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Journal of the European Mathematical Society

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Volume 18, Issue 6, 2016, pp. 1201–1231
DOI: 10.4171/JEMS/611

Published online: 2016-04-25

Representation stability for syzygies of line bundles on Segre–Veronese varieties

Claudiu Raicu[1]

(1) University of Notre Dame, USA

The rational homology groups of packing complexes are important in algebraic geometry since they control the syzygies of line bundles on projective embeddings of products of projective spaces (Segre–Veronese varieties). These complexes are a common generalization of the multidimensional chessboard complexes and of the matching complexes of complete uniform hypergraphs, whose study has been a topic of interest in combinatorial topology. We prove that the multivariate version of representation stability, a notion recently introduced and studied by Church and Farb, holds for the homology groups of packing complexes. This allows us to deduce stability properties for the syzygies of line bundles on Segre–Veronese varieties. We provide bounds for when stabilization occurs and show that these bounds are sometimes sharp by describing the linear syzygies for a family of line bundles on Segre varieties. As a motivation for our investigation, we show in an appendix that Ein and Lazarsfeld’s conjecture on the asymptotic vanishing of syzygies of coherent sheaves on arbitrary projective varieties reduces to the case of line bundles on a product of (at most three) projective spaces.

Keywords: Syzygies, representation stability, Segre varieties, Veronese varieties, chessboard complexes, matching complexes, packing complexes, asymptotic vanishing

Raicu Claudiu: Representation stability for syzygies of line bundles on Segre–Veronese varieties. J. Eur. Math. Soc. 18 (2016), 1201-1231. doi: 10.4171/JEMS/611