Algebra, Geometry, and Combinatorics Seminar

Title: Families of degenerations from mutations of polytopes

Abstract: Theory of Newton-Okounkov bodies has led to the extension of the geometry-combinatorics dictionary from toric varieties to varieties which admit a toric degeneration. In a paper with Harada, we gave a piecewise-linear bijection between Newton-Okounkov bodies of a single variety. This involves a collection of lattices connected by piecewise-linear bijections. Inspired by these ideas in joint work in progress with Harada and Manon we propose a generalized notion of polytopes in $\Lambda=(\{M_i\}_{i\in I},\{\mu_{ij}\}_{i,j\in I})$, where the $M_i$ are lattices and the $\mu_{ij}:M_i\to M_j$ are piecewise-linear bijections. Roughly, these are $\{P_i\mid P_i\subseteq M_i\otimes \mathbb{R}\}_{I\in I}$ such that $\mu_{ij}(P_i)=P_j$ for all $i,j$. In analogy with toric varieties a generalized polytope can encode a compactification of an affine variety as well as toric degenerations for the compactification.

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