The Cox ring of a complexity-one horospherical variety

Cox rings are intrinsic objects naturally generalizing homogeneous coordinate rings of projective spaces. A complexity-one horospherical variety is a normal variety equipped with a reductive group action whose general orbit is horospherical and of codimension one. In this note, we provide a presentation by generators and relations for the Cox rings of complete rational complexity-one horospherical varieties.Comment: 9 pages, to appear in Arch. Mat

Lower and upper bounds for nef cones

The nef cone of a projective variety Y is an important and often elusive invariant. In this paper we construct two polyhedral lower bounds and one polyhedral upper bound for the nef cone of Y using an embedding of Y into a toric variety. The lower bounds generalize the combinatorial description of the nef cone of a Mori dream space, while the upper bound generalizes the F-conjecture for the nef cone of the moduli space \bar{M}_{0,n} to a wide class of varieties.Comment: 25 pages, 4 figures. Final version to appear in IMR

Gorenstein spherical Fano varieties

We obtain a combinatorial description of Gorenstein spherical Fano varieties in terms of certain polytopes, generalizing the combinatorial description of Gorenstein toric Fano varieties by reflexive polytopes and its extension to Gorenstein horospherical Fano varieties due to Pasquier. Using this description, we show that the rank of the Picard group of an arbitrary $d$-dimensional $\mathbb{Q}$-factorial Gorenstein spherical Fano variety is bounded by $2d$. This paper also contains an overview of the description of the natural representative of the anticanonical divisor class of a spherical variety due to Brion.Comment: 22 pages, 3 figure

Common subbundles and intersections of divisors

Let V_0 and V_1 be complex vector bundles over a space X. We use the theory of divisors on formal groups to give obstructions in generalised cohomology that vanish when V_0 and V_1 can be embedded in a bundle U in such a way that V_0\cap V_1 has dimension at least k everywhere. We study various algebraic universal examples related to this question, and show that they arise from the generalised cohomology of corresponding topological universal examples. This extends and reinterprets earlier work on degeneracy classes in ordinary cohomology or intersection theory.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-42.abs.htm

Embedding non-projective Mori Dream Spaces

This paper is devoted to extend some Hu-Keel results on Mori dream spaces (MDS) beyond the projective setup. Namely, \Q-factorial algebraic varieties with finitely generated class group and Cox ring, here called \emph{weak} Mori dream spaces (wMDS), are considered. Conditions guaranteeing the existence of a neat embedding of a (completion of a) wMDS into a complete toric variety are studied, showing that, on the one hand, those which are complete and admitting low Picard number are always projective, hence Mori dream spaces in the sense of Hu-Keel. On the other hand, an example of a wMDS does not admitting any neat embedded \emph{sharp} completion (i.e. Picard number preserving) into a complete toric variety is given, on the contrary of what Hu and Keel exhibited for a MDS. Moreover, termination of the Mori minimal model program (MMP) for every divisor and a classification of rational contractions for a complete wMDS are studied, obtaining analogous conclusions as for a MDS. Finally, we give a characterization of a wMDS arising from a small \Q-factorial modification of a projective weak \Q-Fano variety.Comment: v4: Final version accepted for pubblication in Geometriae Dedicata. Minor changes. Adopting the Journal TeX-macros changed the statements' enumeration. 46 pages, 3 figure