Ideals of varieties parameterized by certain symmetric tensors

TL;DR

This paper extends the understanding of the defining ideals of Segre-Veronese varieties, using weak generic hypermatrices to generalize known results about Segre varieties and their projections.

Contribution

It introduces the concept of weak generic hypermatrices to describe the ideals of a broader class of varieties, including projections of Veronese surfaces and varieties.

Findings

Ideals of Segre-Veronese varieties are generated by 2-minors of hypermatrices.

Weak generic hypermatrices effectively describe these ideals.

Results include cases of projections from generic points and Cohen-Macaulay subvarieties.

Abstract

The ideal of a Segre variety is generated by the 2-minors of a generic hypermatrix of indeterminates. We extend this result to the case of Segre-Veronese varieties. The main tool is the concept of weak generic hypermatrix which allows us to treat also the case of projection of Veronese surfaces from a set of generic points and of Veronese varieties from a Cohen-Macaulay subvariety of codimension 2.