Asymptotic regularity of powers of ideals of points in a weighted projective plane

TL;DR

This paper investigates the long-term behavior of the regularity of symbolic powers of point ideals in weighted projective planes, revealing boundedness and periodicity properties and linking them to geometric conditions like negative curves.

Contribution

It establishes conditions under which the regularity difference is bounded or periodic and connects these properties to the existence of negative curves and Nagata's conjecture.

Findings

Regularity behaves asymptotically like a linear function.

Difference between regularity and linear function is bounded or periodic.

Existence of negative curves implies eventual periodicity of regularity.

Abstract

In this paper we study the asymptotic behavior of the regularity of symbolic powers of ideals of points in a weighted projective plane. By a result of Cutkosky, Ein and Lazarsfeld, regularity of such powers behaves asymptotically like a linear function. We study the difference between regularity of such powers and this linear function. Under some conditions, we prove that this difference is bounded, or eventually periodic. As a corollary we show that, if there exists a negative curve, then the regularity of symbolic powers of a monomial space curve is eventually a periodic linear function. We give a criterion for the validity of Nagata's conjecture in terms of the lack of existence of negative curves.