Asymptotic syzygies of algebraic varieties

Lawrence Ein; Robert Lazarsfeld

doi:10.1007/s00222-012-0384-5

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arXiv · 1103.0483

Asymptotic syzygies of algebraic varieties

Lawrence Ein·Robert Lazarsfeld

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Abstract

This paper studies the asymptotic behavior of the syzygies of a smooth projective variety X as the positivity of the embedding line bundle grows. We prove that as least as far as grading is concerned, the minimal resolution of the ideal of X has a surprisingly uniform asymptotic shape: roughly speaking, generators eventually appear in almost all degrees permitted by Castelnuovo-Mumford regularity. This suggests in particular that a widely-accepted intuition derived from the case of curves -- namely that syzygies become simpler as the degree of the embedding increases -- may have been misleading. For Veronese embeddings of projective space, we give an effective statement that in some cases is optimal, and conjecturally always is so. Finally, we propose a number of questions and open problems concerning asymptotic syzygies of higher-dimensional varieties.

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BibTeX RIS

Lawrence Ein, Robert Lazarsfeld. 2012-02-23. Asymptotic syzygies of algebraic varieties. https://doi.org/10.1007/s00222-012-0384-5

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