Syzygies and singularities of tensor product surfaces of bidegree (2,1)

TL;DR

This paper classifies the syzygies and singularities of tensor product surfaces of bidegree (2,1) by analyzing their associated bigraded ideals and minimal free resolutions, revealing six possible types and explicit geometric properties.

Contribution

It identifies six numerical types of minimal free resolutions for the bigraded ideals of these surfaces and explicitly describes their implicit equations and singular loci.

Findings

Six numerical types of minimal free resolutions identified.

Linear first syzygy cases allow explicit resolution differentials.

Complete description of implicit equations and singularities for the surfaces.

Abstract

Let U be a basepoint free four-dimensional subspace of the space of sections of O(2,1) on P^1 x P^1. The sections corresponding to U determine a regular map p_U: P^1 x P^1 --> P^3. We study the associated bigraded ideal I_U in k[s,t;u,v] from the standpoint of commutative algebra, proving that there are exactly six numerical types of possible bigraded minimal free resolution. These resolutions play a key role in determining the implicit equation of the image p_U(P^1 x P^1), via work of Buse-Jouanolou, Buse-Chardin, Botbol and Botbol-Dickenstein-Dohm on the approximation complex. In four of the six cases I_U has a linear first syzygy; remarkably from this we obtain all differentials in the minimal free resolution. In particular this allows us to describe the implicit equation and singular locus of the image.