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
Mathematical Subject Classification 2010
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