Syzygies of secant ideals of Pl\"ucker-embedded Grassmannians are generated in bounded degree

TL;DR

This paper proves that the defining ideals and syzygies of secant varieties of Grassmannians embedded via Plücker coordinates are generated and structured in degrees bounded independently of the dimensions, using new algebraic and categorical methods.

Contribution

It introduces a novel approach to prove bounded degree generation and syzygy concentration for secant ideals of Grassmannians, independent of dimension parameters.

Findings

Prime ideals of secant varieties are generated in bounded degree independent of dimensions.

Syzygy modules are concentrated in degrees bounded by a constant independent of dimensions.

Develops a new method to prove a poset is noetherian and translates it into functor categories.

Abstract

Over a field of characteristic $0$, we prove that for each $r \geq 0$ there exists a constant $C(r)$ so that the prime ideal of the $r$th secant variety of any Pl\"ucker-embedded Grassmannian ${\bf Gr}(d,n)$ is generated by polynomials of degree at most $C(r)$, where $C(r)$ is independent of $d$ and $n$. This bounded generation ultimately reduces to proving a poset is noetherian, we develop a new method to do this. We then translate the structure we develop to the language of functor categories to prove the $i$th syzygy module of the coordinate ring of the $r$th secant variety of any Pl\"ucker-embedded Grassmannian ${\bf Gr}(d,n)$ is concentrated in degrees bounded by a constant $C(i,r)$, which is again independent of $d$ and $n$.