Abstract. Let I = f 1 , f 2 , f 3 be a bigraded ideal in the bigraded polynomial ring k [s, u; t, v]. Assume that I has codimension 2. Then Z = V(I) ⊂ P 1 × 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).