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A numerical ampleness criterion via Gale duality

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Abstract

The main object of the present paper is a numerical criterion determining when a Weil divisor of a \({\mathbb {Q}}\)–factorial complete toric variety admits a positive multiple Cartier divisor which is either numerically effective (nef) or ample. It is a consequence of \({\mathbb {Z}}\)–linear interpretation of Gale duality and secondary fan as developed in several previous papers of us. As a byproduct we get a computation of the Cartier index of a Weil divisor and a numerical characterization of weak \({\mathbb {Q}}\)–Fano, \({\mathbb {Q}}\)–Fano, Gorenstein, weak Fano and Fano toric varieties. Several examples are then given and studied.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123, Torino, Italy

    Michele Rossi & Lea Terracini

Authors
  1. Michele Rossi
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  2. Lea Terracini
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Correspondence to Lea Terracini.

Additional information

The authors were partially supported by the MIUR-PRIN 2010–2011 Research Funds “Geometria delle Varietà Algebriche”. The first author is also supported by the I.N.D.A.M. as a member of the G.N.S.A.G.A.

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Rossi, M., Terracini, L. A numerical ampleness criterion via Gale duality. AAECC 28, 351–368 (2017). https://doi.org/10.1007/s00200-016-0308-5

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  • DOI: https://doi.org/10.1007/s00200-016-0308-5

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