Related papers: Notes on toric varieties from Mori theoretic viewp…

Under some positivity assumptions, extension properties of rationally connected fibrations from a submanifold to its ambient variety are studied. Given a family of rational curves on a complex projective manifold X inducing a covering…

We prove Vojta's abc conjecture for projective space ${\Bbb P}^n({\Bbb C})$, assuming that the entire curves in ${\Bbb P}^n({\Bbb C})$ are highly ramified over the coordinate hyperplanes. This extends the results of Guo Ji and the…

We consider the set of forms of a toric variety over an arbitrary field: those varieties which become isomorphic to a toric variety after base field extension. In contrast to most previous work, we also consider arbitrary isomorphisms…

From a finite set in a lattice, we can define a toric variety embedded in a projective space. In this paper, we give a combinatorial description of the dual defect of the toric variety using the structure of the finite set as a Cayley sum…

Let $f:X\to S$ be an extremal contraction from a threefolds with terminal singularities onto a surface (so called Mori conic bundle). We study some particular cases of such contractions: quotients of usual conic bundles and index two…

We describe a class of toric varieties in the $N$-dimensional affine space which are minimally defined by no less than $N-2$ binomial equations.

In this paper, the concept of toric difference varieties is defined and four equivalent descriptions for toric difference varieties are presented in terms of difference rational parametrization, difference coordinate rings, toric difference…

We study $G$-equivariant birational geometry of toric varieties, where $G$ is a finite group.

We prove Fujita's spectrum conjecture on the discreteness of pseudo-effective thresholds for polarized varieties.

In this note we consider the problem of determining which Fano manifolds can be realised as fibres of a Mori fibre space. In particular, we study the case of toric varieties, Fano manifolds with high index and some Fano manifolds with high…

We study Fourier-Mukai transforms for smooth projective varieties whose canonical bundles have finite order, and relate them to equivariant transforms on certain finite covering spaces. Our results lead to new equivalences of derived…

In this note we collect some results on the deformation theory of toric Fano varieties.

Here we investigate the birational geometry of projective varieties of arbitrary dimension having defective higher secant varieties. We apply the classical tool of tangential projections and we determine natural conditions for uniruledness,…

We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the…

The real solutions to a system of sparse polynomial equations may be realized as a fiber of a projection map from a toric variety. When the toric variety is orientable, the degree of this map is a lower bound for the number of real…

We give a complete classification of equivariant vector bundles of rank two over smooth complete toric surfaces and construct moduli spaces of such bundles. This note is a direct continuation of an earlier note where we developed a general…

We describe the construction of a class of toric varieties as spectra of homogeneous prime ideals.

We give several structure theorems for certain surjective endomorphisms on Mori fibre spaces, based on the dynamical Iitaka fibration of the ramification divisor. As an application, we prove the Kawaguchi-Silverman conjecture for projective…

The aim of this paper is to extend our old results about Galois action on the torsion points of abelian varieties to the case of (finitely generated) fields of characteristic 2.

We show that a toric Fano contraction associated to an extremal ray whose length is greater than the dimension of its fiber is a projective space bundle.