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

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.

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