Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs

The chromatic polynomial of a graph G counts the number of proper colorings of G. We give an affirmative answer to the conjecture of Read and Rota-Heron-Welsh that the absolute values of the coefficients of the chromatic polynomial form a log-concave sequence. We define a sequence of numerical invariants of projective hypersurfaces analogous to the Milnor number of local analytic hypersurfaces. Then we give a characterization of correspondences between projective spaces up to a positive integer multiple which includes the conjecture on the chromatic polynomial as a special case. As a byproduct of our approach, we obtain an analogue of Kouchnirenko's theorem relating the Milnor number with the Newton polytope.Comment: Improved readability. Final version, to appear in J. Amer. Math. So

Correspondences between projective planes

We characterize integral homology classes of the product of two projective planes which are representable by a subvariety.Comment: Improved readability, 14 page

Positivity of Chern classes of Schubert cells and varieties

We show that the Chern-Schwartz-MacPherson class of a Schubert cell in a Grassmannian is represented by a reduced and irreducible subvariety in each degree. This gives an affirmative answer to a positivity conjecture of Aluffi and Mihalcea.Comment: Improved readability, 18 page

Enumeration of points, lines, planes, etc

One of the earliest results in enumerative combinatorial geometry is the following theorem of de Bruijn and Erd\H{o}s: Every set of points $E$ in a projective plane determines at least $|E|$ lines, unless all the points are contained in a line. Motzkin and others extended the result to higher dimensions, who showed that every set of points $E$ in a projective space determines at least $|E|$ hyperplanes, unless all the points are contained in a hyperplane. Let $E$ be a spanning subset of a $d$-dimensional vector space. We show that, in the partially ordered set of subspaces spanned by subsets of $E$, there are at least as many $(d-k)$-dimensional subspaces as there are $k$-dimensional subspaces, for every $k$ at most $d/2$. This confirms the "top-heavy" conjecture of Dowling and Wilson for all matroids realizable over some field. The proof relies on the decomposition theorem package for $\ell$-adic intersection complexes.Comment: 18 pages, major revisio

Log-concavity of characteristic polynomials and the Bergman fan of matroids

In a recent paper, the first author proved the log-concavity of the coefficients of the characteristic polynomial of a matroid realizable over a field of characteristic 0, answering a long-standing conjecture of Read in graph theory. We extend the proof to all realizable matroids, making progress towards a more general conjecture of Rota-Heron-Welsh. Our proof follows from an identification of the coefficients of the reduced characteristic polynomial as answers to particular intersection problems on a toric variety. The log-concavity then follows from an inequality of Hodge type.Comment: 12 page