Tropical schemes, tropical cycles, and valuated matroids

Algebra Seminar
Monday, November 4, 2013 - 16:05
1 hour (actually 50 minutes)
Skiles 005
University of Warwick
The tropical cycle associated to a subvariety of a torus is the support of a weighted polyhedral complex that that records information about the original variety and its compactifications. In a recent preprint Jeff and Noah Giansiracusa introduced a notion of scheme structure for tropical varieties, and showed that the tropical variety as a set is determined by this tropical scheme structure. I will outline how to also recover the tropical cycle from this information. This involves defining a variant of Grobner theory for congruences on the semiring of tropical Laurent polynomials. The lurking combinatorics is that of valuated matroids. This is joint work with Felipe Rincon.