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A Q\mathbb{Q}--factorial complete toric variety with Picard number 2 is projective

Abstract

This paper is devoted to settle two still open problems, connected with the existence of ample and nef divisors on a Q-factorial complete toric variety. The first problem is about the existence of ample divisors when the Picard number is 2: we give a positive answer to this question, by studying the secondary fan by means of Z-linear Gale duality. The second problem is about the minimum value of the Picard number allowing the vanishing of the Nef cone: we present a 3-dimensional example showing that this value cannot be greater then 3, which, under the previous result, is also the minimum value guaranteeing the existence of non-projective examples.Comment: 10 pages, 5 figures. Minor changes following the referee's advise: list of notation suppressed, few typos fixed, references updated. Final version to appear in Advances in Geometr

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