Classification and syzygies of smooth projective varieties with 2-regular structure sheaf

TL;DR

This paper classifies smooth projective varieties with 2-regular structure sheaf and analyzes their syzygies, providing bounds on Castelnuovo-Mumford regularity and advancing understanding beyond the well-studied 1-regular case.

Contribution

It offers a classification of varieties with 2-regular structure sheaf and investigates their syzygies and Betti tables, extending known results from the 1-regular case.

Findings

Classification of varieties with 2-regular structure sheaf

Sharp bounds for Castelnuovo-Mumford regularity

Analysis of syzygies and Betti tables

Abstract

The geometric and algebraic properties of smooth projective varieties with 1-regular structure sheaf are well understood, and the complete classification of these varieties is a classical result. The aim of this paper is to study the next case: smooth projective varieties with 2-regular structure sheaf. First, we give a classification of such varieties using adjunction mappings. Next, under suitable conditions, we study the syzygies of section rings of those varieties to understand the structure of the Betti tables, and show a sharp bound for Castelnuovo-Mumford regularity.