Big q-ample line bundles

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A recent paper of Totaro developed a theory of q-ample bundles in characteristic 0. Specifically, a line bundle L on X is q-ample if for every coherent sheaf F on X, there exists an integer m 0 such that m ≥ m 0 implies H i (X, F ⊗ O(mL))=0 for i>q. We show that a line bundle L on a complex projective scheme X is q-ample if and only if the restriction of L to its augmented base locus is q-ample. In particular, when X is a variety and L is big but fails to be q-ample, then there exists a codimension-one subscheme D of X such that the restriction of L to D is not q-ample.

Original languageEnglish (US)
Pages (from-to)790-798
Number of pages9
JournalCompositio Mathematica
Issue number3
StatePublished - May 2012
Externally publishedYes


  • augmented base loci
  • linear series
  • partial amplitude

ASJC Scopus subject areas

  • Algebra and Number Theory


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