Linear determinantal equations for all projective schemes

TL;DR

This paper demonstrates that all projective schemes with sufficiently ample line bundles can be described by determinantal equations, extending previous results and providing explicit descriptions for various classes of varieties.

Contribution

It generalizes the determinantal presentation of projective schemes to all connected schemes with ample line bundles and offers effective descriptions for specific classes.

Findings

Every such scheme is defined by 2-minors of a 1-generic matrix of linear forms.

Extended determinantal descriptions to products of projective spaces, Gorenstein toric varieties, and smooth n-folds.

Provided explicit constructions for determinantally presented line bundles.

Abstract

We prove that every projective embedding of a connected scheme determined by the complete linear series of a sufficiently ample line bundle is defined by the 2-minors of a 1-generic matrix of linear forms. Extending the work of Eisenbud-Koh-Stillman for integral curves, we also provide effective descriptions for such determinantally presented ample line bundles on products of projective spaces, Gorenstein toric varieties, and smooth n-folds.