A numerical ampleness criterion via Gale duality

The main object of the present paper is a numerical criterion determining when a Weil divisor of a \Q--factorial complete toric variety admits a positive multiple Cartier divisor which is either numerically effective (nef) or ample. It is a consequence of $\Z$--linear interpretation of Gale duality and se\-con\-dary 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 \Q--Fano, \Q--Fano, Gorenstein, weak Fano and Fano toric varieties. Several examples are then given and studied. \keywords{\Q--factorial complete toric variety \and ample divisor \and nef divisor \and $\Z$-liner Gale duality \and secondary fan \and ampleness criterion \and Cartier index \and \Q-Fano toric variety.Comment: 16 pages, 5 figures. v3: minor changes as indicated by the Referees; among them, total renumbering, abstract rewritten, references updated, Remark 2 added to fix a bug in the proof of Thm. 3, results extended to weak Fano toric varieties. Final version to appear in Journal of Applicable Algebra in Engineering, Communication and Computing. arXiv admin note: text overlap with arXiv:1504.06515, arXiv:1507.0049

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

On singular Fano varieties with a divisor of Picard number one

In this paper we study the geometry of mildly singular Fano varieties on which there is an effective prime divisor of Picard number one. Afterwards, we address the case of toric varieties. Finally, we treat the lifting of extremal contractions to universal covering spaces in codimension 1.Comment: 35 pages. Final version: to appear in Annali della Scuola Normale Superiore, Classe di Scienz

Fibration and classification of smooth projective toric varieties of low Picard number

In this paper we show that a smooth toric variety $X$ of Picard number $r\leq 3$ always admits a nef primitive collection supported on a hyperplane admitting non-trivial intersection with the cone \Nef(X) of numerically effective divisors and cutting a facet of the pseudo-effective cone \Eff(X), that is \Nef(X)\cap\partial\overline{\Eff}(X)\neq\{0\}. In particular this means that $X$ admits non-trivial and non-big numerically effective divisors. Geometrically this guarantees the existence of a fiber type contraction morphism over a smooth toric variety of dimension and Picard number lower than those of $X$, so giving rise to a classification of smooth and complete toric varieties with $r\leq 3$. Moreover we revise and improve results of Oda-Miyake by exhibiting an extension of the above result to projective, toric, varieties of dimension $n=3$ and Picard number $r=4$, allowing us to classifying all these threefolds. We then improve results of Fujino-Sato, by presenting sharp (counter)examples of smooth, projective, toric varieties of any dimension $n\geq4$ and Picard number $r=4$ whose non-trivial nef divisors are big, that is \Nef(X)\cap\partial\overline{\Eff}(X)=\{0\}. Producing those examples represents an important goal of computational techniques in definitely setting an open geometric problem. In particular, for $n=4$, the given example turns out to be a weak Fano toric fourfold of Picard number 4.Comment: 26 pages; 7 figures. Final version for pubblication in International Journal of Mathematics. Minor changes following referee's suggestions: in particular the proof of Lemma 3.2 has been rewritten to making it cleare

Framed sheaves on projective stacks

Given a normal projective irreducible stack X over an algebraically closed field of characteristic zero we consider framed sheaves on X, i.e., pairs (E,\u3d5E), where E is a coherent sheaf on X and \u3d5E is a morphism from E to a fixed coherent sheaf F. After introducing a suitable notion of (semi)stability, we construct a projective scheme, which is a moduli space for semistable framed sheaves with fixed Hilbert polynomial, and an open subset of it, which is a fine moduli space for stable framed sheaves. If X is a projective irreducible orbifold of dimension two and F a locally free sheaf on a smooth divisor D 82X satisfying certain conditions, we consider (D,F)-framed sheaves, i.e., framed sheaves (E,\u3d5E) with E a torsion-free sheaf which is locally free in a neighbourhood of D, and \u3d5E|D an isomorphism. These pairs are \u3bc-stable for a suitable choice of a parameter entering the (semi)stability condition, and of the polarization of X. This implies the existence of a fine moduli space parameterizing isomorphism classes of (D,F)-framed sheaves on X with fixed Hilbert polynomial, which is a quasi-projective scheme. In an appendix we develop the example of stacky Hirzebruch surfaces. This is the first paper of a project aimed to provide an algebro-geometric approach to the study of gauge theories on a wide class of 4-dimensional Riemannian manifolds by means of framed sheaves on "stacky" compactifications of them. In particular, in a subsequent paper [20] these results are used to study gauge theories on ALE spaces of type Ak. \ua9 2014 Elsevier Inc

Algebraic Geometry: Birational Classification, Derived Categories, and Moduli Spaces

The workshop covered a number of active areas of research in algebraic geometry with a focus on derived categories, moduli spaces (of varieties and sheaves) and birational geometry (often in positive characteristic) and their interactions. Special emphasis was put on hyperkähler manifolds and singularity theory

Birational geometry of quiver varieties and other GIT quotients

We introduce a sufficient condition for the Geometric Invariant Theory (GIT) quotient of an affine variety $V$ by the action of a reductive group $G$ to be a relative Mori Dream Space. When the condition holds, we show that the linearisation map identifies a region of the GIT fan with the Mori chamber decomposition of the relative movable cone of $V/\!/_\theta \, G$. If $V/\!/_\theta \, G$ is a crepant resolution of $Y\!\!:= V/\!/_0 \, G$, then every projective crepant resolution of $Y$ is obtained by varying $\theta$. Under suitable conditions, we show that this is the case for Nakajima quiver varieties; in particular, all projective partial crepant resolutions of the affine quiver variety $Y$ are quiver varieties. Similarly, for any finite subgroup $\Gamma\subset \mathrm{SL}(3,\mathbb{k})$ whose nontrivial conjugacy classes are all junior, we obtain a simple geometric proof of the fact that every projective crepant resolution of $\mathbb{A}^3/\Gamma$ is a fine moduli space of $\theta$-stable $\Gamma$-constellations. Our methods apply equally well to nonsingular hypertoric varieties