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Local complete intersections in $\mathbb{P}^2$ and Koszul syzygies

Authors: David Cox and Hal Schenck
Journal: Proc. Amer. Math. Soc. 131 (2003), 2007-2014
MSC (1991): Primary 14Q10; Secondary 13D02, 14Q05, 65D17
Published electronically: November 6, 2002
MathSciNet review: 1963743
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Abstract: We study the syzygies of a codimension two ideal $I=\langle f_1,f_2,f_3\rangle \subseteq k[x,y,z]$. Our main result is that the module of syzygies vanishing (scheme-theoretically) at the zero locus $Z = {\mathbf V}(I)$ 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 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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Additional Information

David Cox
Affiliation: Department of Mathematics and Computer Science, Amherst College, Amherst, Massachusetts 01002-5000

Hal Schenck
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Address at time of publication: Department of Mathematics, Texas A&M University, College Station, Texas 77843

Keywords: Basepoint, local complete intersection, syzygy
Received by editor(s): May 29, 2001
Received by editor(s) in revised form: February 7, 2002
Published electronically: November 6, 2002
Additional Notes: The second author was supported by an NSF postdoctoral research fellowship
Communicated by: Michael Stillman
Article copyright: © Copyright 2002 American Mathematical Society