Vol. 11, No. 10, 2017

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Effective nonvanishing for Fano weighted complete intersections

Marco Pizzato, Taro Sano and Luca Tasin

Vol. 11 (2017), No. 10, 2369–2395
Abstract

We show that the Ambro–Kawamata nonvanishing conjecture holds true for a quasismooth WCI X which is Fano or Calabi–Yau, i.e., we prove that, if H is an ample Cartier divisor on X, then |H| is not empty. If X is smooth, we further show that the general element of |H| is smooth. We then verify the Ambro–Kawamata conjecture for any quasismooth weighted hypersurface. We also verify Fujita’s freeness conjecture for a Gorenstein quasismooth weighted hypersurface.

For the proofs, we introduce the arithmetic notion of regular pairs and highlight some interesting connections with the Frobenius coin problem.

Keywords
weighted complete intersections, nonvanishing, Ambro–Kawamata conjecture
Mathematical Subject Classification 2010
Primary: 14M10
Secondary: 11D04, 14J45
Milestones
Received: 29 March 2017
Revised: 28 July 2017
Accepted: 1 September 2017
Published: 31 December 2017
Authors
Marco Pizzato
I-39100 Bolzano
Italy
Taro Sano
Department of Mathematics, Graduate School of Science
Kobe University
Kobe
Japan
Luca Tasin
Dipartimento di Matematica e Fisica
Roma Tre University
Rome
Italy