Curvilinear base points, local complete intersection and Koszul syzygies in biprojective spaces

Let $I = \langle f_1 , f_2 , f_3\rangle$ be a bigraded ideal in the bigraded polynomial ring $k[s, u; t, v]$. Assume that $I$ has codimension 2. Then $Z = \mathbb{V}(I) \subset \mathbf{P}^{1} \times \mathbf{P}^{1}$ is a finite set of points. We prove that if $Z$ is a local complete intersection, then any syzygy of the $f_i$ vanishing at $Z$, and in a certain degree range, is in the module of Koszul syzygies. This is an analog of a recent result of Cox and Schenck (2003).