Syzygies of adjoint linear series on projective varieties

TL;DR

This paper investigates the syzygies of adjoint linear series on smooth projective varieties, establishing conditions under which certain properties hold and the section ring is Koszul.

Contribution

It proves that $K_X+mA$ satisfies property $N_p$ for $m \\geq n+1+p$ and shows the section ring is Koszul for $m \\geq n+2$, advancing understanding of syzygies in algebraic geometry.

Findings

$K_X+mA$ satisfies property $N_p$ for $m \\geq n+1+p$

The graded ring $R(X, K_X+mA)$ is Koszul for $m \\geq n+2$

Provides new bounds for syzygies of adjoint linear series

Abstract

Let X be a smooth complex projective variety of dimension n and let A be an ample and basepoint free divisor. We prove $K_X+mA$ satisfies property $N_p$ for $m\geqslant n+1+p$. We also show the graded ring of sections $R(X, K_X+mA)$ is Koszul for $m\geqslant n+2$.