Database of curve-enhanced rank 3 matroids
Sam Silver (email: samuelsilver-at-umass-dot-edu)

Let M̅0,n be the moduli space of stable n-pointed rational curves. The cone of curves NE1(M̅0,n) of M̅0,n lives inside the quotient of the ℝ-vector space N1(M̅0,n) with one basis element for each curve in M̅0,n by numerical equivalence (we call curves C1 and C2 numerically equivalent if they have the same intersection number pairing with all divisors D), and is defined as the closure of the convex hull of the cone generated by those classes in N1(M̅0,n) coming from an actual curve in M̅0,n. An F-curve is a curve in M̅0,n given by a 1-parameter family of trees of ℙ1s in which exactly one of the ℙ1s has 4 points and the rest have 3. Fulton conjectured the following:

(F-conjecture.) The effective cone of curves NE1(M̅0,n) is generated by (the classes of) F-curves.

Actually, he made a more general conjecture that included cones of cycles of higher dimension, but it has been disproved for dimensions above 1 [Ver]. By "generated", we mean that any point in NE1(M̅0,n) can be written as a finite linear combination of F-curves with positive coefficients. The main goal of this project is to search, computationally, for a counterexample to the F-conjecture, using a method involving matroids.

For potential counterexamples, we need curves. There is an embedding of the blowup at n points Bln(ℙ2) of the projective plane into M̅0,n, described in Theorem 3.1 of [CT]. We will take curves in the plane, and consider the embedding of their proper transform under this blowup. This embedding depends on essentially matroidal data, that of which triples among the n points are collinear, so we start our search with a given rank-3 matroid M. We can then add the condition that some subset of the points lie on a conic, cubic, etc., and then the class in N1(M̅0,n) of the embedding of its proper transform is dependent only on this combinatorial data. We refer to this data as a "curve-enhanced matroid" (CEM).

To any matroid M is assoicated an affine scheme RS(M), its realization space, which is the moduli space of realizations of M in the projective plane. We want any CEM we consider to be realizable (i.e. the matroid is realizable and a curve of specified degree through the specified points exists), so we at least want RS(M) to be nonempty. The condition that some points lie on a curve is the condition that some polynomials vanish, so we can add these constraints to RS(M) to get a new realization space R for our CEM. Furthermore, we are guided by a "rigidity" heuristic: the principle is that if an irreducible curve is a member of a k-parameter family with k > 0, then in some limit it should "bend and break", and the limit (which will exist in M̅0,n and be numerically equivalent) will be reducible. We only want to consider irreducible curves, since any reducible counterexample will have some component a counterexample as well, which we ought to have already checked if we're going in order of complexity. So we'd like the realization space to have dimension 0. We could have this if dim(RS(M)) = 0 and some points automatically lie on a common curve, or if dim(RS(M)) > 0 and we impose the condition that just enough points lie on a common curve that the dimension drops to 0.

Code has been written to search through lists of n-point matroids sourced from the Polymake database, and to determine whether or not each CEM found is a counterexample [Zhu]. This website lists the results of the search so far. The program will continue running until it becomes slow enough that it would be unreasonable to continue. We hope to check all cases for n = 9, 10, 11, 12, and to try some cases for n = 13, 14, 15.

We're additionally interested in "Pascal-type theorems", in analogy with Pascal's classic "mystical hexagon" theorem, which states that given 6 points in the plane, partitioned into two disjoint triples of pairs, if we draw lines through all the pairs in the two partitions and consider the three intersection points of pairs of these lines that lie outside the set of 6 points, then there exists a conic through the 6 given points if and only if the 3 extra points are collinear. We'd like to try and find any (new) situations where, due to purely matroidal (degree 1) data, some points have a common curve through them that isn't automatic, and the matroid isn't rigid, meaning we have a whole (k-parameter, k>0) family of situations where this happens, and it's realizable in characteristic 0, so that it isn't just a quirk of characteristic p and we can actually draw a picture.

This database contains realizable rank 3 matroids (combinatorial data of n points in ℙ2) augmented with the additional data of a subset of the set of points that must lie on a single curve (here we only consider conics and nodal cubics).

For each matroid, we give its position in the Matroids.Small collection in the Polymake database [PDB], its list of (nontrivial) hyperplanes and its realization space. For each enhancement, we give its realization space with its dimension (possibly just over some prime characteristics), the effective class of the embedding into M̅0,n of the proper transform of the conic under the blowup at the points of the matroid (as in Theorem 3.1 of [CT]) and the result of its test as a possible counterexample to Fulton's conjecture for curves [Zhu].

All stated dimensions are heuristic and could be underestimated (but not overestimated). The dimensions are computed so that we can trim down the set of conics that we test for a given matroid (we only test conics with enough extra points to make the enhanced matroid rigid). Overestimations would cause us to throw out some cases that should be tested (undesirable), whereas underestimation just means we test some extra cases, which is fine.

Code will be made available at a later date.

To keep things organized, we list only 9-point matroids below. Data for other values of n can be found at the following links: n=10, n=11, n=12, n=13, n=14

Last updated 2025-01-02
Bibliography
n=9