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$K$-rings of wonderful varieties and matroids

Shiyue Li (Brown University)
Tuesday, November 8, 2022 - 1:45pm
PDL C-38
Pre-seminar 1:45-2:30
TitleA primer on Chow rings of wonderful varieties and matroids. 
AbstractI will introduce wonderful varieties, matroids and their Chow rings and do some very concrete examples and computations. I will also situate these objects and their properties within the rich and recent developments of combinatorial geometry by giving examples. 
Seminar 2:30-3:20
Title: $K$-rings of wonderful varieties and matroids

The wonderful variety of a realizable matroid and its Chow ring have played key roles in solving many long-standing open questions in combinatorics and algebraic geometry. Yet, its $K$-rings are underexplored until recently. I will be sharing with you some discoveries on the $K$-rings of the wonderful variety associated with a realizable matroid: an exceptional isomorphism between the $K$-ring and the Chow ring, with integral coefficients, and a Hirzebruch–Riemann–Roch-type formula. These generalize a recent construction of Berget–Eur–Spink–Tseng on the permutohedral variety. 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 of an arbitrary matroid. As an application, we present the $K$-rings and compute the Euler characteristic of arbitrary line bundles of the Deligne–Mumford–Knudsen moduli spaces of rational stable curves with distinct marked points. Joint with Matt Larson, Sam Payne and Nick Proudfoot. 

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