Ideals of bounded rank symmetric tensors are generated in bounded degree

TL;DR

This paper proves that the ideals defining secant varieties of Veronese embeddings are generated by polynomials of bounded degree, independent of the embedding dimension, using Hopf ring structures and finite generation techniques.

Contribution

It establishes uniform degree bounds for generators of ideals of secant varieties across all Veronese embeddings, extending to partial flag varieties and projective schemes.

Findings

Ideals of secant varieties are finitely generated in bounded degree.

The approach uses Hopf ring structures to prove finite generation.

Results apply to various projective schemes and multi-graded contexts.

Abstract

Over a field of characteristic zero, we prove that for each r, there exists a constant C(r) so that the prime ideal of the rth secant variety of any Veronese embedding of any projective space is generated by polynomials of degree at most C(r). The main idea is to consider the coordinate ring of all of the ambient spaces of the Veronese embeddings at once by endowing it with the structure of a Hopf ring, and to show that its ideals are finitely generated. We also prove a similar statement for partial flag varieties and, in fact, arbitrary projective schemes, and we also get multi-graded versions of these results.