### Abstract

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 language | English (US) |
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Pages (from-to) | 790-798 |

Number of pages | 9 |

Journal | Compositio Mathematica |

Volume | 148 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2012 |

Externally published | Yes |

### Keywords

- augmented base loci
- linear series
- partial amplitude

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

Brown, M. V. (2012). Big q-ample line bundles.

*Compositio Mathematica*,*148*(3), 790-798. https://doi.org/10.1112/S0010437X11007457