Syzygies of surfaces of general type

TL;DR

This paper investigates the syzygies and projective properties of surfaces of general type embedded by specific line bundles, establishing bounds for when these embeddings satisfy certain algebraic conditions.

Contribution

It provides new criteria for the normality, presentation, and higher syzygies of surfaces of general type using adjoint line bundles, with optimal bounds demonstrated through examples.

Findings

Determines the values of r for which L_r has N_p property.

Relates bounds on r to the regularity of B.

Shows several results are optimal through examples.

Abstract

We prove new results on projective normality, normal presentation and higher syzygies for a surface of general type $X$ embedded by adjoint line bundles $L_r = K + rB$, where $B$ is a base point free, ample line bundle. Our main results determine the $r$ for which $L_r$ has $N_p$ property. In corollaries, we will relate the bounds on $r$ to the regularity of $B$. Examples in the last section show that several results are optimal.