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Asymptotic syzygies of algebraic varieties


We study the asymptotic behavior of the syzygies of a smooth projective variety as the positivity of the embedding line bundle grows. The main result asserts that the syzygy modules are non-zero in almost all degrees allowed by Castelnuovo–Mumford regularity. We also give an effective statement for Veronese varieties that we conjecture to be optimal.

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Correspondence to Robert Lazarsfeld.

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Research of the first author partially supported by NSF grant DMS-1001336.

Research of the second author partially supported by NSF grant DMS-0652845.

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Ein, L., Lazarsfeld, R. Asymptotic syzygies of algebraic varieties. Invent. math. 190, 603–646 (2012).

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  • Vector Bundle
  • Algebraic Variety
  • Smooth Projective Variety
  • Ample Divisor
  • Segre Variety