Duke Mathematical Journal

Syzygies of Segre embeddings and $\Delta$-modules

Andrew Snowden

Abstract

We study syzygies of the Segre embedding of $\mathbf{P}(V_{1})\times\cdots\times \mathbf{P}(V_{n})$, 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.

Article information

Source
Duke Math. J., Volume 162, Number 2 (2013), 225-277.

Dates
First available in Project Euclid: 24 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1359036935

Digital Object Identifier
doi:10.1215/00127094-1962767

Mathematical Reviews number (MathSciNet)
MR3018955

Zentralblatt MATH identifier
1279.13024

Subjects
Primary: 13D02: Syzygies, resolutions, complexes
Secondary: 15A69: Multilinear algebra, tensor products

Citation

Snowden, Andrew. Syzygies of Segre embeddings and $\Delta$ -modules. Duke Math. J. 162 (2013), no. 2, 225--277. doi:10.1215/00127094-1962767. https://projecteuclid.org/euclid.dmj/1359036935

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