Determinantal tensor product surfaces and the method of moving quadrics
Laurent Busé, Falai Chen · Transactions of the American Mathematical Society · 2020
A tensor product surface S \mathscr {S} is an algebraic surface that is defined as the closure of the image of a rational map ϕ \phi from P 1 × P 1 \mathbb {P}^1\times \mathbb {P}^1 to P 3 \mathbb {P}^3 . We provide new determinantal representations of S \mathscr {S} under the assumptions that ϕ \phi is generically injective and its base points are finitely many and locally complete intersections. These determinantal representations are matrices that are built from the coefficients of linear relations (syzygies) and quadratic relations of the bihomogeneous polynomials defining ϕ \phi . Our approach relies on a formalization and generalization of the method of moving quadrics introduced and studied by David Cox and his co-authors.