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

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

Z-linear Gale duality and poly weighted spaces (PWS)

The present paper is devoted to discussing Gale duality from the Z-linear algebraic point of view. This allows us to isolate the class of Q-factorial complete toric varieties whose class group is torsion free, here called poly weighted spaces (PWS), as an interesting generalization of weighted projective spaces (WPS).Comment: 29 pages: revised version to appear in Linear Algebra and Its Applications. Major changes: the paper has been largely rewritten following refree's comments. In particular, main geometric results have been anticipated giving rise to the motivational Section

A Q-factorial complete toric variety is a quotient of a poly weighted space

We prove that every Q-factorial complete toric variety is a finite quotient of a poly weighted space (PWS), as defined in our previous work arXiv:1501.05244. This generalizes the Batyrev-Cox and Conrads description of a Q-factorial complete toric variety of Picard number 1, as a finite quotient of a weighted projective space (WPS) \cite[Lemma~2.11]{BC} and \cite[Prop.~4.7]{Conrads}, to every possible Picard number, by replacing the covering WPS with a PWS. As a consequence we describe the bases of the subgroup of Cartier divisors inside the free group of Weil divisors and the bases of the Picard subgroup inside the class group, respectively, generalizing to every Q-factorial complete toric variety the description given in arXiv:1501.05244, Thm. 2.9, for a PWS.Comment: 25+9 pp. Post-final version of our paper published in Ann.Mat.Pur.Appl.(2017),196,325-347: after its publication we realized that Prop.~3.1 contains an error strongly influencing the rest of the paper. Here is a correct revision (first 25 pp.: this version will not be published) and the Erratum appearing soon in Ann. Mat. Pur. Appl. (last 9 pp.) correcting only those parts affected by the erro

Toric varieties and Gr\"obner bases: the complete Q-factorial case

We present two algorithms determining all the complete and simplicial fans admitting a fixed non-degenerate set of vectors $V$ as generators of their 1-skeleton. The interplay of the two algorithms allows us to discerning if the associated toric varieties admit a projective embedding, in principle for any values of dimension and Picard number. The first algorithm is slower than the second one, but it computes all complete and simplicial fans supported by $V$ and lead us to formulate a topological-combinatoric conjecture about the definition of a fan. On the other hand, we adapt the Sturmfels' arguments on the Gr\"obner fan of toric ideals to our complete case; we give a characterization of the Gr\"obner region and show an explicit correspondence between Gr\"obner cones and chambers of the secondary fan. A homogenization procedure of the toric ideal associated to $V$ allows us to employing GFAN and related software in producing our second algorithm. The latter turns out to be much faster than the former, although it can compute only the projective fans supported by $V$. We provide examples and a list of open problems. In particular we give examples of rationally parametrized families of \Q-factorial complete toric varieties behaving in opposite way with respect to the dimensional jump of the nef cone over a special fibre.Comment: 18 pages, 2 figures. Final version accepted for publication in the special issue of the Journal AAAECC, concerning "Algebraic Geometry from an Algorithmic point of View

A Batyrev type classification of ℚ-factorial projective toric varieties

Abstract The present paper generalizes, inside the class of projective toric varieties, the classification [2], performed by Batyrev in 1991 for smooth complete toric varieties, to the singular ℚ-factorial case.</jats:p

Computational procedures for weighted projective spaces

This is a pdf print of the homonymous Maple file, freely available at http://www.maplesoft.com/applications/view.aspx?SID=127621, providing procedures which are able to produce the toric data associated with a (polarized) weighted projective space i.e. fans, polytopes and their equivalences. More originally it provides procedures which are able to detect a weights vector Q starting from either a fan or a polytope: we will call this process the recognition of a (polarized) weighted projective space. Moreover it gives procedures connecting polytopes of a polarized weighted projective space with an associated fan and viceversa.Comment: 31 page