Determinantal tensor product surfaces and the method of moving quadrics

TL;DR

This paper introduces new determinantal representations for tensor product surfaces in projective space, based on syzygies and quadratic relations, generalizing the method of moving quadrics.

Contribution

It provides novel determinantal formulas for tensor product surfaces using syzygies, extending the method of moving quadrics to broader conditions.

Findings

Determinantal representations constructed from syzygies and quadratic relations.

Applicable under assumptions of injectivity and finitely many base points.

Generalizes the method of moving quadrics for algebraic surface representation.

Abstract

A tensor product surface $\mathscr{S}$ is an algebraic surface that is defined as the closure of the image of a rational map $\phi$ from $\mathbb{P}^1\times \mathbb{P}^1$ to $\mathbb{P}^3$. We provide new determinantal representations of $\mathscr{S}$ under the assumptions that $\phi$ 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 $\phi$. Our approach relies on a formalization and generalization of the method of moving quadrics introduced and studied by David Cox and his co-authors.