Seshadri stratifications and standard monomial theory

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.

Seshadri stratification for Schubert varieties and Standard Monomial Theory

“…As in the case of Newton-Okounkov theory, one of the main problems is to get an explicit description of the semigroups associated to the (quasi-)valuation. To achieve this, we show that the standard monomial theory for Schubert varieties constructed in [24] fits well into the concept for a standard monomial theory for a balanced and normal Seshadri stratification in [5].…”

“…This indication of a connection between standard monomial theory and valuation theory was the starting point for [9], where the second and third author tested out in the simplest (but nontrivial) case, the Grassmann variety, how to join ideas from standard monomial theory and associated semi-toric degenerations [4,7,33] together with the theory of Newton-Okounkov bodies [15] and its associated toric degenerations [1]. The task to find a setup like in [9] for a larger class of embedded projective varieties X ֒→ P(V ) was accomplished in [5] in a surprisingly general framework: Seshadri stratifications.…”

“…Theorem A. The Seshadri stratification of an embedded Schubert variety X(τ ) ֒→ P(V (λ)) arising from its Schubert subvarieties is balanced and normal in the sense of [5].…”

“…2) The standard monomial theory for X(τ ) ֒→ P(V (λ)) constructed in [24] is an example of a standard monomial theory for a balanced and normal Seshadri stratification in [5].…”

“…In Section 7 we show that the quasi-valuation on the path vectors gives the desired values, and the Seshadri stratification is balanced and normal. As a consequence we get a standard monomial theory in the sense of [5], which we describe in Section 8. The associated Newton-Okounkov simplicial complex is described in Section 9.…”