Seshadri stratifications and standard monomial theory

TL;DR

This paper introduces Seshadri stratifications on embedded projective varieties, enabling the construction of Newton-Okounkov complexes and flat degenerations into toric varieties, providing a geometric framework for standard monomial theory and interpreting Lakshmibai-Seshadri paths geometrically.

Contribution

It develops a new geometric structure called Seshadri stratification that links standard monomial theory with Newton-Okounkov bodies and toric degenerations.

Findings

Constructed Newton-Okounkov simplicial complexes.

Established flat degenerations into toric varieties.

Provided geometric interpretation of Lakshmibai-Seshadri paths.

Abstract

We introduce the notion of a Seshadri stratification on an embedded projective variety. Such a structure enables us to construct a Newton-Okounkov simplicial complex and a flat degeneration of the projective variety into a union of toric varieties. We show that the Seshadri stratification provides a geometric setup for a standard monomial theory. In this framework, Lakshmibai-Seshadri paths for Schubert varieties get a geometric interpretation as successive vanishing orders of regular functions.