### 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) |
---|---|

Pages (from-to) | 790-798 |

Number of pages | 9 |

Journal | Compositio Mathematica |

Volume | 148 |

Issue number | 3 |

DOIs | |

State | Published - May 2012 |

Externally published | Yes |

### Fingerprint

### Keywords

- augmented base loci
- linear series
- partial amplitude

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Compositio Mathematica*,

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

**Big q-ample line bundles.** / Brown, Morgan.

Research output: Contribution to journal › Article

*Compositio Mathematica*, vol. 148, no. 3, pp. 790-798. https://doi.org/10.1112/S0010437X11007457

}

TY - JOUR

T1 - Big q-ample line bundles

AU - Brown, Morgan

PY - 2012/5

Y1 - 2012/5

N2 - 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.

AB - 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.

KW - augmented base loci

KW - linear series

KW - partial amplitude

UR - http://www.scopus.com/inward/record.url?scp=84861428949&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84861428949&partnerID=8YFLogxK

U2 - 10.1112/S0010437X11007457

DO - 10.1112/S0010437X11007457

M3 - Article

VL - 148

SP - 790

EP - 798

JO - Compositio Mathematica

JF - Compositio Mathematica

SN - 0010-437X

IS - 3

ER -