Combinatorial Seshadri stratifications on normal toric varieties

TL;DR

This paper introduces a purely combinatorial approach to Seshadri stratifications on normal toric varieties, linking polytope triangulations, orbit closures, and degenerations, and shows equivalence with previous methods.

Contribution

It presents a new combinatorial framework for Seshadri stratifications on toric varieties, independent of prior work, connecting triangulations, stratifications, and degenerations.

Findings

Establishes a connection between polytope triangulations and Seshadri stratifications.

Shows the combinatorial approach yields the same degenerations as previous methods.

Provides a new perspective on degenerations of toric varieties via combinatorics.

Abstract

We apply the theory of Seshadri stratifications to embedded toric varieties $X_P\subseteq \mathbb P(V)$ associated with a normal lattice polytope $P$. The approach presented here is purely combinatorial and completely independent of \cite{CFL}. In particular, we get a close connection between a certain class of triangulations of the polytope $P$, Seshadri stratifications of $X_P$ arising from torus orbit closures, and the associated degenerate semi-toric varieties. In the last section we show that the approach here and the one in \cite{CFL} produce the same quasi-valuations and hence the same degenerations of $X_P$.