EQUIVARIANT CLASSES OF MATRIX MATROID VARIETIES

L. M. FEHÉR, A. NÉMETHI, AND R. RIMÁNYI

Dedicated to the memory of T. Brylawski (1944-2007)

Abstract. To each subset I of {1, …, k} associate an integer r(I). Denote by X the collection of those n × k matrices for which the rank of a union of columns corresponding to a subset I is r(I), for all I. We study the equivariant cohomology class represented by the Zariski closure Y = \overline{X}. This class is an invariant of the underlying matroid structure. Its calculation incorporates challenges similar to the calculation of the ideal of Y, namely, the determination of the geometric theorems for the matroid. This class also gives information on the degenerations and hierarchy of matroids. New developments in the theory of Thom polynomials of contact singularities (namely, a recently found stability property) help us to calculate these classes and present their basic properties. We also show that the coefficients of this class are solutions to problems in enumerative geometry, which are natural generalization of the linear Gromov-Witten invariants of projective spaces.

1. Introduction

1.1. Matroid representation varieties. To each subset I of {1, …, k} associate an integer r(I). Denote by X the collection of those n×k matrices for which the rank of a union of columns corresponding to a subset I is r(I), for all I. Our main object of study in this paper is the Zariski closure Y of this set. This is a version of matroid representation varieties. Other versions (eg. contained in Grassmannians, instead of the affine space of matrices) are also known, and they are closely related to ours. A dual point of view is considering the hyperplanes determined by the column vectors of the matrices. From this point of view Y is the parameter space of certain hyperplane arrangements.

Matroid representation varieties are universal objects in algebraic geometry in the sense that any complication of varieties can be modeled on them. The precise statement of this universality theorem is called Mnëv’s theorem, see [Mnë88], [RG95], or a recent account in [Vak06]. Hence one does not hope that any reasonable question on these varieties has an easy answer. One manifestation of this phenomenon is the determination of the ideal of these varieties. In Section 3 below we will explain with examples how the generators of the ideal encode projective geometry theorems.

The problem we will consider about matroid varieties is an enumerative geometry problem, a generalization of the linear Gromov-Witten invariants of projective spaces. Suppose a matroid variety is given, as above. Consider k generic linear subspaces V_{i} in C^{n}. The question is, how many n×k matrices exist that belong to our matroid variety such that the i’th column vector is in V_{i}. For example, after projectivizing, we can ask the following question: given 8 generic straight lines and a generic point in the projective plane, how many Pappus configurations exist with 8 points of the Pappus configuration belonging to the 8 lines, and the 9’th point coinciding with the given point. The precise definitions, and the answer are given below. In the special case, when the matroid variety is the variety of rank ≤ 2 matrices of size n × k, this enumerative question is equivalent to the determination of k-point linear Gromov-Witten invariants in projective spaces. For general matroid varieties, however, no classical geometric or Gromov-Witten-type methods are known to compute the generalized Gromov-Witten invariants.

The first author was supported by the Alfréd Rényi Institute of Mathematics, and OTKA grants 81203 and 72537, as well as the János Bolyai Scholarship. The second author was supported by OTKA grant K67928. The third author was supported by the Marie Curie Fellowship PIEF-GA-2009-235437, and NSA grant H98230-10-1-0171.

The nature of the matroid Gromov-Witten invariants in P^{2} can be visualized by pictures. Some interactive presentations, created with the Interactive Geometry Software Cinderella [RGK] can be found at www.unc.edu/˜ rimanyi/matroid show.

We will show in Theorem 5.3 that our matroid versions of linear Gromov-Witten invariants can be computed through the equivariant classes [Y]. These are cohomology classes that the varieties Y represent in the GL(n) × GL(1)^{k}-equivariant cohomology ring of the vector space of n × k matrices.

1.2. Equivariant classes represented by invariant varieties of a representation. Let the group G act on the complex vector space V, and let Y ⊂ V be an invariant variety of complex codimension c. Then Y represents a cohomology class [Y] ∈ H_{G}^{2c}(V) in equivariant cohomology. There are various definitions and names for this class, eg. equivariant Poincaré dual, Thom polynomial, multidegree. We will call it the equivariant class of Y. Since H_{G}^{∗}(V) is naturally isomorphic to the ring H^{∗}(BG) of G-characteristic classes, the equivariant class [Y] is simply a G-characteristic class of degree 2c.

The equivariant class of the variety Y encodes a lot of geometric information on Y; let us just allude to the effectiveness of Schubert calculus (the Giambelli formula is such an equivariant class) or the generalization involving classes of quiver loci. Other applications include the enumerative geometry results coming from Thom polynomials of singularities, see eg. [Kle76] for a classic review or [MR07] for a recent addition.

Let us remark that the equivariant class of a matroid representation variety can be interpreted as a class of a quiver locus for the “star quiver” (based on a star shaped graph). However, the equivariant properties of quiver representations are only well understood for quivers of Dynkin type ADE, see [BF99], [FR02], [KMS06], [KS06], [Buc08] and references therein. It would be interesting to compare our results with quiver coefficients defined in [Buc08] for non-Dynkin quivers.

The usual tools to calculate equivariant classes represented by invariant subvarieties involve equivariant resolution, equivariant degeneration, or equivariant localization techniques. These techniques require the understanding of the ideal or the singularities of the variety in question. For matroid varieties we lack this essential information.

Another main approach to calculate equivariant classes, effective for equivariant classes of contact singularities as well, is an interpolation method described in [FR04]. Below we will study an improvement of this interpolation method. In essence, we will describe certain constraints that a particular [Y] must satisfy. Some of these constraints originate from the topological arguments of [FR04], some others from the enumerative interpretation of some coefficients.

Finally, a key advance of this paper, a third set of constraints, follow from the analogue of a stabilization property recently proved for contact singularities [FR07]. This property stems from understanding how an equivariant class changes at non-transversal intersection. In turn, one obtains a bound on the number of factors in each term of such and equivariant class.

Overwhelming experience shows that these three sets of constraints are sufficient to determine the equivariant classes [Y], providing several enumerative applications. However, at the moment, we have no theorem claiming that for a particular matroid a certain set of constraints is sufficient.

In Section 7 we show a certain stabilization property connecting the equivariant classes of matroid varieties in different dimensions n. As a corollary we prove a vanishing theorem on certain coefficients. In Section 8 we outline the method of calculating these invariants.

A particularly interesting question, subject to future study, is whether the matroid Gromov-Witten invariants can be organized as structure constants of an algebraic object with some kind of associativity property—mimicking the construction of the (big) quantum cohomology ring.

1.3. Acknowledgement. The authors would like to express their gratitude to T. Brylawski, S. Fomin, A. Hraskó, B. Sturmfels, and A. Fink for helpful discussions and remarks; and to D. Adalsteinson for letting us use his computer cluster for our calculations.

2. Matrix matroid varieties

We will denote the set of natural numbers {0, 1, …} by N, and the set {1, 2, …, k} by [k]. For a set X let 2^{X} denote its power set, ie. the set of subsets of X. We will identify the vector spaces (C^{n})^{k} and C^{n×k} by the rule

(1)  (v_{1}, …, v_{k}) ↔ ((v_{1}) … (v_{k}))

Elements in (C^{n})^{k} will be referred to as (ordered) vector configurations in C^{n}. For a matrix M ∈ C^{n×k} and subsets U ⊂ [n], V ⊂ [k], let M_{V}^{U} denote the submatrix consisting of the (i, j)-entries of M for i ∈ U, j ∈ V. Let M_{V} = M_{V}^{[n]}.

The vector configuration C = (v_{1}, v_{2}, …, v_{k}) ∈ (C^{n})^{k} defines the rank function r_{C} : 2^{[k]} → N,

r_{C}(V) = dim span{v_{i}}_{i∈V} .

Definition 2.1. For a configuration C we define

X_{C} = {M ∈ C^{n×k} : rank(M_{V}) = r_{C}(V) for all V ⊂ [k]}.

The Zariski closure \overline{X_{C}} ⊂ C^{n×k} will be called the matrix matroid variety associated with C, and will be denoted by Y_{C}.

If we identify n × k matrices with k-tuples of n-vectors as in (1), then X_{C} consists of those configurations whose rank function is the same as that of C. For example, C itself belongs to X_{C}. The matrix matroid variety Y_{C} consists of those configurations that are limits (degenerations) of elements in X_{C}.

Observe that X_{C} and Y_{C} do not change if we re-scale, ie. multiply, any vector v_{i} in C by any non-zero complex number. Hence X_{C} and Y_{C} are determined by the list of points P_{i} := [v_{i}] ∈ P^{n−1} for v_{i} ≠ 0, and the list of those v_{i} which are 0. By abusing language, such a list will also called a configuration.

P_{1}, P_{2}, P_{3}  P_{4}, P_{5}  ∅ P_{6} Figure 1

Example 2.2. Let n = 2, k = 6, and consider the following configuration

P_{1} = P_{2} = P_{3} = (0 : 1) ∈ P^{1},  P_{4} = P_{5} = (1 : 1) ∈ P^{1},  v_{6} = 0.

This configuration is illustrated in Figure 1. Matrices in X_{C} are those 2 × 6 matrices whose

• first three columns are proportional non-zero vectors,

• the fourth and fifth columns are proportional non-zero vectors,

• the first and the fourth columns are non-proportional,

• the sixth column is the zero vector.

It is true that Y_{C} consists of matrices satisfying the “closed” conditions above, but not necessarily the “open” ones. That is, Y_{C} consists of matrices whose first three columns are proportional, fourth and fifth columns are proportional, and sixth column is 0. However, the easy procedure of dropping the open conditions will not specify Y_{C} in general.

Example 2.3. Matrix Schubert varieties. Consider a complete flag of linear spaces

L^{0} ⊂ L^{1} ⊂ … ⊂ L^{n−1} ⊂ L^{n}

in C^{n}. Let ℓ = (l_{0}, l_{1}, …, l_{n}) ∈ N^{n+1} with ∑ l_{i} = k. Choose l_{i} generic points v_{1}^{(i)}, …, v_{l_{i}}^{(i)} in L^{i}. The matrix matroid variety corresponding to the configuration

C_{ℓ} = (v_{1}^{(0)}, …, v_{l_{0}}^{(0)}, v_{1}^{(1)}, …, v_{l_{1}}^{(1)}, …, v_{1}^{(n)}, …, v_{l_{n}}^{(n)})

is studied in the papers [Ful92], [FR03, Sect.5], [KM05], and is called the matrix Schubert variety corresponding to Grassmannian permutation.

For the problems to be considered later in this paper, matrix Schubert varieties will be the simple case. Products of matrix Schubert varieties will also be considered simple. Note that Example 2.2 is such a product of matrix Schubert varieties (after identifying C^{n×k_{1}} × C^{n×k_{2}} with C^{n×(k_{1}+k_{2})}), namely,

(2)  Y_{C} = Y_{C_{(0,3,0)}} × Y_{C_{(1,2,0)}} = Y_{C_{(0,3,0)}} × Y_{C_{(0,2,0)}} × Y_{C_{(1,0,0)}}.

Examples of matrix matroid varieties which are not products of matrix Schubert varieties will be given below.

Remark 2.4. Other candidate names for matrix matroid varieties would be “matroid variety” or “matroid representation variety”. We chose the name “matrix matroid variety”, because of the analogy with matrix Schubert varieties.

P_{1}  P_{2} P_{3} P_{4} P_{5}  P_{6}

Figure 2. The Menelaus configuration C_{M}, and the Ceva configuration C_{C}

3. The ideal of matrix matroid varieties

The rank of a matrix is r if its (r +1)×(r +1) minors vanish, and at least one r ×r minor does not vanish. Hence the algebraic description of X_{C} ⊂ C^{n×k} is a collection of equations (several minors vanish), together with a collection of conditions expressing that certain polynomials (some other minors) do not vanish together (cf. the open and closed conditions of Example 2.2):

X_{C} = {M = (m_{i,j}) ∈ C^{n×k} : p_{u}(m_{i,j}) = 0 for u = 1, 2, …;
q_{1}^{(v)}(m_{i,j}), …, q_{w_{v}}^{(v)}(m_{i,j}) do not vanish together for v = 1, 2, …}.

It follows that

(3)  Y_{C} ⊂ {M = (m_{i,j}) ∈ C^{n×k} : p_{u}(m_{i,j}) = 0 for u = 1, 2, …}.

Although it is tempting to think that we have equality in formula (3), in general, this is not the case. First we give an intuitive reason for this.

3.1. Motivation: the Menelaus configuration. Consider the Menelaus configuration C_{M} of Figure 2, with n = 3, k = 6.

The equations p_{u} of formula (3) are the four 3×3 minors of the 3×6 matrix (m_{i,j}) corresponding to the following triples of column-indices: 126, 135, 234, 456. The right hand side of formula (3) hence contains all 3 × 6 matrices for which these four minors vanish. We claim that there is a matrix for which these minors vanish, but it is not in Y_{C_{M}}, ie. it is not a limit of matrices from X_{C_{M}}. Indeed, consider an affine chart of P^{2}, a line l in it, and the configuration C′ of six generic points on l. If this configuration were in the closure of X_{C_{M}} then there would be a family of configurations belonging to X_{C_{M}}, all in the affine chart, converging to C′. For all these configurations Menelaus’ theorem [Men00] holds, which we recall now.

Theorem 3.1. [Complex affine version of Menelaus’s theorem] Consider the configuration C_{M} of points in C^{2}. Choose an identification of the P_{1}P_{3}P_{5} line with C. Observe that the complex number (P_{5} −P_{1})/(P_{3} −P_{5}) does not depend on the choice; denote this ratio by P_{1}P_{5}/P_{5}P_{3}. Then (using similar notations for the other straight lines) we have

(4)  \frac{P_{1}P_{5}}{P_{5}P_{3}} · \frac{P_{3}P_{4}}{P_{4}P_{2}} · \frac{P_{2}P_{6}}{P_{6}P_{1}} = −1.

Our reasoning is finished by observing that for C′ the Menelaus identity (4) does not hold; this proves that for C_{M} the two sides of Formula (3) are not equal.

One may wonder if there is a complex projective version of Menelaus’s theorem, which would eliminate the need for the affine chart in the geometric proof above. The answer is given in the next section.

What we learned from the Menelaus example is that • the “naive” equations p_{u} are not enough to cut out Y_{C} even set-theoretically from C^{n×k};

• the extra equations needed (besides the naive ones) encode the not-so-obvious geometric theorems of the configuration.

3.2. The ideal of X_{C}, examples. Let I_{C} denote the ideal of the variety Y_{C}, ie. the homogeneous ideal of polynomials vanishing on Y_{C}.

Example 3.2. In Schubert calculus the following statement is well known [Ful92]: For the matrix Schubert variety C_{ℓ} of Example 2.3 the “naive” equations generate I_{C_{ℓ}}:

I_{C_{ℓ}} = (det (M_{{j_{1},...,j_{s}}}^{{i_{1},...,i_{s}}}) = 0)

for i_{1} < … < i_{s}, j_{1} < … < j_{s}, j_{s} ≤ l_{0} + … + l_{s−1}, s = 1, …, n.

Now consider the Menelaus configuration of Section 3.1, and consider the variety corresponding to the naive equations

Y_{naive} = {M ∈ C^{3×6} : det M_{126} = 0, det M_{135} = 0, det M_{234} = 0, det M_{456} = 0}.

Computer algebra packages [GPS01] can be used to find that this variety is the union of two irreducible varieties

(5)  Y_{naive} = Y_{C_{M}} ∪ {M ∈ C^{3×6} : rank(M) ≤ 2}.

This decomposition sheds light on the intuitive reasoning of Section 3.1. As a byproduct, the computer algebra package finds generators of the ideal I_{C_{M}}. It turns out that I_{C_{M}} can be minimally generated by polynomials of degrees

3, 3, 3, 3, 5, 5, 5, \underbrace{6, …, 6}_{12}.

The four degree 3 polynomials can be chosen to be the four naive equations. As a consequence, the collection of the other equations can be considered as the extra, non-trivial complex projective identities holding for Menelaus configurations. We might as well call this set of polynomials the “complex projective Menelaus’ theorem”.

For completeness let us show how to generate degree 6 and degree 5 polynomials in I_{C_{M}} knowing only the usual version of Menelaus’ theorem (Theorem 3.1). In the projective plane (x, y, z) we can choose the y coordinate to be at infinity. In the remaining affine plane we can identify ratios of complex numbers by appropriate projections. Hence, for example, from (4) we can obtain

(6)  \frac{x_{5}/y_{5} − x_{1}/y_{1}}{x_{3}/y_{3} − x_{5}/y_{5}} · \frac{z_{4}/y_{4} − z_{3}/y_{3}}{z_{2}/y_{2} − z_{4}/y_{4}} · \frac{z_{6}/y_{6} − z_{2}/y_{2}}{z_{1}/y_{1} − z_{6}/y_{6}} = −1.

Rearranging this equality we obtain a degree 6 polynomial. Making other choices we may obtain several other degree 6 polynomials. Getting degree 5 ones is more delicate. Consider the degree 6 polynomial obtained from (6), and the ones obtained from the next three similar equalities

\frac{z_{5}/y_{5} − z_{1}/y_{1}}{z_{3}/y_{3} − z_{5}/y_{5}} · \frac{z_{4}/y_{4} − z_{3}/y_{3}}{z_{2}/y_{2} − z_{4}/y_{4}} · \frac{x_{6}/y_{6} − x_{2}/y_{2}}{x_{1}/y_{1} − x_{6}/y_{6}} = −1,

\frac{x_{1}/y_{1} − x_{2}/y_{2}}{x_{2}/y_{2} − x_{6}/y_{6}} · \frac{z_{6}/y_{6} − z_{4}/y_{4}}{z_{4}/y_{4} − z_{5}/y_{5}} · \frac{z_{5}/y_{5} − z_{3}/y_{3}}{z_{3}/y_{3} − z_{1}/y_{1}} = −1,

\frac{z_{1}/y_{1} − z_{2}/y_{2}}{z_{2}/y_{2} − z_{6}/y_{6}} · \frac{z_{6}/y_{6} − z_{4}/y_{4}}{z_{4}/y_{4} − z_{5}/y_{5}} · \frac{x_{5}/y_{5} − x_{3}/y_{3}}{x_{3}/y_{3} − x_{1}/y_{1}} = −1.

It turns out that the sum of these four degree 6 polynomials is y_{4} times

(7)  −x_{5}y_{1}z_{3}z_{6}y_{2} + x_{5}y_{1}z_{3}z_{2}y_{6} + x_{3}y_{5}z_{2}z_{1}y_{6} − x_{5}y_{3}z_{2}z_{1}y_{6} + z_{1}y_{5}z_{3}x_{6}y_{2} − z_{1}y_{5}z_{3}x_{2}y_{6}−
z_{3}y_{5}z_{2}x_{6}y_{1} + z_{5}y_{3}z_{2}x_{6}y_{1} + x_{2}y_{1}z_{6}z_{3}y_{5} − x_{2}y_{1}z_{6}z_{5}y_{3} − x_{6}y_{2}z_{5}z_{1}y_{3}+
x_{2}y_{6}z_{5}z_{1}y_{3} − z_{1}y_{2}z_{6}x_{3}y_{5} + z_{1}y_{2}z_{6}x_{5}y_{3} + z_{6}y_{2}z_{5}x_{3}y_{1} − z_{2}y_{6}z_{5}x_{3}y_{1}.

This latter is one of the degree 5 generators of I_{C_{M}}. The other two can be obtained by similar calculations, or appropriate changes of variables in (7).

For an arbitrary configuration C determining I_{C} seems to be a hopelessly difficult problem. One might start with the ideal generated by the naive equations, and try to get rid of the “fake” components (just like the determinantal variety in (5)) by primary decomposition or by dividing (or saturating) with ideals of extra components. In practice, none of these strategies is feasible in reasonable time for configurations even a little more complicated than the Menelaus configuration.

Below, in Section 4 we will study another invariant of matrix matroid varieties, namely, their equivariant classes, which will be much better computable than I_{C}, and which can answer various questions about these varieties without determining their ideals.

3.3. The codimension of matrix matroid varieties. A consequence of the universality theorem mentioned in the Introduction, is that matrix matroid varieties in general can have multiple components, possibly of different dimensions. Let us mention that it seems a difficult problem to show an example for this phenomenon. Nevertheless, we do not expect any easy general procedure which would give the (co)dimension of a matrix matroid variety corresponding to a given matroid. In practice, however, one may often determine the codimension by imagining the matroid builded up step by step, and keeping track of the degrees of freedom. Consider, for example, the Menelaus configuration of Figure 2. The points P_{1}, P_{2}, P_{4}, P_{5} are projectively free. Adding P_{3} however means a restriction: it can not be anywhere in the plane, it has to be in the intersection of P_{1}P_{5} and P_{2}P_{4}. This is a 2-codimensional restriction. Similarly, adding the point P_{6} is another 2 codimensional restriction. Hence the codimension of Y_{C_{M}} is 2 + 2 = 4. Now consider the Pappus configuration in Figure 3. The subset P_{1}, P_{2}, P_{4}, P_{5} is free. The points P_{3} and P_{6} are on the straight lines P_{1}P_{2} and P_{4}P_{5} respectively. Hence each represents a codimension 1 restriction. These 6 points determine the remaining 3, which hence represent 2-codimensional restrictions each. Therefore the codimension of this matrix matroid variety is 1+1+2+2+2=8.

4. Equivariant classes of matrix matroid varieties.

We will work in the complex algebraic category; cohomology will be meant with integer coefficients; and GL(n) will denote the general linear group GL(n, C).

4.1. Equivariant classes in general. If Y is a complex codimension c subvariety in a compact complex manifold M, then Y represents a cohomology class [Y] in H^{2c}(M). The following equivariant version of this notion is more delicate to define; see e.g. [Kaz97], [EG98], [FR04], [MS04, 8.5], [Ful07].

Let the group G act on the complex vector space V, and let Y ⊂ V be an invariant variety of complex codimension c. Then Y represents a cohomology class [Y] ∈ H_{G}^{2c}(V) in equivariant cohomology. Since H_{G}^{∗}(V) is naturally isomorphic to the ring H^{∗}(BG) of G-characteristic classes, the equivariant class [Y] is simply a G-characteristic class of degree 2c.

4.2. Equivariant classes of matrix matroid varieties. Let D(k) be the group of diagonal matrices of size k. Consider the action of G_{n,k} = GL(n) × D(k) on the vector space C^{n×k} of n × k matrices by

(A, B) · M = AMB^{−1}, A ∈ GL(n), B ∈ D(k), M ∈ C^{n×k} .

Viewing elements of C^{n×k} as vector configurations as in (1), the action of (A, B) ∈ G_{n,k} reparametrizes C^{n} (the action of A) and rescales the vectors one by one (the action of B). Therefore, the spaces X_{C} and hence the varieties Y_{C} are G_{n,k}-invariant.

In the rest of the paper the main concept of interest will be the equivariant class

(8)  [Y_{C}] ∈ H_{G_{n,k}}^{∗}(C^{n×k}) = H^{∗}(BG_{n,k}) = Z[c_{1}, …, c_{n}, d_{1}, …, d_{k}],

where c_{i} are the Chern classes of GL(n), and d_{i} are the first Chern classes of the GL(1) components of D^{k} = GL(1)^{k}. We have deg c_{i} = 2i, deg d_{i} = 2.

4.3. Examples. In Sections 5–9 we will show how to calculate the classes [Y_{C}], and discuss their geometric meaning. Before that, however, we show some examples.

Example 4.1. Consider the configuration C of Example 2.2. We have

[Y_{C}] = (d_{1}d_{2} + d_{1}d_{3} + d_{2}d_{3} − c_{1}(d_{1} + d_{2} + d_{3}) + c_{1}^{2} − c_{2}) (c_{1} − d_{4} − d_{5})(d_{6}^{2} − c_{1}d_{6} + c_{2}).

Let us now consider matrix Schubert varieties C_{ℓ} of Example 2.3. That is, we have ℓ = (l_{0}, …, l_{n}), ∑ l_{i} = k. We may assume without loss of generality that there is an r such that l_{1}, …, l_{r} are all non-zero, while l_{r+1} = … = l_{n} = 0. (Indeed, observe for example that Y_{C_{2,0,2}} = Y_{C_{2,1,1}}, by changing the complete flag.) Define

μ_{i} = \begin{cases}∑_{j=0}^{i−1} l_{j} + 1, & i ≤ r \\ k + i − r & i > r, \end{cases}  λ_{n+1−i} = μ_{i} − i,  for i = 1, …, n.

Let β_{j}^{(i)} be degree j polynomials in the ring (8), defined by

1 + β_{1}^{(i)}t + β_{2}^{(i)}t^{2} + … = \frac{∏_{j<μ_{i}}(1 + d_{j}t)}{1 + c_{1}t + … + c_{n}t^{n}} .

Theorem 4.2. Using the notation above, the matrix Schubert variety C_{ℓ} ⊂ C^{n×k} has complex codimension |λ| = ∑ λ_{i}, and we have

(9)  [Y_{C_{ℓ}}] = (−1)^{|λ|} det (β_{λ_{i}+j−i}^{(n+1−i)})_{i,j=1,...,n} .

P_{1}  P_{2}  P_{3} P_{7}  P_{8}  P_{9} P_{4}  P_{5}  P_{6}

Figure 3. The Pappus configuration C_{P}

Proof. This is, in fact, not a new theorem. Observe, that Y_{C_{ℓ}} is not only invariant under the action of GL(n) × D(k), but under the same action of GL(n) × B(k), where B(k) is the Borel group of upper triangular k ×k matrices associated with the complete flag. The varieties Y_{C_{ℓ}} are, in fact, the orbit closures of this extended action. The equivariant classes of these orbit closures are calculated in [FR03, Thm. 5.1] to be the double Schur polynomials of (9), see also [KM05]. Since the inclusion D(k) ⊂ B(k) is a homotopy equivalence, the G_{n,k}-equivariant classes are the same as the GL(n) × B(k)-equivariant classes. This observation also shows that expression (9) is symmetric in the d_{i} variables. □

Equivariant classes of products of varieties multiply in the obvious sense. For instance, the result of Example 4.1 can be recovered from the factorization (2) and the application of Theorem 4.2 to the three factors. Namely, we have

[Y_{C_{0,3,0}}] = det (\begin{matrix}\frac{∏_{i=1}^{3}(1+d_{i}t)}{1+c_{1}t+c_{2}t^{2}}|_{2} & \frac{∏_{i=1}^{3}(1+d_{i}t)}{1+c_{1}t+c_{2}t^{2}}|_{3} \\ 0 & \frac{∏_{i=1}^{0}(1+d_{i}t)}{1+c_{1}t+c_{2}t^{2}}|_{0} \end{matrix}) = d_{1}d_{2} + d_{1}d_{3} + d_{2}d_{3} − c_{1}(d_{1} + d_{2} + d_{3}) + c_{1}^{2} − c_{2}.

Here, and later, f(t)|_{i} means the i’th coefficient of the Taylor series f(t) in the formal variable t. Similarly,

[Y_{C_{0,2,0}}] = −(d_{4} + d_{5} − c_{1}), [Y_{C_{(1,0,0)}}] = d_{6}^{2} − c_{1}d_{6} + c_{2},

after appropriate shifting of indices.

It is rather difficult to present equivariant classes of matrix matroid varieties which are not products of matrix Schubert varieties. For example the class [Y_{C_{M}}] for the Menelaus configuration of Section 3.1 is a degree 8 polynomial with 173 terms (in c-d-monomials). To indicate how it looks we show this polynomial after we substitute 0 for all the d_{i} variables: [Y_{C}]^{∗} = [Y_{C}]_{d_{i}=0∀i}. Observe that the [Y_{C}]^{∗} class is the GL(n) equivariant class represented by Y_{C}. We have

• [Y_{C_{M}}]^{∗} = 3c_{1}^{2}c_{2} − 2c_{1}c_{3} − c_{2}^{2} = 3Δ_{(211)} + 2Δ_{(22)} + 3Δ_{(31)}. Here the Schur polynomials Δ_{λ} corresponding to a partition (λ_{1} ≥ λ_{2} ≥ … ≥ λ_{r}) are defined

by Δ_{λ} = det(c_{λ_{i}+j−i})_{i,j=1,...,r}. The significance of the Schur basis is presented in Sections 6, 7. The expression in Theorem 4.2 can also be interpreted as a Schur polynomial. Here is a list of similar specializations of [Y_{C}] for the Ceva, Pappus, and Desargues configurations.

• [Y_{C_{C}}]^{∗} = 6Δ_{(2211)} + 4Δ_{(222)} + 3Δ_{(3111)} + 8Δ_{(321)} + Δ_{(33)};

• [Y_{C_{P}}]^{∗} = 11Δ_{(221111)}+16Δ_{(22211)}+8Δ_{(2222)}+12Δ_{(311111)}+28Δ_{(32111)}+28Δ_{(3221)}+17Δ_{(3311)}+

15Δ_{(332)};

• [Y_{C_{D}}]^{∗} = 15Δ_{(222111)} + 20Δ_{(22221)} + 20Δ_{(321111)} + 50Δ_{(32211)} + 30Δ_{(3222)} + 30Δ_{(33111)} +

45Δ_{(3321)} + 10Δ_{(333)}.

The coefficients of these classes in terms of Schur polynomials are all non-negative. We will prove this property for general matroids in Theorem 6.6.

The equivariant classes are rather meaningless formulas unless we find geometric applications. We finish this section with a rather simple application; more delicate geometric meaning will be discussed in Sections 5 and 6.

4.4. The degree of P(Y_{C}). Suppose the group G acts on the vector space V, and Y is an invariant cone. Then the degree of the projective variety P(Y) can be recovered from the equivariant class [Y] ∈ H_{G}^{∗}(V) by the following procedure.

Let T^{n} be a maximal torus of G with corresponding Chern roots α_{i}. If w_{1}, …, w_{n}, w are integers with the property that for any z ∈ C, |z| = 1 we have

(z^{w_{1}}, …, z^{w_{n}}) · v = z^{w}v, (z^{w_{1}}, …, z^{w_{n}}) ∈ T^{n}, v ∈ V,

then

deg P(Y) = [Y](α_{i} = \frac{w_{i}}{w}).

On the right hand side we have the equivariant class, with the number w_{i}/w substituted into the Chern root corresponding to the i’th factor of T^{n}. This theorem easily follows from the study of the change of equivariant classes when the torus action is pulled back to another torus action; see e.g. [MS04, Ex. 8.14, 8.15], [FNR05, 6.4].

For matrix matroid varieties we have two natural choices for the substitution. Either we substitute

c_{i} = \binom{n}{i}, d_{i} = 0, or c_{i} = 0, d_{i} = −1.

Observe that the first substitutions can be carried out for the specialized classes above (d_{i} = 0∀i), hence the following theorem can be checked from the classes presented above.

Theorem 4.3. For the Menelaus, Ceva, Pappus, and Desargues configurations (see Figures 2, 3, 4) we have

deg P(Y_{C_{M}}) = 66,  deg P(Y_{C_{C}}) = 297,  deg P(Y_{C_{P}}) = 2943,  deg P(Y_{C_{D}}) = 4680.

Remark 4.4. The degree of P(Y_{C_{M}}) can also be recovered from the decomposition in (5). Indeed, in this decomposition all three varieties are 4 codimensional; Y_{naive} has degree 3^{4} because of Bézout’s theorem; the determinantal variety \{M ∈ C^{3×6} | rank M ≤ 2\} has degree \binom{6}{4} (see [Ful84, 14.4.14]). Hence deg Y_{C_{M}} = 81 − 15 = 66. The same argument shows that

[Y_{C_{M}}] = (c_{1}−d_{1}−d_{2}−d_{6})(c_{1}−d_{1}−d_{3}−d_{5})(c_{1}−d_{2}−d_{3}−d_{4})(c_{1}−d_{4}−d_{5}−d_{6})−\frac{∏_{j=1}^{6}(1 + d_{j}t)}{1 + c_{1}t + c_{2}t^{2} + c_{3}t^{3}} |_{4}.

For the other varieties in the Theorem we know no other way of determining the degree, but to calculate the equivariant class as in Section 8, then carry out the described substitution.

Figure 4. The Desargues configuration C_{D}

P_{1}  P_{6}  P_{2} P_{3} P_{5}  P_{4}  P(V_{1})  P(V^{2}) P(V_{3}) P(V_{6})  P(V_{4}) P(V_{5})

Figure 5. (a) C, (b) the enumerative problem, (c) the solution = 2

5. Matroid versions of linear Gromov-Witten invariants

In the rest of the paper, for simplicity, we will assume that C is a configuration of k non-zero vectors in C^{n}, and that Y_{C} is pure dimensional.

For nonnegative integers q_{1}, …, q_{k} with ∑ q_{i} = codim_{C}(Y_{C} ⊂ C^{n×k}) we define

N(C; q_{1}, …, q_{k}) = #{([v_{1}], …, [v_{k}]) ∈ (P C^{n})^{k} : (v_{1}, …, v_{k}) ∈ Y_{C}, v_{i} ∈ V_{i}},

where V_{1}, …, V_{k} is a generic collection of linear spaces with dim V_{i} = q_{i} + 1.

More generally, instead of generic linear spaces V_{i}, we could have considered varieties of different dimensions and degree. These generalized enumerative problems can be reduced to the linear version above.

Example 5.1. Consider the configuration C in Figure 5 (a). The number N(C; 1, 1, 1, 0, 0, 0) is the number of solutions to the following problem: given 3 points and 3 straight lines (generically) on the plane P^{2} (Figure 5 (b)). How many triangles exist, whose vertices are on the straight lines, and whose sides pass through the given points? The solution is 2 (Figure 5 (c)) due to the following well known argument: Choosing a point X on P V_{1} we can project it through P V_{6} to P V_{2}, then project further through P V_{4} to P V_{3}, then further through P V_{5} back to P V_{1}, obtaining a point X′. The map X ↦ X′ is a projective transformation of the projective line P V_{1}, whose number of fixed points is the question. Since projective transformations have the form x ↦ (ax + b)/(cx + d) (in affine coordinate), the number of fixed points is 2. (The construction of the fixed points using a compass and straightedge is the famous Steiner construction, see e.g. [PT01, Ad.3].)

Example 5.2. Consider the number N(C_{M}; 1, 1, 1, 1, 0, 0) for the Menelaus configuration of Figure 2. One may try to follow the argument of Example 5.1: choose a point X on P V_{1}, projecting it through P V_{6} to P V_{2}, then further project through the intersection of P V_{4} and the line \overline{P V_{5}, P V_{6}} to P V_{3}, then even further through the point P V_{5}, back to P V_{1}, obtaining X′. The transformation X ↦ X′ is a projective transformation of P V_{1}, hence it has 2 fixed points, suggesting that N(C_{M}; 1, 1, 1, 1, 0, 0) = 2. However, this is wrong, the correct number is N(C_{M}; 1, 1, 1, 1, 0, 0) = 1. One of the two fixed points of the transformation X ↦ X′ is on the “other” component of Y_{naive} in (5), not on Y_{C_{M}}. Geometrically, one of the fixed points of the transformation corresponds to all the points lying on the line of P V_{5} and P V_{6}, which configuration does not belong to Y_{C_{M}}. There are configurations (e.g. the Ceva configuration) for which some “extra” components in Y_{naive} \ Y_{C} have bigger dimensions then Y_{C}. For these, arguments similar to that in Example 5.1 suggest the incorrect N(C; q_{1}, …, q_{k}) = ∞.

If the ideal I_{C} is known, determining the numbers N(C; q_{1}, …, q_{k}) reduces to algebraic calculations, which are, at least theoretically, doable. However, as we mentioned, the ideal I_{C} is not known in general. The equivariant class defined in Section 4.2 provides an answer.

Theorem 5.3. The coefficient of d_{1}^{q_{1}}d_{2}^{q_{2}} … d_{k}^{q_{k}} in [Y_{C}] is (−1)^{codim C}N(C; q_{1}, q_{2}, …, q_{k}).

This follows from a more or less standard intersection theoretic argument, which we show in detail in the next section.

Example 5.4. The matrix matroid variety corresponding to the configuration of Example 5.1 is a product of matrix Schubert varieties. Hence, its equivariant class is computed by Theorem 4.2 to be

(c_{1} − d_{1} − d_{3} − d_{5})(c_{1} − d_{2} − d_{3} − d_{4})(c_{1} − d_{1} − d_{2} − d_{6}).

The coefficient of d_{1}d_{2}d_{3} is −2, reproducing the result of Example 5.1.

Example 5.5. Special cases of N(C; q)’s are solutions to certain so-called Schubert problems. We illustrate this with the prototype of Schubert problems: how many straight lines intersect 4 generic lines in P^{3}? In our language the answer is N(C_{(0,1,3,0,0)}; 1, 1, 1, 1). According to Theorems 5.3 and 4.2 we have N(C_{(0,1,3,0,0)}; 1, 1, 1, 1) = coefficient of d_{1}d_{2}d_{3}d_{4} in

det (\begin{matrix}d_{1}d_{2} + … + d_{4}d_{5} & d_{1}d_{2}d_{3} + … + d_{3}d_{4}d_{5} \\ d_{1} + … + d_{4} & d_{1}d_{2} + … + d_{3}d_{4} \end{matrix}) ,

which is clearly \binom{4}{2} − \binom{4}{1} = 2.

Remark 5.6. Certain N(C; q) invariants are 0 for obvious reasons. For example, if P_{1}, P_{2}, and P_{3} are on one line in the configuration C, then N(C; 0, 0, 0, q_{4}, …, q_{k}) = 0. Indeed, P_{1}, P_{2}, and P_{3}, being on one line, can not be three generic points. Similarly, if there is a subset I = {i_{1}, …, i_{s}} ⊂ [k] such that ∑(q_{i_{j}} + 1) > r_{C}({i_{1}, …, i_{s}}), then obviously N(C; q) = 0. It is easy to see that the existence of such an I is the only reason for vanishing N(C; q).

The method in Section 8 to calculate [Y_{C}] for the configurations C_{M} (Menelaus), C_{C} (Ceva), C_{P} (Pappus), and C_{D} (Desargues) (just like any other configuration we tried) works, leading to the knowledge of all the Gromov-Witten invariants of these configurations. Below we present some information on these invariants.

Menelaus: All non-zero coefficients of the pure d monomials of [Y_{C_{M}}] are 1—for example

the one studied in Example 5.2. Ceva: The same holds for the Ceva configuration: the range of invariants is {0, 1}. Pappus: The range of the N(C_{P}; q) invariants is {0, 1, 2, 3, 4, 5}. Here are some sample

results.

• N(C_{P}; 1, 1, 1, 1, 1, 1, 1, 0) = 5, that is, the number of Pappus configurations on the plane whose i’th vertex is on a pre-described generic line l_{i} for i = 1, …, 8, and whose 9’th point is a pre-described generic point, is 5. We know no other way of finding this number.

• N(C_{P}; 2, 0, 0, 1, 1, 1, 1, 1, 1) = 4. The argument in Example 5.1 would suggest the wrong answer 5.

• N(C_{P}; 1, 1, 0, 2, 1, 1, 0, 0, 2) = N(C_{P}; 1, 1, 1, 1, 1, 0, 0, 1, 2) = 3. Desargues: Again, the range of the invariants N(C_{D}; q) is {0, 1}.

It would be interesting to find a geometric interpretation of the property of a configuration, which is equivalent to the condition that the range of N(C; q) is {0, 1}.

6. Proof of Theorem 5.3

We are going to present a proof of Theorem 5.3 which also proves positivity and enumerative properties of other coefficients.

Let C be a configuration of k vectors in C^{n}, such that Y_{C} is a codimension l subvariety of Hom(C^{k}, C^{n}).

Let τ_{n} be the universal tautological bundle over the Grassmannian Gr_{n} C^{∞} (universal subbundle) which we will approximate with the finite Grassmannian Gr_{n} C^{N} for a large N >> n, k. Below we will refer to certain numbers as ‘large’; by this we mean that those numbers tend to infinity as N → ∞. The cohomology of the finite Grassmannian is a factor of the cohomology H^{∗}(Gr_{n} C^{∞}) = Z[c_{1}, …, c_{n}] by an ideal with large degree generators. In our notations we will ignore this ideal, and identify the two cohomology rings.

Let ϕ : Gr_{n} C^{N} → Gr_{n} C^{N} be the (necessarily non-holomorphic) classifying map of the dual vector bundle τ_{n}^{∗}, and consider the induced diagram

\xymatrix{Σ̂_{C} \ar@{^{(}->}[r] & Hom(τ_{1}^{k}, τ_{n}^{∗}) \ar[r]^{ψ} \ar[d]_{ξ} & Hom(τ_{1}^{k}, τ_{n}) \ar[d] & Σ_{C} \ar@{^{(}->}[l] \\  & Gr_{n} C^{N} ×(P^{N−1})^{k} \ar[r]^{ϕ×id} & Gr_{n} C^{N} ×(P^{N−1})^{k}}

Here ψ is the map induced by ϕ × id, and Σ_{C} is the collection of the copies of Y_{C} in each fiber of the bundle Hom(τ_{1}^{k}, τ_{n}). That is, by definition, the cohomology class represented by Σ_{C} in the cohomology H^{∗}(Hom(τ_{1}^{k}, τ_{n})) = H^{∗}(Gr_{n} C^{N} ×(P^{N−1})^{k}) is the equivariant class [Y_{C}]. We set Σ̂_{C} = ψ^{−1}(Σ_{C}).

The homomorphism (ϕ × id)^{∗} : Z[c_{1}, …, c_{n}, d_{1}, …, d_{k}] → Z[c_{1}, …, c_{n}, d_{1}, …, d_{k}] induced by the map ϕ × id maps c_{i} to (−1)^{i}c_{i}, and d_{i} to d_{i}. Hence we have that

[Σ̂_{C} ⊂ Hom(τ_{1}^{k}, τ_{n}^{∗})] = (ϕ × id)^{∗}[Σ_{C} ⊂ Hom(τ_{1}^{k}, τ_{n})] = [Y_{C}]|_{c_{i}↦(−1)^{i}c_{i}}.

Observe that the bundle ξ = Hom(τ_{1}^{k}, τ_{n}^{∗}) has a large dimensional space of sections. Indeed, let α_{i} : C^{N} → (C^{N})^{∗} be linear maps for i = 1, …, k. Then the map

(10)  s_{α} : (L^{n} ≤ C^{N}, l_{1}^{1}, …, l_{k}^{1} ≤ C^{N}) ↦ ∑_{i=1}^{k} ι_{L}^{∗}α_{i}ι_{l_{i}}

is a section of ξ; where ι_{L} : L → C^{N}, ι_{l_{i}} : l_{i} → C^{N} are the canonical inclusions, and ι_{L}^{∗} : (C^{N})^{∗} → L^{∗} is the adjoint map.

We want to show that for an appropriate choice of α = (α_{1}, …, α_{k}) the section s_{α} of ξ is transversal to Σ̂_{C}. We will use the following straightforward generalization of the classical Bertini theorem.

Proposition 6.1. Let E → X be a smooth vector bundle, B a vector space and φ : B → Γ(E) is a linear family of smooth sections. Suppose that E is generated by the sections φ(b), b ∈ B, i.e. Φ(b, x) := φ(b)(x) : B × X → E is surjective, and Y is a smooth submanifold of the total space E. Then there is a b ∈ B such that φ(b) is transversal to Y.

For the proof we will use two lemmas.

Lemma 6.2. [GG73, II.4.6] Let B, X, Y be smooth manifolds, Φ : B × X → Y smooth and transversal to the submanifold W ⊂ Y. The map Φ encodes a family of maps Φ_{b}(x) = Φ(b, x) : X → Y for b ∈ B. Then

{b ∈ B : Φ_{b} |∩ W }

is dense in B. □

Lemma 6.3. Let E → X be a smooth vector bundle, B a vector space, and φ : B → Γ(E) a linear family of smooth sections. Suppose that E is generated by the sections φ(b), b ∈ B, i.e.

Φ(b, x) := φ(b)(x) : B × X → E

is surjective. Then dΦ|_{(b,x)} : T_{(b,x)}(B × X) → T_{Φ(b,x)}E is surjective for all (b, x) ∈ B × X.

Proof of Lemma 6.3. The statement is local, so we can assume that E = C^{r} ×X. Then Φ(b, x) = (h(b, x), x) for some smooth map h : B × X → C^{r}, which is linear in the b variable. Then Φ is surjective if and only if for all x ∈ X the linear map h(·, x) : B → C^{r} is surjective. On the other hand

dΦ = ( \begin{matrix}∂_{b}h & ∂_{x}h \\ 0 & I \end{matrix} )

and (∂_{b}|_{(b,x)}h) (v) = h(v, x) because of the linearity of h in b. Therefore the surjectivity of h(·, x) implies that the matrix ∂_{b}h has full rank, and hence it also implies the surjectivity of dΦ. □

Proof of Proposition 6.1. We apply Lemma 6.2. and 6.3. Since Φ transversal to all points, it is transversal to all submanifolds. □

Proposition 6.1 immediately implies the following algebraic version.

Proposition 6.4. Let E → X be an algebraic vector bundle, B a vector space, and φ : B → Γ(E) a linear family of algebraic sections. Suppose that E is generated by the sections φ(b), b ∈ B, i.e. Φ(b, x) := φ(b)(x) : B × X → E is surjective, and Y is a subvariety of the total space E. Then there is an open subset U of B such that for all b ∈ U the section φ(b) is transversal to Y.

Transversality to a variety means that φ(b) is transversal to all strata of Y for a complex stratification of Y. If a map f of smooth varieties is transversal to a subvariety Y in this sense then f^{∗}([Y]) = [f^{−1}(Y)] just as if Y were smooth.

We mention that if in Proposition 6.4 we take B = H^{0}(E), then we get that for a vector bundle generated by global sections we always have an open subset of the sections which is transversal to any given subvariety of the total space.

Proposition 6.5. The family of sections s_{α} (defined in (10)) generates the bundle ξ.

Proof. Fix L^{n} ≤ C^{N} and l^{1} ≤ C^{N} and let ψ(β) = ι_{L}^{∗}βι_{l} for β ∈ Hom(C^{N}, (C^{N})^{∗}). It is enough to show that ψ : Hom(C^{N}, (C^{N})^{∗}) → Hom(l, L^{∗}) is surjective, which is clear since ψ(β) is a n × 1 submatrix of β in an appropriate coordinate system. □

Hence we may choose an s_{α} transversal to Σ̂_{C}. We have that V_{C} = s_{α}^{−1}(Σ̂_{C}) is a codimension l subvariety of Gr_{n} C^{N} ×(P^{N−1})^{k}, which represents [Y_{C}]|_{c_{i}↦(−1)^{i}c_{i}} in the cohomology of Gr_{n} C^{N} ×(P^{N−1})^{k}.

Now fix a complete flag in C^{N}, and consider the products of Schubert varieties

S_{λ,q} = S_{λ} × (S_{q_{1}} × … × S_{q_{k}}) ⊂ Gr_{n} C^{N} ×(P^{N−1})^{k}.

Here λ is a partition (with number of parts ≤ n), and S_{i} ⊂ P^{N−1} is a linear space of codimension i, with q = (q_{1}, …, q_{k}). The cohomology classes represented by the S_{λ,q}’s form an additive basis of the cohomology group, and by the Giambelli formula of Schubert calculus we know that

[S_{λ,q}] = Δ_{λ}(−c_{1}, c_{2}, −c_{3}, c_{4}, …, (−1)^{n}c_{n}) · ∏_{i=1}^{k}(−d_{i})^{q_{i}}.

Recall also that this basis is self dual in the sense that ∫ [S_{λ,q}][S_{μ,w}] = 0 unless μ = λ′, the complement of λ in the n × (N − n) rectangle, and w = q′, ie, q + w = (N, …, N) (in which case, the integral is 1).

By appropriate choice of the section s_{α} we may also assume that V_{C} is transversal to the Schubert varieties S_{λ,q}. Let |λ| + ∑ q_{i} = dim(Gr_{n} C^{N} ×(P^{N−1})^{k}) − l. Then we have that

#(V_{C} ∩ S_{λ,q}) = ∫ [V][S_{λ,q}],

which is then the coefficient of [S_{λ′,q′}] when [V_{C}] is written as a linear combination of [S_{μ,w}]’s. By rephrasing we obtain the following theorem.

Theorem 6.6. The coefficient of Δ_{λ}(c_{1}, c_{2}, …, c_{n}) ∏_{i}(−d_{i})^{q_{i}} in [Y_{C}], when the latter is written as a linear combination of classes

Δ_{μ}(c_{1}, c_{2}, …, c_{n}) ∏_{i=1}^{k}(−d_{i})^{w_{i}},

is the intersection number #(V_{C} ∩ S_{λ′,q′}). Hence all coefficients in this linear combination are nonnegative.

Let us now consider the special case of λ being the empty partition, and ∑ q_{i} = l. We obtain that the coefficient of (−1)^{l} ∏_{i} d_{i}^{q_{i}} in [Y_{C}] is the intersection number

#(V ∩ (point × H_{1} × … × H_{k})),

where H_{i} are generic linear subspaces in P^{N} of total codimension l. Identifying the point in Gr_{n} C^{N} with C^{n} this intersection number is the same as the definition of N(C; q). This proves Theorem 5.3.

Theorem 5.3 shows the geometric interpretation of the pure d coefficients of [Y_{C}], but the other extreme, the pure c coefficients are also noteworthy. By the definition of GL(n)-equivariant cohomology we have

Theorem 6.7. Suppose π : E → B is a rank n vector bundle with Chern classes c_{1}, …, c_{n}. Assume that π has k sections satisfying a certain transversality property. Then the cohomology class in B represented by the subvariety

{b ∈ B|s_{1}(b), …, s_{k}(b) form a configuration belonging to Y_{C} ⊂ π^{−1}(b)}

is equal to [Y_{C}]^{∗}.

The transversality property can be easily phrased. Over the real numbers it is a generic property of k-tuples of sections. Thus one obtains a result on the parity of (the cohomology class represented by) the points over which k generic sections of a real projective space bundle form a given configuration C.

Remark 6.8. Certain facts suggest some kind of relations between the c and the d variables of [Y_{C}]. One of these facts is that either one can be used to calculate the degree of Y_{C}—hence they can not be independent. More generally, it can be shown that [Y_{C}] can be written as a polynomial of the weights γ_{i}−d_{j} of the representation in Section 4.2. Another fact is that for matrix Schubert varieties the pure c part determines the whole [Y_{C}] (up to permutation of indexes). Since the pure c part of [Y_{C}] can be presented in a more compact way in general, it would be interesting to see the relation in general. However, we may not expect that the pure c part determines the pure d part for any C. For example, let C_{1} be the configuration of 7 points on the projective plane with the collinearities 123, 145, and 167 (and otherwise general). Let C_{2} be with the collinearities 123, 345, 567 (and otherwise general). The pure c part of the equivariant classes of both of these are c_{1}^{3}. The pure d parts are essentially different (see Theorem 4.2).

7. Stabilization

The interior structure of natural infinite sequences of equivariant cohomology classes of geometrically relevant varieties has remarkable connections with the theory of symmetric functions [Nak99], and iterated residue identities for hyperplane arrangements [BS06, FR08]. In this section we make the first step towards exploring similar relations for the classes [Y_{C}], by showing the property analogous with the so-called d-stability property of Thom polynomials of contact singularities, cf. [FR08, Sect. 7.3]. A byproduct—important in Section 8—is Theorem 7.4 on the vanishing of certain coefficients of [Y_{C}].

Let C be a configuration of k vectors in C^{s}, spanning C^{s}, and let codim(Y_{C} ⊂ C^{s×k}) = l. For n ≥ s let the configuration C^{#n} be obtained from C through the natural embedding of C^{s} into C^{n}. Hence C^{#n} is a configuration of k points in C^{n}, spanning an s dimensional subspace. It is easy to check that codim(Y_{C}_{#n} ⊂ C^{n×k}) = l + (k − s)(n − s). Our goal is to compare the classes

(11)  [Y_{C}] ∈ Z[c_{1}, …, c_{s}, d_{1}, …, d_{k}] and [Y_{C}_{#n}] ∈ Z[c_{1}, …, c_{n}, d_{1}, …, d_{k}].

Clearly Y_{C}_{#n} ∩C^{s×k} = Y_{C}. If this intersection was transversal, or if Y_{C}_{#n} was equal to Y_{C}, then the relation between the two equivariant classes would follow from standard notions of equivariant cohomology. None of these is the case; the relation between the two polynomials does not follow from straightforward applications of equivariant cohomology notions. The relation between the two classes must involve nontrivial algebra, since it involves nontrivial geometry—consider for example the case when C is the collection of 4 generic vectors in C^{2}. In this case [Y_{C}] = 1 while [Y_{C}_{#4}] has a term 2d_{1}d_{2}d_{3}d_{4}, whose coefficient is the solution of the Schubert problem of Example 5.5.

Below we will work with the Chern roots γ_{i} of GL(m) (m = s or n), instead of the Chern classes c_{i}. That is, we identify H^{∗}(BGL(m)) = Z[c_{1}, …, c_{m}] with the symmetric polynomials of γ_{1}, …, γ_{m}; where the i’th elementary symmetric polynomial is c_{i}. In our notation, hence, Y_{C} may be a polynomial in c_{i}’s and d_{i}’s, or a polynomial in γ_{i}’s and d_{i}’s necessarily symmetric in the γ_{i}’s. If S is an s-element subset of [n], then [Y_{C}](γ_{S}) will denote the value of [Y_{C}] if we substitute the variables γ_{i}, i ∈ S for γ_{1}, …, γ_{s}.

Theorem 7.1. Let \binom{[n]}{s} denote the set of s-element subsets of [n]. For such a subset S, let S̄ denote the complement of S in [n]. Then we have

(12)  [Y_{C}_{#n}] = ∑_{S∈\binom{[n]}{s}} \frac{[Y_{C}](γ_{S}) · ∏_{i∈S̄} ∏_{j=1}^{k}(γ_{i} − d_{j})}{∏_{i∈S̄} ∏_{j∈S}(γ_{i} − γ_{j})}.

Proof. Let ε^{k} denote the trivial bundle of rank k, and let τ_{s} be the tautological bundle (or rank s) over Gr_{s} C^{n}. The embedding of bundles τ_{s} ⊂ ε^{n} induces the embedding of bundles i : Hom(ε^{k}, τ_{s}) → Hom(ε^{k}, ε^{n}). The maximal torus U(1)^{n} × U(1)^{k} of G_{n,k} acts on the following diagram

\xymatrix{Y_{C}(ξ) ⊂ & Hom(ε^{k}, τ_{s}) \ar@{^{(}->}[r]^{i} \ar@{-->}[dr]_{ξ} & Hom(ε^{k}, ε^{n}) \ar[d]^{π_{1}} \ar[r]^{π_{2}} & Hom(C^{k}, C^{n}) & ⊃ Y_{C}_{#n} \\  &  & Gr_{s} C^{n},}

where π_{1} are π_{2} the projections of Hom(ε^{k}, ε^{n}) = Gr_{s} C^{n} × Hom(C^{k}, C^{n}), and Y_{C}(ξ) is the collection of the Y_{C}-points in each fiber of ξ. The composition π_{2} ∘ i is a birational map from Y_{C}(ξ) to Y_{C}_{#n}. Therefore we can apply the fibered version of the Atiyah-Bott localization theorem, Theorem (3.8) in [BS06] (see also [FR08, Prop.5.1]), and we obtain the theorem. □

Another relation between the classes (11) stems from the following theorem.

Theorem 7.2. Let n > s, and [Y_{C}_{#n}] = ∑ γ_{n}^{i}p_{i}(γ_{1}, …, γ_{n−1}, d_{1}, …, d_{k}). Then (1) p_{i} = 0 for i > k − s, and

(2) p_{k−s} = k_{C} · [Y_{C}_{#(n−1)}] for an integer k_{C}.

Proof. Apply Theorem 2.1 of [FR07]. □

To explore the algebraic consequences of Theorem 7.2 we define lowering and raising operators on constant width polynomials.

Definition 7.3. The width of a monomial in Z[c_{1}, …, c_{n}] is the number of factors in it. The width of a polynomial is the width of its widest term. Let P_{w}^{n} be the vector space of width ≤ w polynomials in Z[c_{1}, …, c_{n}]. The lowering operator L_{w}^{n} : P_{w}^{n+1} → P_{w}^{n} is defined to be the linear extension of

L_{w}^{n} (c_{i_{1}}c_{i_{2}} … c_{i_{w}}) = c_{i_{1}−1}c_{i_{2}−1} … c_{i_{w}−1},

and L_{w}^{n}(c_{I}) = 0 if the width of c_{I} is less than w (c_{0} is defined to be 1). L_{w}^{n} decreases the degree by w. The raising operator (increasing the degree by w) R_{w}^{n} : P_{w}^{n} → P_{w}^{n+1} is defined by

p = ∑_{λ} a_{λ}Δ_{λ}(c_{1}, …, c_{n})  ↦  ∑_{λ} a_{λ}Δ_{(λ_{1}+1,...,λ_{w}+1)}(c_{1}, …, c_{n+1}),

where ∑_{λ} a_{λ}Δ_{λ} is the unique expression of p as a linear combination of Schur polynomials corresponding to partitions with w parts.

For instance we have L_{3}^{2}(c_{1}c_{2}c_{3}+5c_{3}^{2}) = c_{1}c_{2}, and R_{3}^{2}(c_{1}c_{2}) = R_{3}^{2}(Δ_{210}(c_{1}, c_{2})) = Δ_{321}(c_{1}, c_{2}, c_{3}) = c_{1}c_{2}c_{3} − c_{3}^{2}. We have the one-sided inverse property L_{w}^{n} ∘ R_{w}^{n} =id, but not the other way around.

Theorem 7.4. Let C be a configuration of k vectors in C^{n}, spanning an s dimensional subspace. • The width of [Y_{C}]^{∗} is at most k − s.

• If [Y_{C}]^{∗} is written in the Schur basis Δ_{λ}, then all occurring λ have at most k − s parts.

Proof. If [Y_{C}]^{∗} had a term of width i > k − s, then L_{i}^{n}([Y_{C}]^{∗}) would not be 0, contradicting to Theorem 7.2 (1) (cf. [FR07, Cor.2.5]). This proves the first statement. The second is a combinatorial rephrasing of the first one. □

Theorem 7.5. Let n, k ≥ s, and let us use the notations of Theorem 7.1. Let λ be a partition with at most k − s parts. Then we have

(13)  R_{k−s}^{n−1} ∘ … ∘ R_{k−s}^{s+1} ∘ R_{k−s}^{s} (Δ_{λ}(c_{1}, …, c_{s})) = ∑_{S∈\binom{[n]}{s}} \frac{Δ_{λ}(γ_{S}) · ∏_{i∈S̄} γ_{i}^{k}}{∏_{i∈S̄} ∏_{j∈S}(γ_{i} − γ_{j})}.

Proof. The polynomial Δ_{λ}(c_{1}, …, c_{s}) is the equivariant class [Y_{C}]^{∗} of an appropriate matrix Schubert variety, according to Theorem 4.2. Then Theorem 7.1 gives that the right hand side is the GL(n) equivariant class of another matrix Schubert variety. Checking the indexes, and applying Theorem 4.2 again we obtain the left hand side. □

It would be interesting to find a combinatorial proof of this theorem, possibly along the line of the multivariate Lagrange interpolation formula for symmetric functions [CL96].

Finally, we have the description of the relation between the pure c parts of the equivariant classes (11).

Theorem 7.6. Let C be a configuration of k vectors in C^{s}, spanning C^{s}. Let n ≥ s, and let C^{#n} be obtained from C by the natural embedding C^{s} ⊂ C^{n}. Then [Y_{C}_{#n}]^{∗} has width at most k − s, and

[Y_{C}_{#n}]^{∗} = R_{k−s}^{n−1} ∘ … ∘ R_{k−s}^{s+1} ∘ R_{k−s}^{s} ([Y_{C}]^{∗}) .

Proof. According to Theorem 7.4 we can express [Y_{C}]^{∗} as a linear combination of Δ_{λ} polynomials with each λ having at most k − s parts. Let us apply the operation p ↦ ∑_{S} \frac{p·∏_{i∈S̄} γ_{i}^{k}}{∏_{i∈S̄} ∏_{j∈S}(γ_{i}−γ_{j})} to this expression. Theorem 7.5 gives our result. □

In particular the constant k_{C} above is 1. Furthermore, the pure c part of [Y_{C}] determines [Y_{C}_{#n}] by adding a (k − s) × (n − s) rectangle to each λ in its Schur expansion. The analogous phenomenon for equivariant classes of contact singularities is called the “finiteness of Thom series”, see [FR08]. However, finiteness of Thom series seems more the exception than the rule for contact singularities. The only known finite Thom series correspond to a trivial case (the algebras Z[x_{1}, …, x_{n}]/(x_{1}, …, x_{n})^{2}), the Giambelli-Thom-Porteous formula.

Denoting the Menelaus configuration considered in a subspace of C^{n} (n ≥ 3) by C_{M}^{#n} we obtain that

[Y_{C}{#n \atop M}]^{∗} = 3Δ_{n−1,n−2,n−2} + 2Δ_{n−1,n−1,n−3} + 3Δ_{n,n−2,n−3}.

Remark 7.7. The only property of the representation of Section 4.2 used in this section was that this representation is a quiver representation. Hence the suitable rephrasing of the localization, width, and vanishing results above are valid for all quiver representations.

8. The calculation of equivariant classes of matrix matroid varieties

The standard straightforward methods to calculate equivariant classes of invariant subvarieties—such as the method of resolution or (Gröbner) degeneration—assume more knowledge on the ideal of the variety than we have about the ideal of matrix matroid varieties.

What we can do is list certain properties of the class [Y_{C}], and hope that a computer search proves that there is only one element of the polynomial ring Z[c_{1}, …, c_{n}, d_{1}, …, d_{k}] that satisfies all these properties. The main such property—which we will call Interpolation property—is motivated by methods used in the theory of Thom polynomials of singularities.

8.1. Interpolation. For a configuration D ∈ C^{n×k} let G_{D} denote its stabilizer subgroup in G_{n,k}. The embedding G_{D} → G_{n,k} induces a map between classifying spaces BG_{D} → BG_{n,k}, and, in turn, a homomorphism between rings of characteristic classes ϕ_{D} : H^{∗}(BG_{n,k}) → H^{∗}(BG_{D}).

Theorem 8.1. [FR04, Th.3.2] If D ∉ Y_{C} then ϕ_{D}([Y_{C}]) = 0.

Theorem 8.1 is a homogeneous interpolating condition on [Y_{C}]. To obtain a non-trivial condition, however, we need to find a configuration D, outside of Y_{C}, with reasonably large symmetry group. Let us illustrate the usage of this theorem with an example.

Example 8.2. The calculation of [Y_{C_{M}}]. Consider the following configuration D_{1|2|6}: v_{1}, v_{2}, and v_{6} are three generic vectors in C^{3}, while v_{3} = v_{4} = v_{5} = 0 ∈ C^{3}. Clearly D_{1|2|6} is not in the closure of X_{C_{M}}, since for all configurations in X_{C_{M}} the vectors v_{1}, v_{2}, and v_{6} are coplanar. Therefore ϕ_{D_{1|2|6}}([Y_{C_{M}}]) = 0. The stabilizer subgroup of D_{1|2|6} is U(1)^{6} with the embedding into G_{3,6} via (diag(α, β, γ), diag(α, β, δ, η, θ, γ)). Hence—by abusing language and identifying general elements of a U(1) with the first Chern class of U(1)—the map ϕ_{D_{1|2|6}} : Z[c_{1}, c_{1}, c_{3}, d_{1}, …, d_{6}] → Z[α, β, γ, δ, η, θ] maps

c_{1} ↦ α + β + γ,  c_{2} ↦ αβ + αγ + βγ,  c_{3} ↦ αβγ,
d_{1} ↦ α,  d_{2} ↦ β,  d_{3} ↦ δ,  d_{4} ↦ η,  d_{5} ↦ θ,  d_{6} ↦ γ.

The vanishing of [Y_{C_{M}}] at this map is a non-trivial interpolation property of [Y_{C_{M}}]. In fact, one finds that the degree 4 part of the intersection of the kernels of the ϕ_{D_{1|2|6}}, ϕ_{D_{1|3|5}}, ϕ_{D_{2|3|4}}, and ϕ_{D_{4|5|6}} is 2-dimensional. (This is not surprising in the light of the decomposition (5).) Now let D_{124|356} be the following configuration: v_{1} = v_{2} = v_{4} and v_{3} = v_{5} = v_{6} are two different nonzero vectors in C^{3}. This configuration is not in the closure of X_{C_{M}} because of Menelaus’ theorem. Indeed, the left hand side of (4) is −1 for any configuration in X_{C_{M}}, but it is ∞ for D_{124|356}. As a consequence, [Y_{C_{M}}] must vanish at the map ϕ_{D_{124|356}} : Z[c_{1}, c_{1}, c_{3}, d_{1}, …, d_{6}] → Z[α, β, γ],

c_{1} ↦ α + β + γ,  c_{2} ↦ αβ + αγ + βγ,  c_{3} ↦ αβγ,
d_{1} ↦ α,  d_{2} ↦ α,  d_{3} ↦ β,  d_{4} ↦ α,  d_{5} ↦ β,  d_{6} ↦ β.

It turns out that there is only a 1-dimensional set of degree 4 polynomials in Z[c_{1}, c_{1}, c_{3}, d_{1}, …, d_{6}] vanishing at the five maps ϕ_{D_{1|2|6}}, ϕ_{D_{1|3|5}}, ϕ_{D_{2|3|4}}, ϕ_{D_{4|5|6}}, and ϕ_{D_{124|356}}. Normalization is achieved, for example, by observing that the coefficient of d_{3}^{2}d_{6}^{2} in [Y_{C_{M}}] has to be 1, due to Theorem 5.3.

Observe that the application of Theorem 8.1 to calculate [Y_{C}] resonates to the method of determining the ideal of Y_{C} discussed at the end of Section 3.2. That is, we first deal with the trivial conditions following from the closed conditions on X_{C}, then need to work with some extra equations besides these naive ones. What makes the equivariant cohomology calculation easier is that here we do not have to have a full understanding of all the fake components, or all the extra geometry theorems of the configuration. It is enough to find some of these, use these to find an interpolations constraint. And it is clear when we can stop: as soon as we find enough interpolation constraints to cut down the dimension of the solution set to 1, we can be sure we found [Y_{C}].

8.2. Calculation in practice. The three main conditions we may use to calculate the equivariant class [Y_{C}] are

• the interpolation conditions, Theorem 8.1;

• the enumerative conditions, Theorem 5.3;

• the width condition, Theorem 7.4. The first one depends on the choice of the test configuration D. The second one depends on the choice of the numbers q. For certain choices of D and q these conditions are far from being straightforward, because we do not know whether D belongs to Y_{C}, or the number N(C; q). For some other choices, however, simple arguments answer these questions, and hence we have explicit constraints of [Y_{C}]. See, for example, the calculation of [Y_{C_{M}}] above.

It is quite possible that the interpolation conditions themselves are enough to determine the equivariant class [Y_{C}] up to a scalar. For some other representations the analogous statement is a theorem, eg. [FR04, Thm. 3.5]. However the proof there depends on a condition of the representation (called Euler condition in [FR04, Def. 3.3], closely related to the “equivariantly perfect” condition of [AB83, Sect.1.]). This condition does not hold for the representation of Section 4.2.

What works in practice, is the combination of the three constraints. For all the configurations the authors considered (many more than the ones presented in this paper) there is only one polynomial of degree codim Y_{C} in Z[c_{1}, …, c_{n}, d_{1}, …, d_{k}] satisfying the simple straightforward constraints obtained from interpolation and enumeration, together with the width condition. We conjecture this holds for all configurations.

9. Hierarchy

The interpolation method highlights the importance of the hierarchy of the sets Y_{C}. In fact, the effective usage of the interpolation method to calculate [Y_{C}] assumes that we have another configuration D such that D ∉ Y_{C}. To indicate the non-triviality of this problem we challenge the reader with the problem of deciding whether the configuration D_{134|256} is contained in Y_{C_{C}} (for notations see Figure 2 and Example 8.2).

The hierarchy of the sets X_{C} is not “normal”, in the sense that there are examples of configurations C and D such that certain points of X_{D} are in the closure of X_{C}, some others are not. A small example is C = C_{M}, D =6 points on one line. Hence, we restrict our attention to the case when D is an orbit of the action in Section 4.2. In this case Theorem 8.1 yields the following: if ϕ_{D}([Y_{C}]) ≠ 0 then Y_{D} ⊂ Y_{C}.

The vanishing of ϕ_{D}([Y_{C}]) has no chance of determining the adjacency of Y_{C} and Y_{D} if the stabilizer group G_{D} is trivial (ie. it is the kernel of the representation, U(1)). However, when G_{D} is larger, we have found no counterexample to the following conjecture.

Conjecture 9.1. Let C and D be configurations of k points in C^{n}. Suppose that projectivizations of the non-zero vectors in D form a projectively independent set (hence of cardinality ≤ n). Then

Y_{D} ⊂ Y_{C}  ⇔  ϕ_{D}([Y_{C}]) ≠ 0.

Together with the effective algorithm of Section 8.2 computing [Y_{C}], this conjecture would serve as a computable criterion of hierarchy, cf. [FP07].

References

[AB83] M. F. Atiyah and R. Bott. The Yang-Mills equations over Riemann surfaces. Philos. Trans. Roy. Soc. London Ser. A, 308(1505):523–615, 1983.

[BF99] A. S. Buch and W. Fulton. Chern class formulas for quiver varieties. Invent. Math., 135(3):665–687, 1999.

[BS06] G. Bérczi and A. Szenes. Thom polynomials of Morin singularities. preprint, arXiv, math.AT/0608285, 2006.

[Buc08] A. Buch. Quiver coefficients of Dynkin type. Michigan Math. J., 57:93–120, 2008.

[CL96] W. Chen and J. Louck. Interpolation of symmetric functions. Adv. Math., 117:147–156, 1996.

[EG98] D. Edidin and W. Graham. Equivariant intersection theory. Invent. Math., 131(3):595–634, 1998.

[FNR05] L. M. Fehér, A. Némethi, and R. Rimányi. Degeneracy of two and three forms. Canad. Math. Bull., 48(4):547–560, 2005.

[FP07] L. M. Fehér and Zs. Patakfalvi. The incidence class and the hierarchy of orbits. arXiv:0705.3834, 2007.

[FR02] L. Fehér and R. Rimányi. Classes of degeneracy loci for quivers—the Thom polynomial point of view. Duke Math. J., 114(2):193–213, 2002.

[FR03] L. M. Fehér and R. Rimányi. Schur and Schubert polynomials as Thom polynomials—cohomology of moduli spaces. Cent. European J. Math., 4:418–434, 2003.

[FR04] L. M. Fehér and R. Rimányi. Calculation of Thom polynomials and other cohomological obstructions for group actions. In Real and Complex Singularities (Sao Carlos, 2002), number 354 in Contemp. Math., pages 69–93. Amer. Math. Soc., 2004.

[FR07] L. M. Fehér and R. Rimányi. On the structure of Thom polynomials of singularities. Bull. London Math. Soc., 39:541–549, 2007.

[FR08] L. M. Fehér and R. Rimányi. Thom series of contact singularities. arXiv:0809.2925, 2008.

[Ful84] W. Fulton. Intersection theory. Springer-Verlag, 1984.

[Ful92] W. Fulton. Flags, Schubert polynomials, degeneracy loci, and determinantal formulas. Duke Math. J., 65(3):381–420, 1992.

[Ful07] W. Fulton. Equivariant cohomology in algebraic geometry. Eilenberg lectures, Columbia University, www.math.lsa.umich.edu/˜ dandersn/eilenberg/, 2007.

[GG73] M. Golubitsky and V. Guillemin. Stable mappings and their singularities, volume 14 of Graduate Texts in Mathematics. Springer-Verlag, 1973.

[GPS01] Gert-Martin Greuel, Gerhard Pfister, and Hans Schönemann. Singular 3.0 — a computer algebra system for polynomial computations. In Manfred Kerber and Michael Kohlhase, editors, Symbolic computation and automated reasoning, The Calculemus-2000 Symposium, pages 227–233, Natick, MA, USA, 2001. A. K. Peters, Ltd.

[Kaz97] M. É. Kazarian. Characteristic classes of singularity theory. In The Arnold-Gelfand mathematical seminars, pages 325–340. Birkhäuser Boston, 1997.

[Kle76] S. Kleiman. Enumerative geometry of singularitites. In Real and complex singularities. Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, pages 297–396, 1976.

[KM05] A. Knutson and E. Miller. Gröbner geometry of Schubert polynomials. Annals of Mathematics, 161:1245– 1318, 2005.

[KMS06] A. Knutson, E. Miller, and M. Shimozono. Four positive formulae for type A quiver polynomials. Invent. Math., 166(2):229–325, 2006.

[KS06] Allen Knutson and Mark Shimozono. Kempf collapsing and quiver loci. math/0608327, 2006.

[Men00] Menelaus of Alexandria. Sphaerica, Book III. cc. AD 100. [Mnë88] N. E. Mnëv. The universality theorems on the classification problem of configuration varieties and convex

polytopes varieties. In Topology and geometry—Rohlin Seminar, number 1346, pages 527–543. Springer, 1988.

[MR07] R. Marangell and R. Rimanyi. The general quadruple point formula. arXiv:0712.2418, 2007.

[MS04] E. Miller and B. Sturmfels. Combinatorial Commutative Algebra, volume 227 of Graduate Texts in Mathematics. Springer-Verlag, 2004.

[Nak99] H. Nakajima. Lectures on Hilbert schemes of points on surfaces. Number 18 in University Lecture Series. AMS, 1999.

[PT01] V. V. Prasolov and V. M. Tikhomirov. Geometry. Number 200 in Transl. of Math. Monographs. AMS, 2001.

[RG95] Jürgen Richter-Gebert. Mnëv’s universality theorem revisited. Séminaire Lotharingien de Combinatoire, B34h, 1995.

[RGK] J. Richter-Gebert and U. Kortenkamp. The interactive geometry software Cinderella, version 2.0. http://cinderella.de/.

[Vak06] R. Vakil. Murphy’s Law in algebraic geometry: Badly-behaved deformation spaces. Inventiones Mathematicae, 164(3):569–590, 2006.

Department of Analysis, Eötvös Loránd University Budapest

  E-mail address: lfeher@renyi.hu

Alfréd Rényi Institute of Mathematics Budapest

  E-mail address: nemethi@renyi.hu

Department of Mathematics, University of North Carolina at Chapel Hill USA, and Alfréd Rényi Institute of Mathematics Budapest

  E-mail address: rimanyi@email.unc.edu