Asymptotic Hilbert Polynomial and limiting shapes

TL;DR

This paper introduces a method to determine the limiting shapes of symbolic generic initial systems for higher-dimensional subvarieties in projective space, extending previous work on point ideals.

Contribution

It generalizes the connection between limiting shape volumes and asymptotic multiplicity from points to higher-dimensional sets.

Findings

Established a generalized relationship between limiting shapes and asymptotic multiplicity for higher-dimensional varieties.

Provided a new method for calculating limiting shapes of symbolic generic initial systems.

Extended previous results from point ideals to higher-dimensional subvarieties.

Abstract

The main aim of this paper is to provide a method which allows finding limiting shapes of symbolic generic initial systems of higher-dimensional subvarieties of P^n. M. Mustata and S. Mayes established a connection between volumes of complements of limiting shapes and the asymptotic multiplicity for ideals of points. In the paper we prove a generalization of this fact to higher-dimensional sets.