Local Complete Intersections in P^2 and Koszul Syzygies

TL;DR

This paper characterizes when the module of syzygies vanishing at a zero locus is generated by Koszul syzygies, linking it to the geometric property of being a local complete intersection in P^2.

Contribution

It provides a criterion connecting Koszul syzygies and local complete intersections for codimension two ideals in projective space.

Findings

Syzygies vanish at Z iff Z is a local complete intersection.

The module of Koszul syzygies generates all vanishing syzygies in this case.

The theorem's limitations are illustrated with a counterexample in higher dimensions.

Abstract

We study the syzygies of a codimension two ideal I = <f_1,f_2,f_3> in k[x,y,z]. Our main result is that the module of syzygies vanishing (scheme-theoretically) at the zero locus Z = V(I) is generated by the Koszul syzygies iff Z is a local complete intersection. The proof uses a characterization of complete intersections due to Herzog. When I is saturated, we relate our theorem to results of Weyman and of Simis and Vasconcelos. We conclude with an example of how our theorem fails for four generated local complete intersections in k[x,y,z] and we discuss generalizations to higher dimensions.