Determinantal tensor product surfaces and the method of moving quadrics

A tensor product surface $\mathscr {S}$ is an algebraic that defined as the closure of image a rational map $\phi$ from $\mathbb {P}^1\times \mathbb {P}^1$ to {P}^3$. We provide new determinantal representations under assumptions generically injective and its base points are finitely many locally complete intersections. These matrices built coefficients linear relations (syzygies) quadratic bihomogeneous polynomials defining $\phi$. Our approach relies on formalization generalization method moving quadrics introduced studied by David Cox his co-authors.