Castelnuovo-Mumford regularity bounds for singular surfaces

TL;DR

This paper proves the Eisenbud-Goto regularity conjecture for certain singular surfaces and provides bounds on regularity for zero-dimensional schemes and curves with specific singularities.

Contribution

It establishes the regularity conjecture for normal surfaces with particular singularities and introduces bounds for schemes and curves with embedded or isolated points.

Findings

Proves the Eisenbud-Goto conjecture for normal surfaces with rational, Gorenstein elliptic, and log canonical singularities.

Provides bounds on regularity for zero-dimensional schemes based on Loewy length.

Bounds regularity for curves with embedded or isolated points using arithmetic degree.

Abstract

We prove the regularity conjecture, namely Eisenbud-Goto conjecture, for a normal surface with rational, Gorenstein elliptic and log canonical singularities. Along the way, we bound the regularity for a dimension zero scheme by its Loewy length and for a curve allowing embedded or isolated point components by its arithmetic degree.