Combinatorial Seshadri stratifications on normal toric varieties

TL;DR

This paper provides a purely combinatorial framework for Seshadri stratifications on embedded normal toric varieties by replacing geometric stratifications with triangulations of the defining polytope indexed by flags of faces. It constructs a higher-rank quasi-valuation from a torus-equivariant stratification, shows the image forms a fan of monoids isomorphic to the weight-monoid of the embedding, and proves the associated graded algebra is a fan algebra yielding a reduced, equidimensional degeneration $X_0$ as a union of toric components. A flat one-parameter degeneration is produced via a $\mathbb G_m$-action, interpretable both in a weighted-projective setting and in a shadow inside the original embedding, with a detailed comparison to CFL’s framework. In the integral case, lattice-point triangulations yield explicit toric degenerations and connect the degenerations to normalization relations among maximal-chain components. Overall, the work provides a combinatorial, triangulation-driven route to construct and understand degenerations of toric varieties and their quasi-valuations, with potential generalizations to spherical varieties.

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 $X_P$.

Theorems & Definitions (111)

• Theorem A
• Theorem B
• Theorem C
• Definition 1.1
• Definition 1.2
• Definition 1.3
• Lemma 1.4
• Definition 2.1
• Example 2.2
• Example 2.3