Syzygies of bounded rank symmetric tensors are generated in bounded degree

TL;DR

This paper proves that the syzygies of secant ideals of Veronese subrings have generators bounded in degree, independent of the embedding, revealing uniform algebraic properties across different embeddings.

Contribution

It establishes degree bounds for syzygies of secant ideals of Veronese subrings that are independent of the specific embedding used.

Findings

Degree bounds depend only on syzygy index, secant order, and the scheme, not on the embedding.

Syzygies of secant ideals are generated in degrees bounded by a constant.

Results apply to any projective scheme over a field of characteristic zero.

Abstract

We study the syzygies of secant ideals of Veronese subrings of a fixed commutative graded algebra over a field of characteristic 0. One corollary is that the degrees of the minimal generators of the ith syzygy module of the coordinate ring of the rth secant variety of any Veronese embedding of a projective scheme X can be bounded by a constant that only depends on i, r, and X, and not on the choice of the Veronese embedding.