Chow rings of wonderful varieties and matroids have played key roles in solving many long-standing open questions in combinatorics and algebraic geometry. I will be sharing with you some new discoveries on K-rings of wonderful varieties associated with realizable matroids. We also compute the Euler characteristic of every line bundle on wonderful varieties, and give a purely combinatorial formula. This in turn gives a new valuative invariant for an arbitrary matroid. As an application, we study the K-ring and compute the Euler characteristic of every line bundle of M0,n —- the Deligne-Mumford-Knudsen compactification of the moduli space of rational stable curves of n distinct marked points. Joint work with Matt Larson, Sam Payne and Nicholas Proudfoot.
When studying subvarieties of projective space homological algebra over the standard graded polynomial provides several useful tools (free resolutions, syzygies, Castelnuovo-Mumford regularity, etc.) which capture nuanced geometric information. One might hope that there are analogous tools over multigraded polynomial rings, which provide similar geometric information for subvarieties of other toric varieties. I will discuss recent work developing such tools, as well as some of the subtleties that arise when moving to toric varieties beyond projective space. This is joint work with Lauren Cranton Heller and Mahrud Sayrafi.
I will give an introduction to Oliver Lorscheid’s theory of ordered blueprints – one of the more successful approaches to “the field of one element” – and sketch its relationship to Berkovich spaces, tropical geometry, Tits models for algebraic groups, and moduli spaces of matroids. The basic idea for the latter two applications is quite simple: given a scheme over Z defined by equations with coefficients in {0,1,-1}, there is a corresponding “blue model” whose K-points (where K is the Krasner hyperfield) sometimes correspond to interesting combinatorial structures. For example, taking K-points of a suitable blue model for a split reductive group scheme G over Z gives the Weyl group of G, and taking K-points of a suitable blue model for the Grassmannian G(r,n) gives the set of matroids of rank r on {1,...,n}. Similarly, the Berkovich analytification of a scheme X over a valued field k coincides, as a topological space, with the set of T-points of X, considered as an ordered blue scheme over k. Here T is the tropical hyperfield, and T-points are defined using the observation that a (height 1) valuation on k is nothing other than a homomorphism to T.