#### Vol. 9, No. 6, 2015

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Effective Matsusaka's theorem for surfaces in characteristic $p$

### Gabriele Di Cerbo and Andrea Fanelli

Vol. 9 (2015), No. 6, 1453–1475
##### Abstract

We obtain an effective version of Matsusaka’s theorem for arbitrary smooth algebraic surfaces in positive characteristic, which provides an effective bound on the multiple that makes an ample line bundle $D$ very ample. The proof for pathological surfaces is based on a Reider-type theorem. As a consequence, a Kawamata–Viehweg-type vanishing theorem is proved for arbitrary smooth algebraic surfaces in positive characteristic.

##### Keywords
effective Matsusaka, surfaces in positive characteristic, Fujita's conjectures, Bogomolov's stability, Reider's theorem, bend-and-break, effective Kawamata–Viehweg vanishing
Primary: 14J25
##### Milestones
Received: 24 February 2015
Revised: 16 April 2015
Accepted: 17 May 2015
Published: 7 September 2015
##### Authors
 Gabriele Di Cerbo Department of Mathematics Columbia University New York, NY 10027 United States Andrea Fanelli Department of Mathematics Imperial College London 180 Queen’s Gate London SW7 2AZ United Kingdom