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Given a 4-dimensional vector subspace U = {f0,…,f3} of H0(ℙ1 × ℙ1,(a,b)), a tensor product surface, denoted by XU, is the closure of the image of the rational map λU : ℙ1 × ℙ1 −−→ ℙ3 determined by U. These surfaces arise in geometric modeling and in this context it is useful to know the implicit equation of XU in ℙ3. In this paper, we show that if U ⊆ H0(ℙ1 × ℙ1,(a, 1)) has a finite set of r basepoints in generic position, then the implicit equation of XU is determined by two syzygies of IU = 〈f0,…,f3〉 in bidegrees (a −⌈r 2⌉, 0) and (a −⌊r 2⌋, 0). This result is proved by understanding the geometry of the basepoints of U in ℙ1 × ℙ1. The proof techniques for the main theorem also apply when U is basepoint free.

Implicitization of tensor product surfaces in the presence of a generic set of basepoints