Positivity and complexity of ideal sheaves

TL;DR

This paper introduces a geometric invariant s(I) for ideal sheaves that measures positivity, relates it to asymptotic regularity, and explores its behavior under various operations, providing new insights into ideal complexity.

Contribution

It defines the s-invariant for ideal sheaves, links it to asymptotic Castelnuovo-Mumford regularity, and demonstrates its stability under geometric and algebraic operations.

Findings

s(I) computes the asymptotic regularity of large powers of I

Bounds on generator degrees imply bounds on s(I)

s(I) remains well-behaved under natural operations

Abstract

The problem of bounding the "complexity" of a polynomial ideal in terms of the degrees of its generators has attracted considerable interest, brought into focus by the influential survey of Bayer and Mumford. The present paper examines some of these results and questions from a geometric perspective. Specifically, motivated by work of Paoletti we introduce an invariant s(I) that measures the "positivity" of an ideal sheaf I. Degree bounds on generators of I yield bounds on this s-invariant, but s(I) may be small even when the degrees of its generators are large. We prove that s(I) computes the asymptotic Castelnuovo-Mumford regularity of large powers of I, and bounds the asymptotic behavior of several other measures of complexity. We also show that this s-invariant behaves very well with respect to natural geometric and algebraic operations, which leads to considerably simplified constructions of varieties with irrational asymptotic regularity. The main tools are intersection theory and vanishing theorems.