Syzygies of Segre embeddings and Delta-modules

TL;DR

This paper investigates the syzygies of Segre embeddings, establishing finiteness results and a rational generating function, by introducing the concept of Delta-modules to formalize the emerging structure across all embeddings.

Contribution

It introduces Delta-modules to formalize the structure of syzygies across all Segre embeddings and proves finiteness and rationality results for their generating functions.

Findings

Existence of a finite list of master p-syzygies for all embeddings

The generating function for p-syzygies is rational

Results extend to tangent and secant varieties of Segre embeddings

Abstract

We study syzygies of the Segre embedding of P(V_1) x ... x P(V_n), and prove two finiteness results. First, for fixed p but varying n and V_i, there is a finite list of "master p-syzygies" from which all other p-syzygies can be derived by simple substitutions. Second, we define a power series f_p with coefficients in something like the Schur algebra, which contains essentially all the information of p-syzygies of Segre embeddings (for all n and V_i), and show that it is a rational function. The list of master p-syzygies and the numerator and denominator of f_p can be computed algorithmically (in theory). The central observation of this paper is that by considering all Segre embeddings at once (i.e., letting n and the V_i vary) certain structure on the space of p-syzygies emerges. We formalize this structure in the concept of a Delta-module. Many of our results on syzygies are specializations of general results on Delta-modules that we establish. Our theory also applies to certain other families of varieties, such as tangent and secant varieties of Segre embeddings.