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Local Complete Intersections in P^2 and Koszul Syzygies

David Cox·Hal Schenck
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Abstract

We study the syzygies of a codimension two ideal I = 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.

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David Cox, Hal Schenck. 2001-10-09. Local Complete Intersections in P^2 and Koszul Syzygies. https://arxiv.org/abs/math/0110097

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