Local Complete Intersections in P2 and Koszul Syzygies
Abstract: We study the syzygies of a codimension two ideal I=<f1,f2,f3> \subseteq 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 Simis-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.
If you don't know what some or all of the words above mean, fear not, 'cause I'll define everything. I'll also give the audience something useful to take home - a turbo tutorial on using the Macaulay2 package (a computer program for symbolic algebra).
(joint work with David Cox, Amherst)