Related papers: Toric Fano varieties associated to graph cubeahedr…

We give a necessary and sufficient condition for the nonsingular projective toric variety associated to a finite simple graph to be Fano or weak Fano in terms of the graph.

We give a necessary and sufficient condition for the nonsingular projective toric variety associated to a building set to be weak Fano in terms of the building set.

We characterize building sets whose associated nonsingular projective toric varieties are Fano. Furthermore, we show that all such toric Fano varieties are obtained from smooth Fano polytopes associated to finite directed graphs.

This article gives an overview of toric Fano and toric weak Fano varieties associated to graphs and building sets. We also study some properties of such toric Fano varieties and discuss related topics.

In this paper, we introduce the notion of toric special weak Fano manifolds, which have only special primitive crepant contractions. We study the structure of them, and in particular completely classify smooth toric special weak Fano…

A horospherical variety is a normal algebraic variety where a reductive algebraic group acts with an open orbit which is a torus bundle over a flag variety. For example, toric varieties and flag varieties are horospherical. In this paper,…

We show that some important classes of weak Fano $3$-folds of Picard rank $2$ do not satisfy Bott vanishing. Using this we show that any smooth projective $3$-fold $X$ of Picard rank $2$ with $-K_X$ nef which is the image of a projective…

We classify toric Fano threefolds having at worst terminal singularities such that a rank of a $G$-invariant part of a class group equals one, where $G$ is a group acting on the variety by automorphisms.

We consider a smooth projective morphism between smooth complex projective varieties. If the source space is a weak Fano (or Fano) manifold, then so is the target space. Our proof is Hodge theoretic. We do not need mod $p$ reduction…

We give a necessary and sufficient condition for a generalized Bott manifold to be Fano or weak Fano. As a consequence we characterize Fano Bott manifolds.

We give a structure theorem for n-dimensional smooth toric Fano varieties whose associated polytope has "many" pairs of centrally symmetric vertices.

We completely classify toric weakened Fano 3-folds, that is, smooth toric weak Fano 3-folds which are not Fano but are deformed to smooth Fano 3-folds. There exist exactly 15 toric weakened Fano 3-folds up to isomorphisms.

The purpose of this paper is to give basic tools for the classification of nonsingular toric Fano varieties by means of the notions of primitive collections and primitive relations due to Batyrev. By using them we can easily deal with…

We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by Mirror Symmetry, we give conditions for the limit toric variety to be a…

We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we show that the Fano varieties are always birational to…

One can associate to a bipartite graph a so-called edge ring whose spectrum is an affine normal toric variety. We characterize the faces of the (edge) cone associated to this toric variety in terms of some independent sets of the bipartite…

Classical toric varieties are among the simplest objects in algebraic geometry. They arise in an elementary fashion as varieties parametrized by monomials whose exponents are a finite subset $\mathcal{A}$ of $\mathbb{Z}^n$. They may also be…

Let $X$ be a normal projective variety and $f:X\to X$ a non-isomorphic polarized endomorphism. We give two characterizations for $X$ to be a toric variety. First we show that if $X$ is $\mathbb{Q}$-factorial and $G$-almost homogeneous for…

Let $X$ be a smooth Fano fourfold admitting a conic bundle structure. We show that $X$ is toric if and only if $X$ admits an amplified endomorphism; in this case, $X$ is a rational variety.

We study the Picard variety of the Fano surface of nodal and mildly cuspidal cubic threefolds in arbitrary characteristic by relating divisors on the Fano surface to divisors on the symmetric product of a curve of genus 4.