Syzygies of Segre embeddings and Δ-modules

Abstract

We study syzygies of the Segre embedding of $\mathbf{P}\left(V_1\right)\times \cdots \times \mathbf{P}\left(V_n\right)$, 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.