Title:
Geometric constructions from abstract wall-crossing
Abstract:
Theory of Newton-Okounkov bodies has led to the extension of the geometry-combinatorics dictionary from toric varieties to certain varieties which admit a toric degeneration. In a recent paper with Megumi Harada, we gave a wall-crossing formula for the Newton-Okounkov bodies of a single variety. Our wall-crossing involves a collection of lattices $\{M_i\}_{i\in I}$ connected by piecewise-linear bijections $\{\mu_{ij}\}_{i,j\in I}$. In addition, Kiumars Kaveh and Christopher Manon analyzed valuations into semifields of piecewise linear functions and explored their connections to families of toric degenerations. Inspired by these ideas in joint work in progress with Megumi Harada and Christopher 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 these generalized polytopes can encode compactifications of affine varieties as well as some of their geometric properties. In this talk, we illustrate these ideas.
