1 February 2013 Syzygies of Segre embeddings and Δ-modules
Andrew Snowden
Duke Math. J. 162(2): 225-277 (1 February 2013). DOI: 10.1215/00127094-1962767


We study syzygies of the Segre embedding of P(V1)××P(Vn), and prove two finiteness results. First, for fixed p but varying n and Vi, 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 fp with coefficients in something like the Schur algebra, which contains essentially all the information of p-syzygies of Segre embeddings (for all n and Vi), and show that it is a rational function. The list of master p-syzygies and the numerator and denominator of fp 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 Vi vary) certain structure on the space of p-syzygies emerges. We formalize this structure in the concept of a Δ-module. Many of our results on syzygies are specializations of general results on Δ-modules that we establish. Our theory also applies to certain other families of varieties, such as tangent and secant varieties of Segre embeddings.


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Andrew Snowden. "Syzygies of Segre embeddings and Δ-modules." Duke Math. J. 162 (2) 225 - 277, 1 February 2013. https://doi.org/10.1215/00127094-1962767


Published: 1 February 2013
First available in Project Euclid: 24 January 2013

zbMATH: 1279.13024
MathSciNet: MR3018955
Digital Object Identifier: 10.1215/00127094-1962767

Primary: 13D02
Secondary: 15A69

Rights: Copyright © 2013 Duke University Press


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Vol.162 • No. 2 • 1 February 2013
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