We study syzygies of the Segre embedding of , and prove two finiteness results. First, for fixed but varying and , there is a finite list of master -syzygies from which all other -syzygies can be derived by simple substitutions. Second, we define a power series with coefficients in something like the Schur algebra, which contains essentially all the information of -syzygies of Segre embeddings (for all and ), and show that it is a rational function. The list of master -syzygies and the numerator and denominator of can be computed algorithmically (in theory). The central observation of this paper is that by considering all Segre embeddings at once (i.e., letting and the vary) certain structure on the space of -syzygies emerges. We formalize this structure in the concept of a -module. Many of our results on syzygies are specializations of general results on -modules that we establish. Our theory also applies to certain other families of varieties, such as tangent and secant varieties of Segre embeddings.
"Syzygies of Segre embeddings and -modules." Duke Math. J. 162 (2) 225 - 277, 1 February 2013. https://doi.org/10.1215/00127094-1962767