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

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.