov (see his appendix in [3]) examinesthe regularity condition for the three-web of characteristics on solutions of a systemof equations of hydrodynamic type. Here, the regularity condition means that thesystem is weakly nonlinear and semi-Hamiltonian. The paper [4] is concerned witha three-web consisting of straight lines, which appears in the neighbourhood ofa singular point of a smooth mapping. In [5], the same authors prove that anycontinuous mapping that carries a smooth three-web into a smooth one is smooth.

The web equation z = f(x, y) that relates the parameters of the lines of the webthat pass through one point may be regarded as a binary operation in a smoothlocal quasigroup; this paves the way for using the results of the theory of websin the theory of quasigroups and loops. In particular, to the subclass of regularwebs there corresponds the subclass of loops that are isotopic to the Abelian groupz = x + y (see [3]).

Considering the above, the problem of finding the subclass of regular webs ina given class of webs is important. However, this problem is quite challenging.For example, if a three-web on the plane is formed by three families of straightlines (not necessarily parallel!), then the regularity condition reads as follows: thestraight lines of the web must belong to the same cubic family ([2], Ch. I, § 4).In particular, the webs formed by three pencils of straight lines (we denote themby W0) are regular.

In the class of three-webs formed by algebraic curves, next in order of complica-tion (after rectilinear webs) are the circle webs formed by pencils of circles. Ingeneral, such webs are not regular. In [1], [6] the authors solved Blaschke’s problemof describing all projectively distinct regular circle three-webs. In the present paper,

On triangulations of the plane by pencils of conics. II 871

we extend Blaschke’s problem by considering a certain class of regular three-websformed by pencils of conics (second order curves).

Let V be a cubic surface (a cubic) in the three-dimensional projective space P3;ℓi, i = 1, 2, 3, are the three real straight lines lying on V . We denote by Li thepencil of planes with axes ℓi, by λi the family of conics that are cut out by theplanes of the pencils Li on V , and by W the three-web formed by the families λi.Upon projecting the web W to a plane we obtain the three-web W , which is alsoformed by families of conics that are perspectively equivalent to the three-web W .If one of the webs W or W is regular, then so is the other. This follows from thefact that under projection the closed hexagonal figures are sent to closed ones.

Such a seemingly exotic way of obtaining a three-web from conics on the plane isaccounted for by the fact that a three-web W is regular if the lines ℓi are in generalposition (that is, they are pairwise skew). Burau seems to have been the first toprove this fact (for more details, see Ch. III, § 47 of [2]; a simple proof will be givenin § 2). Hence the three-webs cut out on cubic surfaces in the manner describedabove will be referred to as Burau webs.

However, a Burau three-web will not in general be regular if the lines ℓi are ina special arrangement. The corresponding three-webs will be called special Burauwebs. In the present paper, we examine all special cases of arrangements of lines ℓi

on a cubic surface: two of the lines ℓi intersect; one of the lines intersects the twoothers; all three lines lie in one plane. In each of these cases we shall identifythe regular webs and subdivide them into basic subclasses, for each of which thecorresponding cubic surface is characterized by relations on the coefficients and interms of singularities (Theorems 3.1, 4.1, 5.1, 5.2).

Further, we project the so-obtained regular three-webs from the cubic surface tothe plane from a singular point of the surface V . In this case —and in this caseonly —the web obtained by projection consists of pencils of conics. (As will beclear from the proofs, a cubic admitting a special regular Burau three-web has atleast one singular point.) Enumerating all possible projections, we arrive at themain result formulated in Theorem 6.1: if a cubic does not decompose into a planeand a quadric, then there exist 38 projectively distinct types of webs, as formed onthe plane by pencils of conics, that are perspectively equivalent to regular specialBurau webs.

In the present paper we put forward the following projective classification ofpencils of conics. An arbitrary pencil is determined by two base conics, whichintersect at four points common to all the conics of the pencil. These are calledthe vertices of the pencil. A pencil is said to be elliptic (an E-pencil) if all itsvertices are distinct (that is, simple or 1-fold) and real; hyperbolic (an H-pencil)if the vertices are distinct complex conjugate pairs; hemielliptic (an HE-pencil) iftwo vertices are distinct and real, and two vertices are imaginary. If two verticescoincide, then all the conics of the pencil touch each other at the double vertex.Such a pencil will be referred to as a parabolic pencil (a P -pencil) of elliptic orhyperbolic type (a PE-pencil or a PH-pencil), depending on whether the simplevertices are real or imaginary. If the vertices coincide in pairs, then a pencil will becalled twice parabolic (a WP -pencil). If three vertices of the pencil coincide, thenall the conics of the pencil will have a second order contact at the triple (3-fold)vertex. Such a pencil will be called strongly parabolic (an SP -pencil). Finally, if all


four vertices coincide, then all the conics of the pencil will have a contact of orderthree at this vertex. Such a pencil will be said to be ultraparabolic (a UP -pencil).

The following result is verified immediately on computation.

Proposition 0.1. A PE-pencil (PH-pencil) of conics is uniquely determined byits vertices and the common tangent at the double vertex. A WP -pencil of conics isuniquely determined by its vertices and the common tangents of conics of the pencilat these vertices.

There exist ∞1 SP -pencils (an SP -bundle) of conics with prescribed vertices anda common tangent at the triple vertex.

There exist ∞2 UP -pencils (a UP -hyperbundle) of conics having a prescribed4-fold vertex and a common tangent at this vertex.

An arbitrary SP -pencil has the form Q + λTℓ = 0, where Q = 0 is an arbitraryconic, T = 0 is a tangent to Q, ℓ = 0 is the straight line passing through thetangent point, and λ is the parameter of the pencil. In some projective coordinates,the equation of the SP -bundle has the form x1x3 + a(x2)2 + λx2x3 = 0, where A1

is the triple vertex, A3 is the simple vertex, and A1A2 is the common tangent.An arbitrary UP -pencil has the form Q + λT 2 = 0, where the notation is the

same. The equation of a UP -hyperbundle may be written as follows : x1x3+a(x2)2+bx2x3 + λ(x3)2 = 0, where A1 is the 4-fold vertex.

§ 1. Regularity conditions for a curvilinear three-web

In order to describe the regular circle webs in [1] we employed the theoremon boundaries, according to which the boundaries of the domain of definition ofa regular web are lines of this web. In this paper we shall employ a differentregularity condition, most convenient for webs defined by an algebraic equation.

The families λi of lines forming an arbitrary curvilinear three-web W are givenin most general form by the equations

F1(x, y, u1) = 0, F2(x, y, u2) = 0, F3(x, y, u3) = 0, (1.1)

where Fi, i = 1, 2, 3, are smooth functions and ui are parameters of the families.Eliminating the variables x and y from (1.1), we get the equation of the web W

F (u1, u2, u3) = 0. (1.2)

The function F is called the web function. Note that all the variables are essentiallyinvolved in F .

Equation (1.2) determines the image of the original three-web W in the spaceof parameters. For example, by narrowing the range of the parameter u3, the webequation (1.2) can be written as follows:

u3 = f(u1, u2). (1.3)

On the two-dimensional manifold with local coordinates u1 and u2, equation (1.3)defines a three-web W ′ that is equivalent to the original three-web W . Thefoliations of the web W ′ are given by the equations u1 = const, u2 = const,


u3 = f(u1, u2) = const. Equations (1.2) and (1.3) are not defined here uniquely,but up to the following changes of the parameters

u1 → α(u1), u2 → β(u2), u3 → γ(u3), (1.4)

where α, β, γ are local diffeomorphisms.It is known (see, for example, [2]) that if the web equation (1.2) is polylinear,

then this web is regular. Hence, the regularity condition of an arbitrary web W canbe phrased as follows: a web W is regular if and only if equation (1.2) is reducibleto the form

Au1u2u3 + B1u2u3 + B2u1u3 + B3u1u2 + C1u1 + C2u2 + C3u3 + D = 0

by transformations of the form (1.4). In this case we shall say that the transform-ation ui → α(ui) linearizes the equation (1.2) in the variable ui.

Theorem 1.1. Equation (1.2) can be linearized in the variable u1 by a transfor-mation of the form u1 = α(u1) if and only if F is the composition

F (u1, u2, u3) = F (u1, ϕ(u2, u3)). (1.5)

Proof. Assume that a substitution u1 = α(u1) linearizes equation (1.2), that is, theequation F (α(u1), u2, u3) = 0 is linear in u1. From this equation, u1 = ϕ(u2, u3),whence u1 = α(ϕ(u2, u3)). This means that (1.5) holds. Conversely, if (1.5) holds,then F = 0 implies that ϕ(u2, u3) = α(u1) ≡ u1.

Hereafter we shall be concerned with the webs for which F is a polynomial. Wethen write equation (1.2) as follows:

un1P0 + un−1

1 P1 + · · ·+ Pn = 0, (1.6)

here the Pi are polynomials of the variables u2 and u3. The polynomials Pi aredefined up to a common factor, which is a linear fractional function λ of variablesu2 and u3. By Theorem 1.1, the linearization condition in u1 means that thepolynomials Pi are, up to a factor, polynomials of some fractional linear functionϕ = U/V of variables u2 and u3: λPi = ai0(U/V )k + ai1(U/V )k−1 + · · ·+ aik, andhence

Pi = ai0(U)k + ai1(U)k−1V + · · ·+ aikV k. (1.7)

Conversely: if the polynomials Pi are as in (1.7), then equation (1.6) is solvablewith respect to ϕ = U/V : U/V = α(u1) ≡ u1. This establishes the following

Theorem 1.2. The algebraic equation (1.6) is linearizable in the variable u1 bya substitution u1 = α(u1) if and only if the polynomials Pi are as in (1.7).

In the space of variables u2 and u3, the web W ′, as defined by equation (1.6)under conditions (1.7), consists of the Cartesian net u2 = const, u3 = const anda family of algebraic curves, as defined by equation (1.6), where u1 is the parameterof the family. The base curves of the family are defined by the equations Pi = 0. Theleft-hand side of equality (1.6) factors (over the field R) into the product of linearand quadratic factors of the forms, aU + bV and aU2 + bUV + cV 2, respectively, sothat each of the base curves is a union of algebraic curves of the form aU + bV = 0lying in the same pencil with base U = 0 and V = 0.


Corollary 1. If the Pi are second-degree polynomials, then the following twovariants of linearization of equation (1.6) are possible.

A) The first linearization condition : U and V are second-degree polynomials.Then the polynomials Pi are their linear combinations, that is, they lie in the pencilwith base U , V . In this case, equation (1.6) is linearizable in the variable u1.

B) The second linearization condition : U and V are first-degree polynomials.Then each of the second order curves Pi = 0 decomposes into a pair of straight lineslying in one pencil. In this case, equation (1.6) is reducible to the form U/V = u1.This equation is polylinear in the variables u1, u2, u3, and hence, the web definedby it is regular.

§ 2. Constructions on a cubic surface related to the Burau three-webs

2.1. For future purposes we point out some simple properties of cubic surfacesin the projective space. Recall that the three-dimensional projective space P3 isa four-dimensional vector space L4 with equivalence relation a ∼ λa. To the basisvectors eα from L4 there correspond in P3 the points of the projective frame, whichare denoted by Aα, α = 1, 2, 3, 4. An arbitrary point M of the projective spacemay be written as M = xαAα, the projective homogeneous coordinates xα beingdefined up to a nonzero multiple.

The equation of an arbitrary cubic V in some projective basis Aα will be writtenas follows:

aαβγxαxβxγ = 0, (2.1)

where the xα are the projective coordinates, α, β, γ = 1, 2, 3, 4. The following resultis immediate.

Property 1. A vertex Aα of the projective frame lies on the cubic V if and only ifaααα = 0; it is a singular point of the cubic V if and only if aααβ = 0, β = 1, 2, 3, 4.

In what follows, the straight lines on V will be referred to as generators; theymay either be real or imaginary.

Property 2. Let ℓ1 and ℓ2 be two skew generators of the cubic V . Then on Vthere exist 5 generators that intersect ℓ1 and ℓ2.

Proof. We place the vertices A1 and A2 of the projective frame on ℓ1, and A3 andA4 on ℓ2. If these lines lie on V , then

a111 = a112 = a122 = a222 = 0, a333 = a334 = a344 = a444 = 0. (2.2)

Consider the line ℓ3 = A1 + aA2, A3 + bA4, which intersects ℓ1 and ℓ2. We shallrequire that any point M = A1 + aA2 + t(A3 + bA4) of this line lies on V . Then,in view of the relations (2.2), we obtain the identity in t

a113 + a114b + a133t + a233at + a223a2 + a224a

2b + a144b2t

+ a244ab2t + 2a123a + 2a124ab + 2a134bt + 2a234abt ≡ 0,


whence follow the relations

a113 + a114b + a223a2 + a224a

2b + 2a123a + 2a124ab = 0,

a133 + a233a + a144b2 + a244ab2 + 2a134b + 2a234ab = 0.

Eliminating b, we obtain a fifth-order equation in a.

Corollary 2. If ℓ1 and ℓ2 are two skew generators of the cubic V , then V containsat least one real generator that crosses ℓ1 and ℓ2.

One may immediately verify the following result.

Property 3. Assume that three noncoplanar generators of the cubic V pass throughthe point M . Then M is a singular point.

Property 4. Let M be a singular point of the cubic V . Thena) there pass through M six (not necessarily real) generators of this cubic lying

in the cone of tangents to V at the point M ;b) if V contains yet another generator not passing through M , then it intersects

some two of the above six generators.

Proof. We place the vertex A1 of the frame at the singular point M . Then, byProperty 1, a11α = 0, α = 1, 2, 3, 4, and the equation of the cubic becomes linearin x1:

Φ3(A1) + x1Φ2(A1) = 0, (2.3)

where Φ3(A1) and Φ2(A1) are, respectively, the cubic and quadratic forms depend-ing on the variables x2, x3, x4. The equations Φ3(A1) = 0 and Φ2(A1) = 0 define,respectively, the cones of third and second order with vertex A1. These cones inter-sect in six straight lines, and it is easy to see that each of these lines lies on thesurface V .

One may immediately verify that the cone Φ2(A1) = 0 (for brevity we shallsometimes write: the cone Φ2(A1)) is formed by tangents at the point A1 to curveslying on V . If this cone is nondegenerate, then the singular point A1 is calleda conic knot; if it decomposes into a pair of planes, then the singular point A1 iscalled a biplanar knot; if these planes coincide, then it is called a uniplanar knot.

Let us prove assertion b). Assume that the line A2A3 lies on V . Then a222 =a223 = a233 = a333 = 0. In view of these relations, Φ3(A1) = x4Φ2(A1); that is,the third-order cone Φ3(A1) decomposes into a plane and a second-order cone. Theequation of the cubic V now becomes

x4Φ2(A1) + x1Φ2(A1) = 0.

The plane x4 = 0, which contains the singular point A1 and the generator A2A3,intersects the cubic V in the line x4 = 0, Φ2(A1) = 0, or

x4 = 0, a122(x2)2

+ 2a123x2x3 + a133(x3)

2= 0.

This system defines two straight lines (not necessary real).

Note that if the discriminant of the last equation vanishes, then the cones Φ3(A1)and Φ2(A1) touch along the generator. In this case, there pass through the singularpoint A1 a total of 5 generators (one of which is double).


Property 5. If A and B are singular points of the cubic V , then the line ABlies on V and is a double common generator of the cones of tangents Φ2(A) andΦ2(B). Through each of the singular points A and B there pass five generators ofthe cubic V , including the common double generator AB.

Proof. Let A1 and A2 be singular points on V . In view of Property 1, the equationof the cubic has the form

6a123x1x2x3 + 6a124x

1x2x4 + 6a134x1x3x4 + 6a234x

2x3x4

+ 3a133x1(x3)2 + 3a233x

2(x3)2 + 3a144x1(x4)2 + 3a244x

2(x4)2

+ 3a334x4(x3)2 + 3a344x

3(x4)2 + a333(x3)3 + a444(x3)3 = 0.

We see that the equations x3 = x4 = 0 of the line A1A2 satisfy the equation of thecubic.

Further, writing the equation of the cubic V in the form (2.3), we find that thecones of tangents Φ2(A1) and Φ2(A2) at the points A1 and A2 are given, respec-tively, by the equations

2a123x2x3 + 2a124x

2x4 + 2a134x3x4 + a133(x3)2 + a144(x4)2 = 0,

2a123x1x3 + 2a124x

1x4 + 2a234x3x4 + a233(x3)2 + a244(x4)2 = 0.

At an arbitrary point of the common generator A1A2, the gradients of these conesare proportional to the direction (0, 0, a123, a124). Hence, the common generatorA1A2 is double, and, if the cones do not decompose, this means that they touchalong A1A2.

Finally, let us find the intersection of the cubic cones Φ3(A1) and Φ3(A2) withcommon tangent plane a123x

3 + a124x4 = 0 of quadratic cones. Solving the cor-

responding systems, we arrive in both cases at the multiple solution x3 = x4 = 0.This means that in both cases the line A1A2 is the double intersection of the cubicand quadratic cones, that is, among the six generators passing through the pointA1 or A2, one is double.

One may immediately verify the following result.

Property 6. Let ℓ1, ℓ2 and ℓ3 be the generators of the cubic V , where ℓ1 and ℓ2intersect, and ℓ3 does not intersect ℓ1 and ℓ2. Then the third generator which liesin the plane of the lines ℓ1 and ℓ2 intersects ℓ3.

Property 7. Let S be a singular point of the cubic V and ℓ be a generator of Vthat does not pass through S. We let m1, m2, . . . ,m6 denote the generators of Vthat pass through S (see Property 4); let m5 and m6 be the generators which (inview of Property 4) intersect the generator ℓ. Then the conics cut out on V bythe planes of the pencil with axis ℓ by projection onto the plane from the point S,are mapped to the pencil of conics that pass through the four points m1, m2, m3,m4, which are the images of the lines m1, m2, m3, m4. In particular, this pencilcontains two conics, of which one is defined by the cone of tangents Φ2(S), and theother by the cone Φ2(S) (Property 4).

Proof. The generators m1, m2, . . . ,m6 intersect all the planes of the pencil withaxis ℓ, the first four of these intersecting all the conics that are cut out by these


planes on V . When projecting from the point S, all intersection points that lie onone generator are projected into the same point. Hence, the images of all the conicspass through the points m1, m2, m3, m4.

In view of Property 4, the cubic cone Φ3(S) decomposes into a plane and thesecond-order cone Φ2(S), which also contains the generators m1, m2, m3 and m4.Hence its image also passes through these points.

2.2. To end this section, let us return to the Burau three-web W generated by thepairwise skew generators ℓi of the cubic V . We claim that the web W is regular.

We place the vertices A1, A2 of the projective frame on the generator ℓ1 anddraw through them the straight lines t1 and t2 intersecting ℓ2 and ℓ3. We place,respectively, the vertices A3 and A4 of the frame at the points of intersection of thelines t1 and t2 with the generator ℓ2. Further, we normalize the frame so that thepoints of intersection of the lines t1 and t2 with the generator ℓ3 assume the formsA1 +A3 and A2 +A4, respectively. Requiring that the lines ℓ1 = A1A2, ℓ2 = A3A4,ℓ3 = A1 + A3, A2 + A4 lie on the cubic, we find the equation of the cubic V :

a113x1x3(x1 − x3) + a114(x1)2x4 + 2a123x

1x2x3 + 2a124x1x2x4

+ 2a134x1x3x4 + a144x

1(x4)2 − (a144 + 2a124 + 2a234)(x2)2x3

+ a224x2x4(x2 − x4)− (a114 + 2a123 + 2a134)x2(x3)2 + 2a234x

2x3x4 = 0.

We define the pencils of the planes Li with axes ℓi by the equations

L1 : x3 = λx4, L2 : x1 = µx2, L3 : x1 − x2 = ν(x3 − x4),

where λ, µ and ν are the parameters of the pencils. These parameters are alsoparameters of the conics that are cut out by the planes of the pencils on the cubicsurface. Eliminating the variables xα from the equation of the cubic, a little manip-ulation yields the equation of the Burau three-web in question:

a114(−λµ + νµ + νλ) + 2a123λν − 2a134λ(µ− ν)− 2a124(λ− ν)+ a144(ν − λ− ν)− 2a234λ− a113λν + a224 = 0.

We see that this equation is polylinear with respect to all parameters. Hence, theweb is regular.

In what follows we shall refer to the properties of the cubic surface listed in thissection without indication of the section number.

§ 3. Regular special Burau three-webs of type W1

and perspectively equivalent webs of type W1.Case 1: two of the generators ℓi have a common point

3.1. Basic webs of type W1. Assume that the generators ℓ1 and ℓ2 intersect atthe point A4 and lie in the plane π. We choose the frame as follows: ℓ1 = A1A4,ℓ2 = A2A4, ℓ3 = A3, A1 + A2. Notice that the line ℓ3 meets the plane π at thepoint A1 + A2. The special Burau webs with this arrangement of the lines ℓi willbe denoted by the symbol W1.


In this frame the equation of the cubic surface is as follows:

(x1 − x2)(a112x1x2 + a113x

1x3 − a223x2x3 + a133(x3)2) + 2a124x

1x2x4

+ 2a134x1x3x4 + 2a234x

2x3x4 + a334(x3)2x4 + a344x3x42

= 0. (3.1)

We define the pencils of planes Li by the equations

L1 : x2 = λx3, L2 : x1 = µx3, L3 : x4 = ν(x1 − x2). (3.2)

Eliminating the variables xα from (3.1) and (3.2) gives the equation of the three-webW1 in question:

a344ν2(µ− λ) + 2a124λµν + a112λµ + 2a134µν + 2a234λν

+ a113µ− a223λ + a334ν + a133 = 0. (3.3)

Equation (3.1) is linear in the parameters λ and µ. We rewrite this equation in theform ν2P + νQ + R = 0, where

P ≡ a344(µ− λ), Q ≡ 2a124λµ + a134µ + 2a234λ + a334,

R ≡ a112λµ + a113µ− a223λ + a133.

By the first linearization condition (Corollary 1), the three-web under considerationis regular if the polynomials P , Q, R belong to one pencil, that is,

rank

0 a344 −a344 02a124 2a134 2a234 a334

a112 a114 −a223 a133

= 2.

Two cases are possible: eithera344 = 0 (3.4)

or

rank(

2a124 2a134 + 2a234 a334

a112 a114 − a223 a133

)= 1. (3.5)

Proposition 3.1. The point A4 of intersection of the lines ℓ1 and ℓ2 is a singularpoint of the cubic V if and only if relation (3.4) holds. The line ℓ3 has two singularpoints if and only if condition (3.5) is satisfied.

Proof. 1) The first part of this result is a direct consequence of Property 1. We letW11 denote the webs defined by relation (3.4).

2) The singular points of the surface V are given by the system

a112x2(2x1 − x2) + a113x

3(2x1 − x2)− a223x2x3 + a133(x3)2

+ 2a124x2x4 + 2a134x

3x4 = 0,

a112x1(x1 − 2x2)− a113x

1x3 + a223x3(2x2 − x1)− a133(x3)2

+ 2a124x1x4 + 2a234x

3x4 = 0,

(x1 − x2)(a113x1 − a223x

2 + 2a133x3) + 2a134x

1x4 + 2a234x2x4

+ 2a334x3x4 + a344(x4)2 = 0,

2a124x1x2 + 2a134x

1x3 + 2a234x2x3 + a334(x3)2 + 2a344x

3x4 = 0.

(3.6)


Substituting the equations x1 = x2, x4 = 0 of the line ℓ3 = A3, A1 + A2 we obtainonly 2 equations

a112(x1)2 + (a113 − a223)x1x3 + a133(x3)2 = 0,

2a124(x1)2 + 2(a134 + a234)x1x3 + a334(x3)2 = 0.(3.7)

In view of conditions (3.5) these equations coincide.

Let us examine the second case in more detail. To begin with, we note thatat least one of the singular points, as defined by system (3.7), does not lie in theplane π, for otherwise the cubic V decomposes into a plane and a quadric. Hence,one of the singular points may be placed at the vertex A3 of the frame. But then(see Property 1)

a133 = a334 = 0. (3.8)

There are two geometrically distinct subcases.Subcase 1: the generator ℓ of the cubic V , a112(x1 − x2) + 2a124x

4 = 0, whichintersects all the lines ℓi and lies in the plane π (see Property 6), does not passthrough the point A4. Then we may place the vertices A1 and A2 of the frame on ℓ,which gives

a112 = 0. (3.9)

Using the regularity condition (3.5), this gives a124(a113 − a223) = 0. If a124 = 0,then the cubic V decomposes. Hence,

a113 − a223 = 0. (3.10)

We let W12 denote the regular webs defined by the relations (3.8)–(3.10).Subcase 2: the generator ℓ passes through the point A4. In this case,

a124 = 0 (3.11)

and the equation of the line ℓ assumes the form x1 − x2 = 0, this is the lineA4, A1 + A2. Thus, in the case in question there pass through the point A4 threegenerators of the conic V that lie in one plane π. Using the regularity condition(3.5), we have

a134 + a234 = 0. (3.12)

We let W13 denote the webs defined by the relations (3.8), (3.11), (3.12).In view of these relations, equation (3.3) of the web W13 reads as follows:

a344ν2(µ− λ) + 2νa134(µ− λ) + a112λµ + a113µ− a223λ = 0. (3.13)

This equation can be simplified if points A1 and A2 (they are still not fixed) arechosen to be polar-conjugate to the point A4 with respect to the cone of tangentsΦ2(A3) to the surface V at the singular point A3. Hence, the following relationholds:

a134 = 0. (3.14)

We let W ′12 denote the webs W12 for which condition (3.4) also holds. Thus, the

web W ′12 is defined by the relations (3.4), (3.8)–(3.10). These webs are characterized


by the fact that the cubic V contains only three singular points: A4 and two pointson the line ℓ3 — A3 and M(a134 + a234, a134 + a234,−a124, 0). We let W ′′

12 denotethe webs W ′

12 for which the relation (3.12) also holds. In this case, A3 ≡ M andthere exists one double singular point on the line ℓ3.

We claim that the second linearization condition (see Corollary 1) does not leadto new classes of webs. Indeed, in this case the polynomials Q and R (see above)must decompose into pairs of lines with a common intersection point. But they maydecompose only into lines of the form λ = const and µ = const, and hence if theircentres coincide, then the lines themselves coincide. It follows that the polynomialsQ and R are proportional, so that all the polynomials P , Q, R belong to one pencil,and we thus arrive at the case (3.5) already examined.

Consider the second linearization condition, by rewriting equation (3.3) in termsof the powers of µ: µP + Q = 0, where

P = µ(a344ν2 + 2a124λν + 2a134ν + a113),

Q = −a344λν2 + 2a234λν − a223λ + a334ν + a133.

Elementary arguments show that these polynomials may be written in the form(1.7) only when a344 = 0. Assume further that, in accordance with the linearizationcondition, the curves P = 0 and Q = 0 decompose. In view of the structure of thepolynomials P and Q, each of these must have a component of the form aλ + b.But since the decomposed curves must have a common centre, these componentsare proportional. It follows that the web equation (3.3), and hence, the equationof the cubic V , decomposes into the product of two factors, and thus the cubic Vdecomposes. We obtain the same result by rewriting the web equation (3.3) withrespect to the powers of λ.

The following theorem summarizes the results obtained.

Theorem 3.1. Let V be a cubic surface defined by equation (2.1). Assume that Vdoes not decompose into a plane and a quadric. Let ℓi be 3 straight lines on V , ofwhich ℓ1 and ℓ2 have a common point, and ℓ3 does not intersect ℓ1 and ℓ2; also letW1 be the special Burau three-web defined by these lines. We choose a projectiveframe such that ℓ1 = A1A4, ℓ2 = A2A4, ℓ3 = A3, A1 + A2. Then the equationof the cubic V assumes the form (3.1), and the equation of the three-web W1, theform (3.3). There exist 5 classes of regular three-webs of type W1: W11, W12, W13,W ′

12 and W ′′12, as singled out in the given frame by, respectively, the relations : (3.4);

(3.8)–(3.10); (3.8), (3.11), (3.12) and (3.14); (3.4), (3.8)–(3.10); (3.4), (3.8)–(3.10)and (3.12).

The cubic V carrying the web W11 is characterized by the fact that the point A4

of intersection of the lines ℓ1 and ℓ2 is a singular point of the cubic V ; the cubicV carrying the web W12 or the web W13 is characterized by the fact that the lineℓ3 contains 2 singular points. Moreover, for the webs W13, there passes throughthe point of intersection of the generators ℓ1 and ℓ2 yet another generator of thecubic V lying in the same plane as the lines ℓ1 and ℓ2; the cubic V carrying theweb W ′

12 is characterized by the fact that it contains in total 3 singular points : A4

and 2 points on the line ℓ3; the cubic V carrying the web W ′′12 is characterized by

the fact that it contains in total 2 singular points : A4 and the double point A3.


Notice that if on the web W12 one of the singular points lying on the line ℓ3 isthe point A1 + A2 = ℓ3 ∩ π, then a124 = 0 and so the cubic V decomposes.

These 5 classes of regular special Burau three-webs of type W1, as well as theperspectively equivalent webs of type W1 (see below), will be referred to as the basicclasses of type W1 or, respectively, of type W1. A detailed projective classificationof the webs W1 will be given in § 6. In the following sections we shall stick toanalogous definitions and notation.

3.2. Basic webs of type W1.

2-1. The web W11 is obtained by projecting the web W11 from the singular point A4

onto the plane A1A2A3. The planes passing through the lines ℓ1 = A1A4 andℓ2 = A2A4 will project into the pencils of straight lines with vertices A1 and A2

(the families λ1 and λ2, respectively). In this case, the equation of the cubic V

reads as follows: (x1 − x2)Φ2(A4) + x4Φ2(A4) = 0, or

(x1 − x2)(a112x1x2 + a113x

1x3 − a223x2x3 + a133(x3)2)

+ x4(2a124x1x2 + 2a134x

1x3 + 2a234x2x3 + a334(x3)2) = 0. (3.15)

The equation of the third family of the web W11 is obtained by eliminating thevariable x4 from equation (3.15) and the third equation in (3.2):

a112x1x2 + a113x

1x3 − a223x2x3 + a133(x3)2

+ ν(2a124x1x2 + 2a134x

1x3 + 2a234x2x3 + a334(x3)2) = 0.

We see that the family λ3 of the web W11 is the pencil of conics with base conicsΦ2(A4) = 0 and Φ2(A4) = 0, which intersect at the points A1, A2, m3 and m4

which correspond to the common generators of the cones Φ2(A4) and Φ2(A4) (seeProperty 7). Thus, the web W11 is formed by the pencil of conics having at least2 real vertices A1 and A2, and two pencils of lines with vertices at these points.

2-2. The web W12 is obtained by projecting the web W12 from the singular pointA3 onto the plane π = A1A2A4. In view of (3.8)–(3.10), the equation (3.1) of thecubic V becomes x3Φ2(A3) + Φ3(A3) = 0, or

x3(a113(x1−x2)2 +2a134x1x4 +2a234x

2x4 + a344(x4)2)+2a124x1x2x4 = 0. (3.16)

Eliminating the variable x3 by using the first or second equations in (3.2), we obtainthe equations of the first two families of the web W12:

λ1 : Φ2(A3) + 2λa124x2x4 = 0, λ2 : Φ2(A3) + 2µa124x

1x4 = 0. (3.17)

The family λ3 is obtained by projecting the pencil of planes passing through theline A3, A1 + A2, from the point A3 onto the plane containing the point A1 + A2.Hence, λ3 is the pencil of straight lines with vertex A1 + A2.

We see that the first family is the pencil of conics with base Φ2(A3) = 0 andx2x4 = 0, and the second is the pencil of conics with base Φ2(A3) = 0 and x1x4 = 0.The vertices of the first pencil are as follows: A1 + A2 (a double point) and the


points of intersection of the conic Φ2(A3) = 0 with the line x2 = 0; we denotethe latter points by m1, m2. Hence, the family λ1 is a parabolic pencil of conics(a P -pencil) with vertices m3, m4 and double vertex A1 + A2.

Similarly, the family λ2 of lines of the web W12 is also the P -pencil of conicswith double vertex A1 +A2 and simple vertices m1, m2. The conics of both pencilshave the same tangent A1A2 at the point A1 + A2 (Fig. 2).

Figure 2

It is easily verified that the equation Φ2(A3) = 0 defines an arbitrary conic ofthe plane passing through the point A1 + A2 and touching the line A1A2 at thispoint. Consequently, the above properties completely characterize the web W12.

Remark. The line ℓ3 contains, in addition to A3, yet another singular point S,whose coordinates are found from the second equation of (3.7) (in view of (3.4)):S(a134 +a234, a134 +a234,−a124, 0). One may show that the three-web, as obtainedby projecting the web W12 from the singular point S, is projectively equivalent tothe web W12.

2-3. We let W ′12 denote the three-web obtained by projecting the web W ′

12 from thesingular point A3 to the plane A1A2A4. The families of lines of this web will be thesame as in the previous subsection, but now taking into account the relation (3.4).Considering this, the conics of both pencils λ1 and λ2 pass through the point A4.This property characterizes the webs W ′

12.

2-4. When projecting the web W ′12 from the singular point A4 onto the plane

A1A2A3 we obtain the web W11 formed by the pencil of conics having at least2 real vertices A1 and A2, and two pencils of lines with vertices at these points.

2-5. Consider the web W obtained by projecting W ′12 from the singular point

M(a134 +a234, a134 +a234,−a124, 0) onto the plane A1A2A4. We change to the newprojective frame {A1, A2, A3 ≡ M,A4}, the coordinates transforming as follows:

x1 = x1 + (a134 + a234)x3, x2 = x2 + (a134 + a234)x3,

x3 = −a124x3, x4 = x4.


In the new coordinates the equations of the first two families of the web W assumethe form

a113(x1 − x2)2 + 2a134x4(x1 − x2) + 2λa124x

1x4 = 0,

a113(x1 − x2)2 + 2a234x4(x2 − x1) + 2µa124x

2x4 = 0,

and the family λ3 is the pencil of lines with vertex A1 + A2.As in case 2-4, the first two families of lines of the web W are parabolic pencils

of conics with common double vertex A1 + A2, and the conics of both pencils passthrough the point A4. Hence, W ≡ W ′

12.Note that for the webs W ′′

12 (adding condition (3.12)) we obtain the same typesof webs W1 as for the webs W ′

12.

2-6. The web W13 is obtained by projecting the web W13 from the singular pointA3 onto the plane A1A2A4. By virtue of (3.8), (3.11) and (3.12), the equation (3.1)of the cubic V assumes the form

x3Φ2(A3) + a112x1x2(x1 − x2) = 0, (3.18)

where the cone Φ2(A3) is defined by the equation

(x1 − x2)(a113x1 − a223x

2 + 2a134x4) + a344(x4)2 = 0. (3.19)

The cone Φ2(A3) is an arbitrary cone with vertex A3 that contains the line ℓ3 =A3, A1 +A2, and the tangent plane to this cone along this generator passes throughthe point A4. The web W13, as well as the web W12, consists of two parabolicpencils of conics with the same tangent A1 + A2, A4 at the common double vertexA1 + A2 and the pencil of straight lines with vertex at this point. The pencils ofconics have the same base conic Φ2(A3). The difference from the webs W12 is thatthe lines ℓ1 = A1A4 and ℓ2 = A2A4 meet the common tangent of the conics ofpencils at the point A4.

Remark. If we project the three-web W13 from the singular point S, then we againobtain the web of type W13.

So, in total there are 4 basic classes of regular webs of type W1: these are thewebs W11, W12, W ′

12 and W13.

§ 4. Regular special Burau three-webs of typeW2 and projectively equivalent webs of type W2.

Case 2: the generator ℓ2 intersects ℓ1 and ℓ3

4.1. Basic webs of type W2. Consider the frame ℓ1 = A3A1, ℓ2 = A1A2,ℓ3 = A2A4. Equation (2.1) of the cubic V containing the lines ℓi reads:

a114(x1)2x4 + 2a123x1x2x3 + 2a124x

1x2x4

+ 2a134x1x3x4 + a144x

1(x4)2 + a223(x2)2x3

+ a233x2(x3)2 + 2a234x

2x3x4 + a334(x3)2x4 + a344x3(x4)2 = 0. (4.1)


The pencils of planes Li are defined by the equations

L1 : x2 = λx4; L2 : x3 = µx4; L3 : x1 = νx3.

Eliminating the variables xα from the equation of the cubic and the pencils of theplanes Li we arrive at the equation of the Burau web W2

a114µν2 + ν(2a123λµ + 2a124λ + 2a134µ + a144)

+ a223λ2 + a233λµ + 2a234λ + a334µ + a344 = 0. (4.2)

This equation is linear in µ. The first linearization condition in ν (see Corollary toTheorem 1.2) gives the condition

rank

a223 a233 2a234 a334 a344

0 2a123 2a124 2a134 a144

0 0 0 a114 0

= 2. (4.3)

Rewriting equation (4.2) in the form λ2P + λQ + R = 0, we obtain a similarlinearization condition in λ

rank

a114 2a134 a334 a144 a344

0 2a123 a233 2a124 2a234

0 0 0 0 a223

= 2. (4.4)

So, excluding the cases when the cubic V decomposes, we find the regularity con-ditions.

1) The web W21:a223 = a114 = 0. (4.5)

2) Assume, for example,

a223 = 0, a114 = 0. (4.6)

(In the symmetric case a223 = 0, a114 = 0, the geometry is analogous and hencethis case is eliminated from consideration.)

Condition (4.4) is satisfied, in view of (4.6), and condition (4.3) leads to thefollowing variants:

2.1) the web W22:a234

a124=

a344

a144=

a233

2a123≡ s; (4.7)

2.2) the web W25:a124 = a144 = a123 = 0; (4.8)

2.3) the web W26:

a124 = a234 = 0,a344

a144=

a233

2a123≡ s; (4.9)

2.4) the web W27:

a124 = a234 = 0, a144 = a123 = 0; (4.10)


2.5) the web W28:

a144 = a344 = 0,a234

a124=

a233

2a123≡ s; (4.11)

2.6) the web W29:

a144 = a344 = 0, a123 = a124 = 0. (4.12)

3) Now assume thata223 = 0, a114 = 0. (4.13)

Then conditions (4.3) and (4.4) yield the following relations:

a123 = 0, a233 = 0, a124 = 0, a144 = 0. (4.14)

We let W23 denote the corresponding regular web.We now employ the second linearization condition (see Corollary 1) to find classes

of regular webs. Assume that the web equation is written in the form (4.2). Bythe indicated condition, the web is regular, provided the equations P ≡ a114µ = 0,Q ≡ 2a123λµ + 2a124λ + 2a134µ + a144 = 0, R ≡ a223λ

2 + a233λµ + 2a234λ +a334µ + a344 = 0 define three degenerate second order curves with common centre.If a114 = 0, then the curve P = 0 decomposes into the lines µ = 0 and ℓ∞. Hence,the centre of the curve lies on ℓ∞, and so the two remaining curves Q = 0 andR = 0 must decompose only into lines of the form µ = const. This gives a123 = 0,a124 = 0, a223 = 0, a233 = 0, a234 = 0, and so the cubic (4.1) decomposes.

If a114 = 0, then each of the curves Q = 0 and R = 0 contains (in the case that itdecomposes) a factor of the form aλ + b = 0. But since the centres of these curvesmust coincide, these factors must be proportional. It follows that the left-handside of equation (4.2), and hence of equation (4.1), factorizes and so the cubic Vdecomposes.

We now rewrite equation (4.2) in powers of λ:

a223λ2 + λ(2a123µν + 2a124ν + a233µ + 2a234)

+ a114µν2 + 2a134µν + a144ν + a334µ + a344 = 0,

and again apply the second linearization condition. Here, P = a223 is a polynomialof zero degree, say U (see (1.7)), and Q = 2a123µν + 2a124ν + a233µ + 2a234 isa polynomial of second degree, say V . Then, the polynomial R = a114µν2 +2a134µν + a144ν + a334µ + a344 must be expressed in terms of U and V by formula(1.7), which is possible only when a114 = 0 and the polynomials Q and R lie in onepencil. But then we arrive at the first linearization condition.

Finally, we rewrite equation (4.2) in the form µQ + R = 0, where

Q ≡ a114ν2 + 2a123λν + 2a134ν + a233λ + a334 = 0,

R ≡ a223λ2 + 2a124λν + 2a234λ + a144ν + a344 = 0.

(4.15)

According to the second linearization condition, a web is regular if the conicsQ = 0, R = 0 are pairs of lines with a common intersection point. This leads us to


the following conditions:

a233 = −2aa123, a144 = −2ba124,

a134 = −aa114 − ba123, a234 = −ba223 − aa124,

a334 = a2a114 + 2aba123, a344 = b2a223 + 2aba124.

(4.16)

We let W24 denote the corresponding regular web.

4.2. Characterization of cubics carrying regular webs of type W2 interms of singularities.

2-1. The cubic surfaces carrying the web W21 are characterized by the fact thatthe points A1 and A2 at which the lines ℓi meet are singular. It is easily verifiedthat these singular points are, in general, conic knots.

2-2. The following result holds.

Proposition 4.1. The cubic V carrying the web W22 has two singular points onthe line ℓ3, of which one point (the point A2) is a biplanar knot. The tangentplanes π1 and π2, into which the cone of tangents at the point A2 decomposes, havethe following properties : one of these contains the line ℓ2, and the other one, theline ℓ3. Taken together the above properties are characteristic.

Proof. That the point A2 is singular follows from (4.6) and Property 1. In order tofind the remaining singular points, consider the system defining the singular pointsof the cubic (4.1):

2a114x1x4 + 2a123x

2x3 + 2a124x2x4 + 2a134x

3x4 + a144(x4)2 = 0,

2a123x1x3 + 2a124x

1x4 + 2a223x2x3 + a233(x3)2 + 2a234x

3x4 = 0,

2a123x1x2 + 2a134x

1x4 + a223(x2)2 + 2a233x2x3 + 2a234x

2x4

+ 2a334x3x4 + a344(x4)2 = 0,

a114(x1)2 + 2a124x1x2 + 2a134x

1x3 + 2a144x1x4 + 2a234x

2x3

+ a334(x3)2 + 2a344x3x4 = 0.

(4.17)

Letting x1 = x3 = 0, we obtain equations for the coordinates of the singular pointslying on the line ℓ3 = A2A4. In view of (4.6) and (4.7) we obtain the single equation2a124x

2x4 +a144(x4)2 = 0. If x4 = 0, we have the point A2; if 2a124x2 +a144x

4 = 0,we obtain yet another singular point. Note that a124 = 0, for otherwise (4.7) wouldgive a234 = 0 and so we arrive at the webs W26. Hence, the second singular pointis distinct from A2. Placing the vertex A4 of the frame at this point, we obtaina144 = a344 = 0 by Property 1, and so the relations (4.7) are mapped into therelations (4.11) that define the webs W28. Thus, the web W28 is geometricallyequivalent to the web W22; however, they are analytically different due to thedistinct position of the point A3.

As the result of this canonization the equation of the cubic V assumes the formΦ3(A4) + x2Φ2(A2) = 0, where

Φ2(A2) = 2(x1 + sx3)(a123x3 + a124x

4),

Φ3(A4) = x4(a114(x1)2 + 2a134x1x3 + a334(x3)2).

(4.18)


The cone of tangents Φ2(A2) is seen to decompose into two planes: π1 (x1+sx3 = 0)and π2 (a123x

3 + a124x4 = 0). Of these planes, the first contains the line ℓ3, and

the second, the line ℓ2.It remains to shows that there are no other singular points on the cubic V . To

start with, we make a further specialization of the frame by placing the vertex A3

of the frame at the point where the line ℓ1 meets the plane π1. Then s = 0 and therelations (4.6) and (4.7) that define the web W22 in question assume the followingsimple form:

a223 = a144 = a233 = a234 = a344 = 0. (4.7′)

It is now easily verified that the cubic V contains no other singular points, exceptfor the A2 and A4 already found.

Conversely, assume that the surface V is given by the equation (4.1), and thaton the line ℓ3 there are two real singular points, of which one (the point A2) isa biplanar knot. We place the vertex A4 of the frame at the second point. Thena223 = a144 = a344 = 0 by Property 1. Under these conditions,

Φ2(A2) = 2a123x1x3 + 2a124x

1x4 + a233(x3)2 + 2a234x3x4.

Assume that the singular point A2 is a biplanar knot. Then the cone Φ2(A2)decomposes, which gives the relation

a124(a124a233 − 2a123a234) = 0.

If a124 = 0, then the cone Φ2 decomposes into two planes, of which one (x3 = 0)contains the line ℓ2, and the other does not in general contain the lines ℓ2 and ℓ3.

If the second factor is zero, we obtain the relation (4.6), the cone Φ2(A2) decom-posing into two planes, of which one contains the line ℓ2, and the other, the line ℓ3.

2-3. In view of the relations (4.14) that characterize the webs W23, the system(4.17), which determines the singular points of the cubic V , assumes the form

(a114x1 + a134x

3)x4 = 0, (a223x2 + a234x

4)x3 = 0,

a134x1x4 + a223(x2)2 + 2a234x

2x4 + 2a334x3x4 + a344(x4)2 = 0,

a114(x1)2 + 2a134x1x3 + 2a234x

2x3 + a334(x3)2 + 2a344x3x4 = 0.

This system has four solutions: two singular points on ℓ3, defined by the equationa223(x2)2 + 2a234x

2x4 + a344(x4)2 = 0, and two singular points on the line ℓ1,defined by the equation a114(x1)2 + 2a134x

1x3 + a334(x3)2 = 0. Also, in view ofinequalities (4.17), none of these points coincides with the points A1 and A2. Simplecalculations show that this property characterizes the case in question.

By placing the vertices A3 and A4 of the frame at singular points, we obtaina334 = a344 = 0. Thus, in the canonical frame the web W23 is determined by therelations

a123 = a124 = a144 = a233 = a334 = a344 = 0. (4.14′)


2-4. In view of the relations (4.16) characterizing the webs W24, the equation ofthe cubic V assumes the form

a114x4(x1−ax3)2 +a223x

3(x2− bx4)2 +2(a123x3 +a124x

4)(x1−ax3)(x2− bx4) = 0.(4.19)

We see that the line x1 − ax3 = 0, x2 − bx4 = 0 lies on the cubic V , all its pointsbeing singular. It crosses the lines ℓ1 and ℓ3, but does not pass through the pointsA1 and A2. Hence, we may place the vertices A3 and A4 of the frame on this line. Inthis case, its equation is as follows: x1 = x2 = 0, that is, a = b = 0. Consequently,the relations (4.16) assume the following simple form:

a134 = 0, a233 = 0, a334 = 0, a234 = 0, a144 = 0, a344 = 0. (4.16′)

Conversely, assume that on the cubic V there exists a line m of singular pointsthat meets the lines ℓ1 and ℓ3, but does not contain the points A1 and A2. Weset A3 ≡ ℓ1 ∩m, A4 ≡ ℓ3 ∩m. Then the equations x1 = x2 = 0 must transformidentically all equalities in the system (4.17) that defines the singular points. Let-ting x1 = x2 = 0 in (4.17), we obtain the identities 2a134x

3x4 + a144(x4)2 ≡ 0,a233(x3)2 + 2a234x

3x4 ≡ 0, a334x3x4 ≡ 0, a334(x3)2 ≡ 0, whence also follow the

equalities in (4.16′).This being so, the existence of a line of singular points is a characteristic property

of the surface V carrying the special Burau web W24.

2-5. By analyzing the system (4.17) under the conditions (4.6) and (4.8) that definethe webs W25, we find that V contains two singular points: B(0, a344, 0,−2a124)and A2, which lie on ℓ3. In this case, the cone of tangents at the point A2 is asfollows: Φ2(A2) = x3(a233x

3 +2a234x4), that is, the point A2 is a biplanar knot. In

this case, as distinct from the webs W22, the plane π1 (x3 = 0) contains the linesℓ2 and ℓ3, and the plane π2 (a233x

3 + 2a234x4 = 0) contains the line ℓ2.

As in Para. 2-2, it may be checked that the last-mentioned properties are char-acteristic.

By placing the vertex A4 of the frame at the singular point, we obtain a344 = 0.In this frame the web W25 is determined by the relations

a223 = a124 = a144 = a123 = a344 = 0. (4.8′)

These equations coincide with the equations (4.12), hence the classes W25 and W29

coincide.Let us single out some special cases.

2-5-1. The webs W 125: the conditions (4.8′) are augmented by the relation

a234 = 0, (4.20)

which means that the singular point A2 is a uniplanar knot.

2-5-2. The webs W 225: the conditions (4.8′) are augmented by the relation

a114a334 − a2134 = 0, (4.21)

which means that the cubic V has yet another singular point P (2a134a234, 0,−2a114a234, a114a233).


2-5-3. The webs W 325: the conditions (4.8′) are augmented by the relations (4.20)

and (4.21).

2-6. Under conditions (4.6) and (4.9), which define the webs W26, the system (4.17)assumes the form

2a114x1x4 + 2a123x

2x3 + 2a134x3x4 + a144(x4)2 = 0,

2a123x1x3 + 2sa123(x3)2 = 0,

2a123x1x2 + 2a134x

1x4 + 2sa123x2x3 + 2a334x

3x4 + sa144(x4)2 = 0,

a114(x1)2 + 2a134x1x3 + 2a144x

1x4 + a334(x3)2 + 2sa144x3x4 = 0.

Since a123 = 0, it follows from the second equation that either x3 =0 or x1+sx3 =0.If x3 = 0, then from the first and last equations,

(2a114x1 + a144x

4)x4 = 0, (a114x1 + 2a144x

4)x1 = 0.

Since a114a144 = 0, it follows that x1 = x4 = 0, that is, we obtain the point A2.If x1 + sx3 = 0, but x3 = 0, then the last equation of the system implies that

a114s2 − 2a134s + a334 = 0. Using the last relation, the cubic V decomposes into

the plane x1 + sx3 = 0 and a quadric. Hence, there are no other singular points,save the point A2.

In the case in question, the cone of tangents at the singular point A2 is as follows:Φ2(A2) = 2a123x

3(x1 + sx3), that is, the point A2 is a biplanar knot. The plane π1

(x3 = 0) contains the lines ℓ2 and ℓ3, and the plane π2 (x1 + sx3 = 0) contains theline ℓ3. It is readily verified that these properties are characteristic.

For the webs W26, we may specialize the frame by letting A3 = π2 ∩ ℓ1. In thiscase s = 0, and the relations that define the webs W26 assume the form

a223 = a124 = a234 = a233 = a344 = 0. (4.9′)

2-7. By analyzing the system (4.17) under the conditions (4.6) and (4.10) thatdefine the webs W27 it follows that the surface V in this case has only one singularpoint A2. Since Φ2(A2) = a233(x3)2, the point A2 is a uniplanar knot, and soπ1 ≡ π2 ∋ ℓ2, ℓ3.

Conversely, assume that the cubic defined by (4.1) has a uniplanar knot A2 andπ1 ≡ π2 ∋ ℓ2, ℓ3. Then Φ2(A2) = a233(x3)2, which gives a223 = a123 = a124 =a234 = 0. After the change of variables 1/µ → µ, the web equation (4.2) assumesthe form

a114(ν)2 + 2a134ν + a233λ + a334 + µ(a144ν + a344) = 0,

in view of these relations. Using the linearization conditions, we find the regularitycondition: a144 = 0. This, together with the above relations, gives the conditions(4.10) that define the webs W27.

This establishes the following

Theorem 4.1. Let V be a cubic surface that does not decompose into a plane anda quadric. Assume that V contains the straight lines ℓ1 = A3A1, ℓ2 = A1A2, ℓ3 =A2A4. A necessary and sufficient condition that the special Burau three-web W2,defined by the lines ℓi, be regular is that the cubic V have the following singularities(up to geometrically symmetric cases):


• the points A1 and A2 are singular points of the cubic V (the basic class W21);• the line ℓ3 contains two singular points, of which one, A2, is a biplanar

knot with the following property : one of the planes, into which the cone oftangents at A2 decomposes, contains the line ℓ2, and the other, the line ℓ3(the basic class W22);

• on each of the lines ℓ1 and ℓ3 there are two singular points not coincidingwith the points A1 and A2 (the basic class W23);

• V contains a line of singular points that crosses the lines ℓ1 and ℓ3, but whichdoes not pass through the points A1 and A2 (the basic class W24);

• the line ℓ3 contains two singular points, and the point A2 is a planar knot ;moreover, one of the planes, into which the cone of tangents at A2 decom-poses, contains the lines ℓ2 and ℓ3, and the other, the line ℓ2 (the basic classW25);

• the line ℓ3 contains one biplanar knot A2; moreover, one of the planes, intowhich the cone of tangents at A2 decomposes, contains the lines ℓ2 and ℓ3,and the other, the line ℓ3 (the basic class W26);

• the line ℓ3 contains one uniplanar knot A2, which contains the lines ℓ2 and ℓ3(the basic class W27).


3-1. The web W21 is obtained by projecting the web W21 from one of the singularpoints A1 or A2. Both cases are geometrically equivalent. One may immediatelyverify that the web W21 coincides with the web W11, which was examined in theprevious section.

3-2. The web W ′22 is obtained by projecting the web W22 from the biplanar knot

A2 to the plane π = A1A3A4. Assume that the relations that define the three-webW22 are written in the form (4.7′). We then have (see (4.18))

Φ2(A2) = 2x1(a123x3 + a124x

4) ≡ 2x1T1,

Φ3(A2) = x4(a114(x1)2 + 2a134x1x3 + a334(x3)2) ≡ x4Φ2(A2).

(4.18′)

The pencil λ1 is given by the equation

2λT1x1 + Φ2(A2) = 0. (4.22)

It contains two degenerate conics: T1x1 = 0 and Φ2(A2) = 0, which intersect at the

double point A4 and the points m1 and m2 (T1 = 0, Φ2(A2) = 0). Thus, λ1 is theP -pencil of conics with simple vertices m1, m2 and the double vertex A4. Since A4

is a double point, all the conics of the pencil have the same tangent A3A4 (x1 = 0)at this point. The family λ2 of lines of the web W ′

22 is the pencil of straight lineswith vertex A1, and the family λ3 is the pencil of lines with vertex A4.

3-3. The web W ′′22 is obtained by projecting the web W22 from the conic knot A4 to

the plane π = A1A2A3. We write the equation of the cubic V (using the relations(4.7′)) in the form Φ3(A4) + x4Φ2(A4) = 0, where

Φ2(A4) = a114(x1)2 +2a134x1x3 + a334(x3)2 +2a124x

1x2, Φ3(A4) = 2a123x1x2x3.(4.23)


The equation of the pencil λ1 is as follows:

Φ2(A4) + 2λa123x1x3 = 0. (4.24)

The straight line A2A3 (x1 = 0) touches the conic Φ2(A4) at the point A2; thestraight line x3 = 0 intersects the conic Φ2(A4) at the points A2 and m3 (x3 = 0,a114x

1 + 2a124x2 = 0). Since A2 is a triple point, λ1 is the SP -pencil of conics

with triple vertex A2 and simple vertex m3. All conics of this pencil have the sametangent A2A3 at the point A2 (Fig. 3).

Figure 3

The equation of the pencil λ2 is as follows:

Φ2(A4) + 2µa123x1x2 = 0. (4.25)

All conics of this pencil also pass through the point A2 and have the same tangentA2A3 at this point. In addition, they pass through the points m1 and m2 (x2 = 0,Φ2(A4) = 0). Hence, λ2 is the P -pencil of conics with simple vertices m1 and m2

and a double vertex A2.The family λ3 of the web W ′′

22 is the pencil of straight lines with vertex A2.

3-4. For the webs W23, all 4 singular points of the cubic V are geometricallyequivalent. We let W23 denote the web that is obtained by projecting the webW23, for example, from the conic knot A4 to the plane π = A1A2A3. We writeequation (4.1) of the cubic V in the form Φ3(A4) + x4Φ2(A4) = 0, where, in viewof (4.14′),

Φ2(A4) = a114(x1)2 + 2a134x1x3 + 2a234x

2x3, Φ3(A4) = a223(x2)2x3.

The family λ1 is defined by the equation Φ2(A4) + λa223x2x3 = 0 and consists

of the conics that pass through the points A2, A3 and m2, with the same tangentℓ2 = A2A1 at A2. Hence, λ1 is the PE-pencil of conics with double vertex A2

(Fig. 4).The family λ2 is defined by the equation Φ2(A4)+µa223(x2)2 = 0 and consists of

the conics that pass through the points A3 and m2, with the same tangent (t1 andt2, say) at these points. Hence, λ2 is the WP -pencil of the conics. The third familyof lines of the web W23 is the pencil of lines with vertex A2.


Figure 4

3-5. Assume that the three-web W24 is defined by (4.16′). Then each point on theline A3A4 is the conic knot. We let W24 denote the projection of the web W24, forexample, from the point A4 on the plane A1A2A3. In view of (4.16′), the equationof the cubic V is written as follows: Φ3(A4)+x4Φ2(A4) = 0, where Φ2(A4) = x1T1,Φ3(A4) = x2x3T2, T1 = a114x

1 + 2a124x2, T2 = 2a123x

1 + 2a223x2.

The first family λ1 of lines of the web W24 is defined by the equation x1T1 +λx3T2 = 0 and consists of the conics that pass through the points A2, m1 and thedouble point A3. Hence, λ1 is the PE-pencil of the conics. All conics of the pencilhave the same tangent T2 = 0 at A3.

The family λ2, as defined by the equation x1T1 + µx2T2 = 0, is the family ofpairs of straight lines with common vertex A3.

The third family of lines of the web W24 is the pencil of straight lines withvertex A2.

3-6. We let W ′24 denote the three-web obtained by projecting the web W24 from

an arbitrary point P = A4 + pA3 to the line of singular points A3A4. One mayimmediately verify that the first and third families of lines of the web W ′

24 are givenby the equations

λ1 : Φ2(P ) + λx3(2a123x1 + a223x

2) = 0,

λ3 : Φ2(P )− νx3(a114x1 + 2a124x

2) = 0,

where Φ2(P ) = a114(x1)2 + 2x1x2(pa123 + a124) + pa223(x2)2, and the second fam-ily is the pencil of lines with vertex A3. The families λ1 and λ3 are P -pencilswith common double vertex A3 and common simple vertices m1 and m2, whichare defined on the line A1A2 by the equation Φ2(P ) = 0. As is seen from theabove equations of pencils, the conics of the pencil λ1 have the same tangentℓ1 ≡ 2a123x

1 + a223x2 = 0 at the vertex A3, and the conics of the pencil λ3 have

the same tangent ℓ3 ≡ a114x1 + 2a124x

2 = 0 (Fig. 5).


Figure 5

3-7. We let W ′25 denote the projection of the web W25 from the biplanar knot A2

to the plane A1A3A4. In view of (4.8′) the equation of the cubic V assumes theform x4Φ2(A2) + x2Φ2(A2) = 0, where

Φ2(A2) = x3(a233x3 + 2a234x

4), Φ2(A2) = a114(x1)2 + 2a134x1x3 + a334(x3)2.

The first family is defined by the equation Φ2(A2) + λΦ2(A2) = 0. This is theP -pencil of conics with multiple vertex A4 and the same tangent x3 = 0 at A4.Simple vertices of the pencils (we denote them by m1 and m2) are the points ofintersection of the straight line a233x

3 + 2a234x4 = 0 with the pair of straight lines

Φ2(A2) = 0. The families λ2 and λ3 are pencils of straight lines with vertices A1

and A4.Note that the web in question differs from the web W ′

22 in that the vertex A1 ofthe pencil of lines (the family λ2) lies on the tangent to the pencil of conics λ1 atthe double point A4.

In the case a234 = 0 (the webs W 125, see (4.20)), the family λ1 becomes the pencil

of lines with vertex A4. Hence, λ1 ≡ λ3, and in this case the web W ′25 does not

exist.In the case a114a334 − a2

134 = 0 (the webs W 225, see (4.21)), the single vertices of

the pencil λ1 coincide and λ1 becomes a parabolic (WP ) pencil.

3-8. We let W ′′25 denote the projection of the web W25 from the conic knot A4 to

the plane A1A2A3. In view of (4.8′), the equation of the cubic V is written in theform Φ3(A4) + x4Φ2(A4) = 0, where

Φ2(A4) = a114(x1)2+2a134x1x3+2a234x

2x3+a334(x3)2, Φ3(A4) = a233x2(x3)2.

The straight line x3 = 0 touches the conic Φ2(A4) at the point A2; the straight linex2 = 0 intersects it at the points m1 and m2, which are defined by the equationa114(x1)2 + 2a134x

1x3 + a334(x3)2 = 0.The family λ1 is given by the equation Φ2(A4) + a233λ(x3)2 = 0. This is the

UP -pencil of conics with 4-fold vertex A2. At this vertex all conics of the pencilhave the same tangent x3 = 0.


The family λ2 is given by the equation Φ2(A4) + a233µx2x3 = 0. This is theP -pencil of conics with double vertex A2, simple vertices m1 and m2, and commontangent x3 = 0.

The third family of straight lines of the web W ′′25 is the pencil of lines with

vertex A2.In the case a234 = 0 (the webs W 1

25), the family λ1 becomes the pencil of straightlines with vertex A4 and the web W ′

25 does not exist in this case. In the casea114a334 − a2

134 = 0 (the webs W 225), the single vertices of the pencil λ1 coincide,

and λ1 becomes a parabolic (WP ) pencil.

3-9. We let W ′′′25 denote the projection of the web W 2

25 from the singular pointP (2a134a234, 0,−2a114a234, a114a233) onto the plane A1A2A4 (see Para. 2-5-2). Wechange to the new frame by moving the vertex A3 to the point P . In this case, thecoordinates transform as follows:

px1 + x3 = x1, a233x3 + 2a234x

4 = x4, x2 = x2, x3 = x3. (4.26)

Taking (4.8′) into account, in the new coordinates the equation of the cubic Vbecomes (we drop the tilde over the new coordinates)

a334(x1)2x4 − a334a233(x1)2x3 + 2a234x2x3x4 = 0, (4.27)

and the equations of the pencils Li that cut out the Burau web (see § 3.1) read

2a234x2 = λ(x4 − a233x

3), 2a234x3 = µ(x4 − a233x

3), x1 = ν(p + 1)x3.

Recall that the equation of the family λi of the web W ′′′25 is obtained by dropping

the variable x3 from equation (4.27) and the equation of the family Li. After somecalculation, we obtain, respectively,

a334a233(x1)2 − 2x2x4 + λ(x4)2 = 0, a334(x1)2 + µx2x4 = 0,

a334(pν + 1)x1x4 − a334a233(x1)2 + 2a234x2x4 = 0.

The first family is the UP -pencil with base conics (x4)2 = 0 and a334a233(x1)2 −2x2x4 = 0, the 4-fold vertex of the pencil lies at the point A2, the straight line x4 = 0is the common tangent to the conics of the pencil. All conics of the pencil passthrough the point A4. The second family is the WP -pencil of conics with commonmultiple vertices A2 and A4, the tangents at these points being x4 =0 and x2 = 0,respectively. The third family is an SP -pencil with base conics a334a233(x1)2 −2x2x4 = 0 and x1x4 = 0, with triple vertex A2, and simple vertex A4, the straightline x4 = 0 being the common tangent to the conics of the pencil.

3-10. We let W26 denote the projection of the web W26 from the planar knot A2

onto the plane A1A3A4. In view of (4.9′), the equation of the cubic V becomes2a123x

1x2x3 + x4Φ2(A2) = 0, where

Φ2(A2) = a114(x1)2 + 2a134x1x3 + a144x

1x4 + a334(x3)2.

The family λ1 is given by the equation 2a123λx1x3 + Φ2(A2) = 0. This is theSP -pencil of conics with the triple vertex A4 and the simple vertex m1, which is


defined by the equations x3 = 0, Φ2(A2) = 0. At this vertex, all the conics of thepencil have the same tangent x1 = 0.

The second family of lines of the web W24 is the pencil of lines with vertex A1,and the third, the pencil of straight lines with vertex A4.

3-11. We let W27 denote the projection of the web W27 from the planar knot A2

to the plane A1A3A4. In view of (4.6) and (4.10), the equation of the cubic V

becomes a233x2(x3)2 + x4Φ2(A2) = 0, where

Φ2(A2) = a114(x1)2 + 2a134x1x3 + a334(x3)2 + a344x

3x4.

The family λ1 is defined by the equation a233λ(x3)2 + Φ2(A2) = 0. This is theUP -pencil of conics with 4-fold vertex A4, at which all the conics of the pencil havethe same tangent x3 = 0.

The family λ2 is the pencil of straight lines with vertex A1, and λ3 is the pencilof straight lines with vertex A4.

Thus, in total there are 10 basic classes of regular webs of type W2 that differfrom the basic classes of type W1. These are the webs W ′

22, W ′′22, W23, W24, W ′

24,W ′

25, W ′′25, W ′′′

25 , W26 and W27.

§ 5. Regular special Burau three-webs of type W3 and perspectivelyequivalent webs of type W3. Case 3: all 3 generators ℓi lie in one plane

5.1. Basic webs of type W3. We let π denote the plane that contains all thegenerators ℓi and choose the frame as follows: ℓ1 = A2A3, ℓ2 = A1A3, ℓ3 = A1A2.Further, we place the vertex of the projective frame A4 on the cubic V . Theequation of V now becomes

a114(x1)2x4 + a144x1(x4)2 + a224(x2)2x4 + a244x

2(x4)2

+ a334(x3)2x4 + a344x3(x4)2 + 2a123x

1x2x3 + 2a124x1x2x4

+ 2a134x1x3x4 + 2a234x

2x3x4 = 0. (5.1)

Clearly, the cubic V decomposes into a quadric and the plane π containing the linesℓi if and only if a123 = 0. In this section it will be assumed that a123 = 0.

The pencils of planes Li will be given by the equations

x1 = λx4, x2 = µx4, x3 = νx4. (5.2)

Eliminating the variables xα from (5.1) and (5.2), we obtain the equation of theweb W3 under consideration:

2a123λµν + a114λ2 + a224µ

2 + a334ν2 + 2a124λµ + 2a134λν

+ 2a234µν + a144λ + a244µ + a344ν = 0. (5.3)

Case I. Suppose that all points Ai are singular. Then

a114 = 0, a224 = 0, a334 = 0, (5.4)

and equation (5.3) becomes polylinear, hence the web is regular. We denote itby W31.


Case II. Only two of the points Ai are singular (A1 and A3, say). Then

a114 = 0, a334 = 0, a224 = 0, (5.5)

and the web equation (5.3) assumes the form

a224µ2 + µ(2a123λν + 2a124λ + 2a234 + a244) + 2a134λν + a144λ + a344ν = 0. (5.6)

Using the first linearization condition, we arrive at the regularity condition

rank

0 0 0 a224

2a123 2a124 2a234 a244

2a134 a144 a344 0

= 2. (5.7)

Proposition 5.1. A necessary and sufficient condition that the singular points A1

and A3 be biplanar knots is that condition (5.7) be satisfied.

Proof. Condition (5.7) is equivalent to the relations

a134

a123=

a144

2a124=

a344

2a234≡ p. (5.8)

In view of (5.8), equation (5.1) of the cubic V reads as follows:

a224(x2)2x4+a244x2(x4)2+2(px4+x2)(a124x

1x4+a234x3x4+a123x

1x3) = 0. (5.9)

It is seen from this equation that the cone of tangents at the point A1 decomposesinto the planes px4 + x2 = 0 and a124x

4 + a123x3 = 0, and, at A3, into the planes

px4 + x2 = 0 and a234x4 + a123x

1 = 0.Conversely, if the points A1 and A3 lying on surface (5.1) are biplanar knots

then we arrive at condition (5.8).

We place the vertex A4 of the frame onto the intersection of the planes a124x4 +

a123x3 = 0 and a234x

4 + a123x1 = 0. Then the relations a124 = a234 = 0 hold, and

the regularity condition (5.7) reduces to the relations

a124 = a234 = a144 = a344 = 0. (5.10)

Analyzing the system for the singular points of the surface V in case (5.10), itfollows that, in addition to A1 and A3, there may be another singular point A4,provided that

a244 = 0. (5.11)

We let W32 (respectively, W ′32) denote the regular three-web defined by the condi-

tions (5.5) and (5.10) ((5.5), (5.10) and (5.11)).A similar argument to that in the previous cases may apply to show that the

second linearization condition does not give rise to new classes of regular webs.Case III. Only one of the points Ai is singular (A1, say). Then

a114 = 0, a224 = 0, a334 = 0. (5.12)

Writing down the web equation (5.3) in this case in terms of powers of thevariables µ or ν and applying the first linearization condition, we arrive at thecondition a123 = 0, under which the cubic V decomposes.


We write equation (5.3) in the form λQ + R = 0,

λ(2a123µν+2a124µ+2a134ν+a144)+a224µ2+a334ν

2+2a234µν+a244µ+a344ν) = 0,(5.13)

and employ the second linearization condition. Recall that this condition meansthe decomposition of the conics Q = 0 and R = 0 and agreement of their cen-tres. These conics may decompose only as follows: 2a123(µ + a)(ν + b) = 0 and(Aµ+Bν+C)(A1µ+B1ν) = 0, respectively. The centres of these conics coinciding,the point µ = −a, ν = −b lies on each of the lines into which R decomposes. Inview of this, equation (5.13) becomes either

2a123λ(µ + a)(ν + b) + p(A(µ + a) + B(ν + b))(bµ− aν) = 0, (5.14)

provided a and b do not vanish simultaneously, or

2a123λµν + (Aµ + Bν)(A1µ + B1ν) = 0, (5.14′)

provided a = b = 0. Substituting into (5.14) the variables λ, µ, ν from formula(5.2), we obtain the equation of the cubic V :

2a123x1(x2+ax4)(x3+bx4)+p(A(x2+ax4)+B(x3+bx4))(bx2−ax3)x4 = 0. (5.15)

Proposition 5.2. A necessary and sufficient condition that the equation of thecubic (5.1) be reducible to the form (5.15) is that the cubic contain a straight lineof singular points passing through A1.

Proof. It is verified directly that the straight line m, as given by the equations

x2 + ax4 = x3 + bx4 = 0, (5.16)

lies on the cubic (5.15), all its points being singular.Before proceeding with the converse assertion, we specialize the frame by placing

on m the vertex A4 of the frame; then m ≡ A1A4, a = b = 0, the web equationassuming the form (5.14′). Comparing this equation with (5.1), we find that

a114 = 0, a124 = 0, a134 = 0, a144 = a244 = a344 = 0. (5.17)

We may proceed to complete the proof of Proposition 5.2. Assume that cubic (5.1)has a straight line of singular points that passes through A1. We place the pointA4 on this line. Then the system defining the singular points of the cubic (5.1)must be satisfied for x2 = x3 = 0 and arbitrary x1 and x4. This yields preciselythe relations (5.17). Note that there are no other singular points on cubic (5.17).

Let W33 denote the regular web defined by the relations (5.17).We now rewrite equation (5.3) in the form Pµ2 + Qµ + R = 0 with

P = a224, Q = 2a123µλ + 2a124λ + 2a234ν + a244,

R = a334ν2 + 2a134λν + a144λ + a344ν,

and apply the second linearization condition. Comparing with (1.7) we see thatonly one case is possible: one of the polynomials U , V must be of degree 0, and the


remaining one, of degree 2. Hence P , Q, R belong to the same pencil, that is, wehave arrived at the first linearization condition. A similar conclusion follows if weexpand (5.3) in powers of ν.Case IV.

a114 = 0, a224 = 0, a334 = 0.

Applying, as in the previous cases, the first linearization condition to (5.3),we obtain the matrix that contains nonzero minors of order 3, which equal2a114a224a123, and so on. As a result, no regular webs will be obtained in this way.

The second linearization condition also does nothing— the proof is the same asabove.

The following theorem summarizes the results obtained.

Theorem 5.1. Assume that the lines ℓ3 = A1A2, ℓ1 = A2A3 and ℓ2 = A3A1 lie onthe cubic surface V . Then the equation of V is of the form (5.1), and the equationof the special Burau three-web W3, as defined by the line ℓi, is of the form (5.3).Further, a necessary and sufficient condition that a three-web W3 be regular is thatone of the following series of relations be satisfied : (5.4) (the basic class W31); (5.5)and (5.10) (the basic class W32); (5.5), (5.10) and (5.11) (the basic class W ′

32);(5.16) (the basic class W33). In any of these cases, the cubic V is characterized,respectively, by the following geometric conditions : the points Ai at which the linesℓi meet are singular points of the cubic V ; two of these points are biplanar knots ; twoof these points are biplanar knots and there exists yet another singular point on V ;there exists a straight line of singular points passing through one of the points Ai.


2-1. Let W31 be the projection of the web W31 to the plane A2A3A4 from thesingular point A1. By virtue of (5.6), the equation (5.1) of the cubic V readsΦ3(A1) + x1Φ2(A1) = 0, where

Φ2(A1) = a144(x4)2 + 2a123x2x3 + 2a124x

2x4 + 2a134x3x4,

Φ3(A1) = x4(a244x2x4 + a344x

3x4 + 2a234x2x3) = x4Φ2(A1).

The family λ1, as given by the equation λΦ2(A1) + Φ2(A1) = 0, is the pencil ofconics passing through the points A2, A3, m1 and m2. The families λ2 and λ3 arepencils of lines with vertices A2 and A3. Clearly, the three-web W31 coincides withthe three-web W11.

2-2. Let W32 be the projection of the web W32 onto the plane A2A3A4 from thesingular point A1. By virtue of (5.7) and (5.9), the equation (5.1) of the cubic Vreads in this case as follows: Φ3(A1) + x1Φ2(A1) = 0, where Φ2(A1) = 2T1T2,T1 = x2 + px4, T2 = a123x

3 + a124x4,

Φ3(A1) = x4(x2(a224x2 + a244x

4) + 2a234x3T1) ≡ x4Φ2(A1).

The family λ1 is given by the equation λΦ2(A1) + Φ2(A1) = 0. This is theparabolic pencil of conics with double vertex A3 and simple vertices m1 and m2

(T2 = 0, Φ2(A1) = 0). All the conics of the pencil λ1 have the same tangent T2 = 0at A3.


The families λ2 and λ3 are pencils of straight lines with vertices A2 and A3.From this description it is seen that the web W32 coincides with the web W ′

22.

2-3. Let W ′32 be the projection of the web W ′

32 onto the plane A2A3A4 from thesingular point A1. By virtue of (5.5), (5.10) and (5.11), equation (5.1) of thecubic V reads in this case as follows:

a224(x2)2x4 + 2a123x1x2x3 + 2a134x

1x3x4 = 0.

The equation of the family λ1 reads

a224(x2)2 + 2λx3(a123x2 + a134x

4) = 0,

the families λ2 and λ3 being pencils of lines with vertices A2 and A4, respectively.The family λ1 is the WP -pencil of conics with double vertices A4 and A3, thetangents at these points being x3 = 0, a123x

2 + a134x4 = 0, respectively.

2-4. Let W ′′32 be the projection of the web W ′

32 onto the plane A1A2A3 from thesingular point A4. The equation of the cubic V will be the same as in the previouscase, the equations of families reading

λ1 : a224(x2)2 + 2a134x1x3 + 2λa123x

2x3 = 0,

λ2 : a224(x2)2 + 2a134x1x3 + 2µa123x

1x3 = 0,

λ3 : a224(x2)2 + 2a134x1x3 + 2νa123x

1x2 = 0.

Here, λ1 is the SP -pencil of conics with vertices A1 (triple) and A3, the straightline x3 = 0 being the common tangent to the conics of the pencil at A1; λ2 is theWP -pencil of conics with double vertices A1 and A3 and tangents x3 = 0, x1 = 0at these points, respectively; λ3 is the SP -pencil of conics with vertices A1 and A3

(triple), the straight line x1 = 0 being the common tangent to the conics of thepencil at A3. All 3 pencils contain the base conic a224(x2)2 + 2a134x

1x3 = 0.

2-5. Let W33 be the projection of the web W33 onto the plane A1A2A3 from thesingular point A4. By (5.17), equation (5.1) of the cubic V assumes the form

a224(x2)2x4 + a334(x3)2x4 + 2a123x1x2x3 + 2a234x

2x3x4 = 0. (5.18)

Excluding the variable x4 from the equations of the pencils (5.2) and of the cubicequation (5.18), it follows that the family λ1 is the pencil of degenerate conics —these are the pairs of straight lines with common vertex A1; the family λ2 is theparabolic pencil of conics with the same tangent ℓ3 = A1A2 at the double vertexA1 and the simple vertices m1 and m2 (lying on the line ℓ1 = A2A3); the familyλ3 is the parabolic pencil of conics with the same tangent ℓ2 = A1A3 at the doublevertex A1 and the simple vertices m1 and m2. Comparing this with the result of§ 4.3-6 it is seen that the web W33 coincides with the web W ′

24.


2-6. By projecting the web W33 from the singular point A1 onto the plane A2A3A4

we obtain the web W0 consisting of three pencils of lines with vertices A2, A3, A4.This being so, in total there are 3 basic classes of regular webs of type W3 that are

distinct from the webs of types W1 and W2: these are the webs W ′32, W ′′

32 and W0.The following theorem summarizes the results of §§ 3–5.

Theorem 5.2. In total, there are 17 basic classes of regular webs that are per-spectively equivalent to special regular Burau three-webs on nondegenerate cubicsurfaces : these are the webs W11, W12, W ′

12, W13, W ′22, W ′′

22, W23, W24, W ′24,

W ′25, W ′′

25, W ′′′25 , W26, W27, W ′

32, W ′′32 and W0.

§ 6. Projective classification of regular three-webs that areperspectively equivalent to special regular Burau three-webs

The projective classification is carried out modulo projective transformations.Recall that a projective (fractional linear) transformation of the plane is uniquelyspecified by eight parameters or four pairs of corresponding points, while the crossratio of four points on a straight line or four straight lines of a pencil is a projectiveinvariant. Let K be some configuration on the plane. We let p, i and a denote,respectively, the number of parameters that specify this configuration, the numberof its projective invariants, and the dimension of the maximum group of projectiveautomorphisms of the configuration K. Clearly, a = 8 − p, and if there existtwo configurations, identical in form, with equal invariants that depend on p < 9parameters, then these configurations are projectively equivalent.

6.1. Projective classification of three-webs W1.

1-1. Projective classification of three-webs W11.In accordance with § 3.2-1, the family λ3 of the three-web W11 is the pencil of

conics with base Φ2(A4) and Φ2(A4) and vertices A1, A2, m3 and m4. The familiesλ1 and λ2 are pencils of straight lines with vertices A1 and A2. We let W11.1 andW11.2 denote the webs for which the points m3 and m4 are, respectively, real anddistinct or purely imaginary. Further, let W11.3 be the webs for which the points m3

and m4 coincide. In this case, the family λ3 is the PE-pencil with simple verticesA1 and A2 and double vertex m ≡ m3 ≡ m4.

Proposition 6.1. Any two webs of the same type W11.k, k = 1, 2, 3, are projec-tively equivalent. The three-web W11.3 admits a one-parameter group of projectiveautomorphisms.

Proof. The webs W11.1 and W11.2 are completely determined by the configurationK of the vertices A1, A2, m3 and m4 of which no three lie on the same straight line.Here, p = 8, the vertex coordinates being the parameters. Consequently, a = 0.Since there is a unique projective transformation that carries over such a quadrupleof points into a similar one, it follows that any two webs of the type in questionare equivalent. The configuration K has no projective invariants (i = 0), becauseit has no quadruples of points lying on a straight line and no quadruples of straightlines belonging to one pencil.


A web of type W11.3 is entirely determined by the points A1, A2, m and thecommon tangent of the conics of the pencil λ3 at the point m (see Proposition 0.1).The configuration K is specified by seven parameters (p = 7): the coordinates ofthe points and the tangent parameter in the pencil of straight lines with vertex m.Hence, a = 8− 7 = 1, and there exists a projective transformation that carries overthe configuration K into an arbitrary one analogous to K.

Consider the metric model of the web W11.2. To do so we remove the line m3m4

to infinity and take the complex conjugates m3 and m4 as Laguerre points. Thenthe conics become circles. Hence, the three-web W11 in question is projectivelyequivalent to the circle web formed by an elliptic pencil of circles and two pencilsof straight lines with vertices at vertices of the pencil of circles (class 4 accordingto Lazareva’s classification of circle webs, see [1]).

We let W11.4 denote the webs for which the point m3 coincides with the point A1.These webs are characterized by the relation a112a134 = a113a124. In this case λ3 isthe PE-pencil with simple vertices A2, m4 and a double vertex A1. Let W11.5 be thewebs for which the point m3 agrees with the point A1 and the point m4 coincideswith the point A2. In this case λ3 is a WP -pencil with vertices A1, A2. Such websare characterized by the relations a112a134 = a113a124 and 2a112a234 = −a223a124.

The proof of the following result is similar to that of Proposition 6.1.

Proposition 6.2. Any two webs of the same type W11.4 or W11.5 are projectivelyequivalent. A three-web of type W11.4 admits a one-parameter group of projectiveautomorphisms. A three-web of type W11.5 admits a two-parameter group of pro-jective automorphisms.

Let W11.6 be the webs for which the points m3 and m4 coincide with the point A1.The relations characterizing these webs can be put in the following form

a124 = pa112, a134 = pa113, a234 = qa112 − pa223, a334 = 2qa113 + 2pa133.

In this case λ3 is an SP -pencil with simple vertex A2 and triple vertex A1. Thebase of the pencil is formed by the conic Φ2(A4) (see (3.15)) and the degenerateconic A1A2 ∪ t, where t is the tangent to the conic Φ2(A4) at A1.

Proposition 6.3. Any two webs of type W11.6 are projectively equivalent. A three-web of type W11.6 admits a two-parameter group of projective automorphisms.

Proof. By Proposition 0.1 there exist ∞1 SP -pencils (an SP -bundle) of conics withassigned vertices and common tangent at the triple vertex. In order to single outa pencil in the bundle, one needs to fix its parameters in the bundle. Thus, theentire structure is determined by six parameters: the coordinates of the verticesA1 and A2, the parameter of the tangent in the pencil with vertex A1, and theparameter of a pencil in the bundle.

1-2. Projective classification of three-webs W12.Recall that a three-web W12 is specified in the plane A1A2A4 by a common base

conic Φ2(A3) of two P -pencils with common vertex A1 + A2 on it, and the linesℓ1 = A1A4 and ℓ2 = A2A4 that meet the conic Φ2(A3) at the points m1 and m2,m3 and m4, respectively. The straight line A1A2 is tangent to the conic Φ2(A3) at


the point A1 + A2 (see Fig. 2). We have the following projectively distinct classesof webs depending on the position of lines with respect to the conics.

The three-webs W12.1: the points m1, m2, m3 and m4 are real and distinct. Inthis case, the family λ1 (λ2) is the PE-pencil of conics with simple vertices m3, m4

(m1, m2) and the double vertex A1 + A2. The family λ3 is the pencil of straightlines with the vertex A1 + A2.

The three-webs W ′12.1: these are three-webs W12.1 for which the parabolic pencils

of conics λ1 and λ2 have the common vertex m ≡ m1 = m3 = A4. These webs, assingled out by the additional relation a344 = 0, are exactly the webs W ′

12; howeverwe retain the notation W ′

12.1 for the sake of uniformity.The webs W12.2: the pairs of points m1 and m2, m1 and m2 are complex conju-

gates. In this case, the families λ1 and λ2 are PH-pencils of conics.The webs W12.3: the points m1 and m2 are real, and m1 and m2 are complex

conjugates (or vice versa).The webs W12.4: one of the lines, ℓ1 say, is tangent to the conic Φ2(A3) at the

point m ≡ m1 = m2, and the other line meets this conic at two real points. In thiscase, λ1 is the WP -pencil of conics with double vertices m and A1 + A2, and thefamily λ2 is the PE-pencil of conics. The case in question is characterized by therelation a113a344 = (a134)2.

The webs W12.5: one of the lines, ℓ1 say, is tangent to the conic Φ2(A3), and theremaining one meets this conic at two complex conjugate points.

The webs W12.6: ℓ1, ℓ2 are tangent to the conic Φ2(A3). The relations charac-terizing these webs read as follows:

a113 = a2, a344 = b2, a134 = a234 = ab.

In this case, the families λ1 and λ2 are WP -pencils of conics with vertices m ≡m1 = m2 and A1 + A2, n ≡ m3 = m4 and A1 + A2, respectively.

Proposition 6.4. 1) There exist ∞2 projectively distinct webs of type W12.k, k =1, 2, 3.

2) There exist ∞1 projectively distinct webs of types W ′12.1 and W12.k, k = 4, 5.

3) Any two webs of type W12.6 are projectively equivalent.

Proof. 1) The cross ratios of the points I1 = (A1, A4, m1, m2) and I2 = (A2, A4,

m3, m4) are projective invariants of a web W of type W12.k, k = 1, 2, 3. Clearly,if two webs are equivalent, then the corresponding cross ratios coincide. We claimthe converse. Let W ′ be a different three-web of the same type, and let P bethe projective transformation that carries over the 4 points m1, m2, m3 and m4

(pertaining to the three-web W ) into the four primed points (pertaining to thethree-web W ′). In this case (see Fig. 2) P (A4) = A′4, and since I1 = I ′1 andI2 = I ′2, we have P (A1) = A′1 and P (A2) = A′2. We now observe that the commonbase conic Φ2(A3) of the pencils λ1 and λ2 is tangent to the line A1A2. Thisuniquely determines this conic in the pencil of conics with vertices m1, m2, m3

and m4. Similarly, the conic Φ2′(A3) for the web W ′ is also uniquely determined.

Finally, since incidence is preserved under projective transformations, it followsthat P (Φ2(A3)) = Φ2

′(A3), and hence, P (A1 +A2) = A′1 +A′2, giving P (W ) = W ′.


2) The web W ′12.1 has a unique invariant I: this is the cross ratio of the four

lines passing through the point A1 + A2:

A2A1; A1 + A2, m; A1 + A2, m2; A1 + A2, m4.

We claim that if two webs W and W ′ of type W ′12.1 have the same invariants, then

these webs are projectively equivalent. Let P be a (unique) projective transforma-tion carrying the 4 points A1+A2, m1 ≡ m3, m2, m4 into the corresponding primedpoints. We have I = I ′, and so P (A1A2) = A′1A

′2, that is, the common tangent of

pencils of conics is transformed into the same tangent. By Proposition 0.1, P -pencilsare uniquely recovered from the tangent and vertices, and so P (W ) = W ′.

The three-webs of type W ′12.4 and W12.5 have a unique projective invariant I:

this is the cross ratio of four lines A2A1; A1 + A2, m; A1 + A2, m3; A1 + A2, m4.A similar analysis as for the webs of type W ′

12.1 shows that two webs of type W ′12.4

(or W12.5) are projectively equivalent whenever their invariants coincide.3) A three-web of type W12.6 is completely determined by the conic Φ2(A3) with

three fixed points m, n and A1 + A2 on it. In turn, the conic Φ2(A3) is uniquelydetermined by three points m, n and A1 + A2 and two tangents ℓ1 and ℓ2 at thepoints m and n, respectively. Indeed, by Proposition 0.1, from multiple verticesm, n and the common tangents ℓ1 and ℓ2 one may uniquely recover the WP -pencilof conics, which also contains the conic Φ2(A3). The latter is uniquely determinedfrom the condition that it passes thought the point A1 + A2.

This analysis shows that the web W12.6 is uniquely determined by the quadrupleof points m, n, A1 + A2 and A4 = ℓ1 ∩ ℓ2. But it now follows that two websof type W12.6 are projectively equivalent, because there always exists a projectivetransformation that carries over this quadruple of points into a similar quadrupleof primed points.

It worth noting that the web W12 is not equivalent to any circle web, becausethere is no pair of complex conjugates through which conics of both pencils λ1

and λ2 would pass.

1-3. Projective classification of three-webs W13.We recall yet another distinction between the webs of types W12 and W13: for

the latter the lines ℓ1, ℓ2 and the common tangent A1A2 to the conics of the pencilsλ1 and λ2 at the point A1 + A2 meet at a single point A4. Hence, in the projectiveclassification of the webs W13 we also, as in the previous case, have 5 types of webs,except for the cases W ′

12.1 and W12.6, which are impossible, because the lines ℓ1, ℓ2and A4, A1 + A2 are different. The proof of the following result is similar to thatof Proposition 6.4.

Proposition 6.5. 1) There exist ∞1 projectively distinct webs of type W13.k,k = 1, 2, 3.

2) Any two webs of type W13.4 or of type W13.5 are projectively equivalent.

6.2. Projective classification of three-webs W2. It was pointed out in § 4.3-1that the web W21 coincides with the web W11.


2-1. Projective classification of three-webs W ′22.

Recall (see § 4.3-2) that the first family λ1 of lines of the web W ′22 is given

by equation (4.22). This is the P -pencil of conics with simple vertices m1 andm2 and the double vertex A4, all the conics of the pencil having at A4 the sametangent A4A3. The families λ2 and λ3 are pencils of lines with vertices A1 andA4, respectively, moreover A1 ∈ m1m2. We let W ′

22.1 denote the web for whichthe vertices m1 and m2 are real and distinct; W ′

22.2 and W ′22.3 denote the webs for

which these vertices are, respectively, purely imaginary or coincide.If one of the points m1 or m2 coincides with the point A1, then (see (4.18′)) we

obtain a114 = 0, contradicting condition (4.6).If one of the points m1 or m2 coincides with the point A4, then it follows from

(4.18′) that a334 = 0 and the cubic V decomposes. Thus, in total there are threeof the above types of webs W ′

22.

Proposition 6.6. 1) There exist ∞1 projectively distinct webs W ′22.1 and W ′

22.2,and any web of these types admits a one-parameter group of projective automor-phisms.

2) Any two webs of type W ′22.3 are projectively equivalent, and any web of this

type admits a one-parameter group of projective automorphisms.

Proof. 1) Given a web W ′22.1, there are 4 lines that pass through the double vertex

A4 of the pencil λ1: A4A1, the tangent A4A3, A4m1 and A4m2; their cross ratio(we denote it by I) is an invariant of the web.

We note further that a web W of type W ′22.1 is uniquely determined by the

points A4, m1 and m2, the common tangent x1 = 0 to the conics of the pencil λ1 atthe point A4 (altogether 7 parameters) and the invariant I. Since p = 7, we havea = 8− 7 = 1. The proof is the same for the webs W ′

22.2.2) For the webs W ′

22.3 the pencil λ1 is twice parabolic with double vertices A4 andm ≡ m1 ≡ m2. We let T denote the point of intersection of the common tangentsto the conics of the pencil at double vertices. A web is completely determined bythe quadruple of points A1, A4, m and T .

On the other hand, there exist ∞1 projective transformations that carry sucha quadruple into a similar one. It follows, first, that any two webs of type W ′

22.3

are projectively equivalent, and second, that the web W ′22.3 admits a one-parameter

family of projective automorphisms.

A metric model of a web W ′22.2 is obtained by removing the line m1m2 to infin-

ity. If we regard the complex conjugates m1 and m2 as Laguerre points, then thePH-pencil of conics λ1 becomes a parabolic pencil of circles, and the web W ′

22.2

becomes a circle three-web. This is class 6.1 according to Lazareva’s classification,see [1].

2-2. Projective classification of three-webs W ′′22.

We let W ′′22.1 (respectively W ′′

22.2) denote the webs W ′′22 for which the points m1

and m2 are real (purely imaginary); let W ′′22.3 denote the webs for which m1 =

m2 ≡ m. If, for example, m1 = m3, then we have a114 = 0, which contradicts (4.6).Thus, in total there are three types of webs W ′′

22.


Proposition 6.7. 1) There exist ∞1 projectively distinct webs of types W ′′22.1

and W ′′22.2.

2) Any two webs of type W ′′22.3 are projectively equivalent.

3) A web W ′′22.3 admits a one-parameter group of projective automorphisms.

Proof. 1) There is one invariant for the webs W ′′22.1 and W ′′

22.2: the cross ratio ofthe points A1, A3, m1, m2 on the line A1A3 (we denote it by I). This establishesthe first part of Proposition 6.7.

In order to specify the web W ′′22.1, one needs to specify the coordinates of the

points A2, A3, m3 (6 parameters), the parameter of the line A1A3 in the pencil oflines with vertex A3 and the coordinates of one of the vertices of the PE-pencil,for example, m1 (8 parameters in total). Then the point A1 is defined as theintersection of the lines A2m3 and A3m1, and the second vertex m2 of the PE-pencilis found using the cross ratio.

A similar argument applies to the web W ′′22.2.

The web W ′′22.3 is defined by specifying the points A2, A3, m3 and the parameter

of the base conic Φ2(A4) in the SP -pencil λ1 (the pencil itself is defined by the triplevertex A2, the tangent A2A3 at this vertex, and the simple vertex m3). Thus, thereare in total 7 parameters, which gives us a = 8− 7 = 1.

2-3. Projective classification of three-webs W23.An arbitrary three-web W23 is formed by the PE-pencil of conics λ1 with double

vertex A2 and simple vertices A3 and m2, the WP -pencil of conics λ2 with multiplevertices A3 and m2, and the pencil of lines with vertex A2 (see § 4.3-4 and Fig. 4).The pencils λ1 and λ2 have a common base conic Φ2(A4). If the points A3 andm2 are distinct, then we denote the corresponding three-web by W23.1. If thesepoints coincide, then λ1 becomes a WP -pencil with multiple vertices A2 and A3,and λ2, a UP -pencil with 4-fold vertex A3. The conics of both pencils have thesame tangent A1A3 at the vertex A3. We denote the webs of this kind by W23.2.

The webs W23.1 and W23.2 have projective invariant I, which is the cross ratio ofthe points A1, A2, T1, T2 on the tangent A1A2, where T1 and T2 are points on thetangents t1 and t2 to the conics of a WP -pencil at the multiple vertices A3 and m2.

Proposition 6.8. Any two webs of type W23.1 (W23.2) with the same invariant areprojectively equivalent. A web W23.2 admits a one-parameter group of projectiveautomorphisms.

Proof. The web W23.1 is uniquely determined by 8 parameters: the coordinatesof the points A1, A3, the coordinates of the point m2 on the line A1A3, and theparameters of the tangents A1A2, t1 and t2. The vertex A2 of the P -pencil on A1A2

is determined by the invariant I.The web W23.2 is determined by 7 parameters: the coordinates of the points A1,

A2, A3 and the parameter of the base conic Φ2(A4) in the WP -pencil λ1.

The web W24 coincides with the web W11.4, and the web W ′24 with the web W33,

which we will consider below.


2-4. Projective classification of three-webs W ′25.

In accordance with § 4, the webs W ′25 subdivide into three types according to

whether the vertices m1 and m2 of the P -pencil λ2:a) are real and distinct (the webs W ′

25.1);b) are imaginary (the webs W ′

25.2);c) coincide (the webs W ′

25.3).In the first two cases the families of a web are given by the quadruple of points

A4, A1, m1 and m2, of which the last three lie on one line. This structure isdetermined by 7 parameters. In the case where the points m1 and m2 coincide (thewebs W ′

25.3), it is determined by 6 parameters. This establishes the following

Proposition 6.9. Any two webs of type W ′25.i, i = 1, 2, 3, are projectively equiva-

lent. Any web of type W ′25.1 or W ′

25.2 admits a one-parameter group of projectiveautomorphisms. A web of type W ′

25.3 admits a two-parameter group of projectiveautomorphisms.

The web W ′25.2 is equivalent to the circle web formed by two parabolic pencils

of circle and an elliptic pencil whose vertices coincide with the vertices of thepencils, all the pencils having a common circle (class 6.2 according to Lazareva’sclassification).

2-5. Projective classification of three-webs W ′′25.

In view of § 4, the webs W ′′25 subdivide into three types according to whether the

vertices m1 and m2 of the P -pencil λ2:a) are real and distinct (the webs W ′′

25.1);b) are imaginary (the webs W ′′

25.2);c) coincide.In the last case we obtain the web W23.2.

Proposition 6.10. Any two webs of types W ′′25.1 or W ′′

25.2 are projectively equiva-lent.

Proof. The web W ′′25.1 is completely determined by the pair of points A1 and A2

(4 parameters), the parameter of the line λ1λ2 in the pencil of lines with vertex A1,the parameters of the points m1 and m2 on the line λ1λ2, and the parameter of thebase conic Φ2(A4) in the P -pencil λ2: in total, 8 parameters. A similar argumentapplies for webs of type W ′′

25.2.

2-6. Projective classification of three-webs W ′′′25.

In accordance with § 4.3-9, the second and third families of the web W ′′′25 are

completely determined by the triple of points A1, A2 and A4. However, the firstfamily (the UP -pencil of conics) is not given uniquely by its 4-fold vertex A2 andthe tangent x4 = 0 at this point (see Proposition 0.1). In order to define this pencil,we need to fix its base conic λ = 0, which enters the second family (the WP -pencil).Thus, the web is determined by 7 parameters: the coordinates of the points A1,A2, A4 and the parameter of the conic in the WP -pencil. We thus have

Proposition 6.11. Any two webs of type W ′′′25 are projectively equivalent ; the web

W ′′′25 admits a one-parameter group of projective automorphisms.


2-7. Projective classification of three-webs W26 and W27.According to § 4.3-10, 3-11, the webs W26 and W27 do not admit projectively

different types.

Proposition 6.12. 1) Any two webs of type W26 are projectively equivalent ; theweb W26 admits a one-parameter group of projective automorphisms.

2) Any two webs of type W27 are projectively equivalent ; the web W27 admitsa two-parameter group of projective automorphisms.

Proof. The three-web W26 is formed by the SP -pencil of conics λ1 with triple vertexA4 and simple vertex m1 and two pencils of lines with vertices A1 and A4; here,m1 ∈ A1A4 and the line A3A4 is the common tangent to the conic of the pencilλ1 at the vertex A4, see § 4.3-10. This construction depends on 6 parameters: thecoordinates of the points A1 and A4, the coordinates of the point m1 on the lineA1A4 and the parameter of the tangent A3A4 at the pencil with vertex A4.

In accordance with § 4.3-11, the web W27 is formed by the UP -pencil of conicsλ1 with 4-fold vertex A4 and two pencils of lines with vertices A1 and A4; here,A1A4 is the common tangent to the conics of the pencil λ1 at the vertex A4.This construction is determined by 6 parameters: the coordinates of the pointsA1, A4 and the two parameters of the UP -pencil λ1 in the hyperbundle of conicsthat is determined by the 4-fold vertex A4 and the common tangent A1A4 (seeProposition 0.1).

6.3. Projective classification of three-webs W3. The web W ′32 (see § 5.5.2)

coincides with the web W ′′22.3.

3-1. The web W ′′32 (see § 5.5.2) is completely determined by the triple of points Ai,

i=1, 2, 3 (six parameters) and the choice of the base conic a224(x2)2+2a134x1x3 =0

in the WP -pencil (one parameter), that is, by 7 parameters in total.The three-web W33 (see § 5.2-5) is completely determined by the two lines ℓ2 =

A3A1 and ℓ3 = A1A2 through A1 and two points m1 and m2 lying on the lineℓ1 = A2A3. We denote by W33.1 (respectively, W33.2, W33.3) the webs for whichthe points m1 and m2 are real and distinct (complex conjugates, coincide); the webW33.3 is singled out by the relation (a234)2 = a224a334 for the coefficients of thecubic (5.18). The cross ratio of the lines ℓ2, ℓ3, m1 and A1m2 are invariants of thewebs W33.1 and W33.2.

This establishes the following

Proposition 6.13. 1) Any two webs of type W ′′32 are projectively equivalent ; the

web W ′′32 admits a one-parameter family of projective automorphisms.

2) There exist ∞1 projectively distinct webs W33.1 and W33.2; any web of thesetypes admits a one-parameter group of projective automorphisms.

3) Any two webs of type W33.3 are projectively equivalent ; the web W33.3 admitsa two-parameter group of projective automorphisms.

Since all conics of the web W33.2 pass through a pair of complex conjugate points,the web W33.2 is projectively equivalent to the circle web. The latter is formed bytwo parabolic pencils of circles with common vertex A1 and an elliptic pencil of

908 V.B. Lazareva and A.M. Shelekhov

circles, one vertex of which coincides with A1. Hence, all 3 pencils belong to oneparabolic bundle of circles, and we thus obtain class 0 according to Lazareva’sclassification.

The following theorem summarizes the results of this section.

Theorem 6.1. There exist 38 projectively distinct types of triangulation of theplane by pencils of conics corresponding to projectively distinct types of webs W thatare perspectively equivalent to regular special Burau webs on nondegenerate cubicsurfaces. These are the webs : W11.k, k = 1, . . . , 6; W ′

12.1; W12.k, k = 1, . . . , 6;W13.k, k = 1, . . . , 5; W ′

22.k, k = 1, 2, 3; W ′′22.k, k = 1, 2, 3; W23.1, W23.2; W ′

25.k,k = 1, 2, 3; W ′′

25.k, k = 1, 2; W26; W27; W ′′32, W33.k, k = 1, 2, 3; W0.

Of these, the webs W ′12.1, W12.4, W12.5, W13.k (k = 1, 2, 3), W ′

22.1, W ′22.2, W ′′

22.1,W ′′

22.2, W23.1, W33.1, W33.2 have one projective invariant, that is, each of thesetypes contains ∞1 projectively distinct webs ; the webs W12.k, k = 1, 2, 3, have twoprojective invariants.

The webs W11.3, W11.4, W ′22.i, i = 1, 2, 3, W ′′

22.3, W23.2, W ′25.1, W ′

25.2, W ′′32,

W33.i, i = 1, 2, 3, admit a one-parameter group of projective automorphisms ; thewebs W11.5, W11.6, W ′

25.3, W26, W27, W0 admit a two-parameter group of projectiveautomorphisms.

The webs W11.2, W ′22.2, W33.2 and W ′

25.2 are projectively equivalent to circle webs(classes 4, 6.1, 0 and 6.2, respectively, according to Lazareva’s classification).

Remark. 1. The projective classification of webs with invariants can be finessed.2. We have indicated the maximal groups of automorphisms; the structure of

these groups may also be finessed. In addition, some webs admit discrete groups ofautomorphisms.

3. The scope of the paper did not allow us to describe the structure of cubicsurfaces containing special regular Burau webs and illustrate graphically all classesof webs.

Bibliography

[1] V.B. Lazareva and A.M. Shelekhov, “On triangulations of the plane by pencils ofconics”, Math. Sb. 198:11 (2007), 107–134; English transl. in Sb. Math. 198:11(2007), 1637–1663.

[2] W. Blaschke, Einfuhrung in die Geometrie der Waben, Birkhauser, Berlin 1955.

[3] M.A. Akivis and A.M. Shelekhov, Geometry and algebra of multidimensionalthree-webs, Math. Appl. (Soviet Ser.), 82, Kluwer Acad. Publ., Dordrecht 1992.

[4] J.-P. Dufour, “Familles de courbes planes differentiables”, Topology 22:4 (1983),449–474.

[5] J.-P. Dufour and P. Jean, “Rigidity of webs and families of hypersurfaces”,Singularities and dynamical systems (Iraklion 1983), North-Holland Math. Stud.,vol. 103, North-Holland, Amsterdam–New York 1985, pp. 271–283.

http://dx.doi.org/10.4213/sm3843

http://dx.doi.org/10.4213/sm3843

http://dx.doi.org/10.1070/SM2007v198n11ABEH003899

http://dx.doi.org/10.1070/SM2007v198n11ABEH003899

http://www.zentralblatt-math.org/zmath/search/?an=Zbl 0068.36501



http://dx.doi.org/10.1016/0040-9383(83)90037-X

http://dx.doi.org/10.1016/0040-9383(83)90037-X





[6] V.B. Lazareva, “Classification of regular circle three-webs up to circulartransformations”, Fundam. Prikl. Mat. 16:1 (2010), 95–107; English transl. inJ. Math. Sci. (N.Y.) 177:4 (2011), 579–588.

V.B. LazarevaTver’ State UniversityE-mail : [email protected]

A.M. ShelekhovTver’ State UniversityE-mail : [email protected]

Received 3/APR/08 and 5/FEB/13Translated by A. ALIMOV

http://mi.mathnet.ru/eng/fpm1293

http://mi.mathnet.ru/eng/fpm1293

http://dx.doi.org/10.1007/s10958-011-0483-7

http://dx.doi.org/10.1007/s10958-011-0483-7

mailto:[email protected]

mailto:[email protected]

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On triangulations of the plane by pencils of conics. II