Related papers: On derived categories of arithmetic toric varietie…
Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this thesis we explore this correspondence to classify smooth lattice…
The main result of this paper is that every (separated) toric variety which has a semigroup structure compatible with multiplication on the underlying torus is necessarily affine. In the course of proving this statement, we also give a…
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…
It was shown in [S. Kaliman, M. Zaidenberg, Gromov ellipticity of cones over projective manifolds, Math. Res. Lett. (to appear), arXiv:2303.02036 (2023)] that the affine cones over flag manifolds and rational smooth projective surfaces are…
We generalized the construction of deformations of affine toric varieties of K. Altmann and our previous construction of deformations of weak Fano toric varieties to the case of arbitrary toric varieties by introducing the notion of…
We give a short proof of the Zariski-Lipman conjecture for toric varieties: any complex toric variety with locally free tangent sheaf is smooth.
We associate to each toric vector bundle on a toric variety X(Delta) a "branched cover" of the fan Delta together with a piecewise-linear function on the branched cover. This construction generalizes the usual correspondence between toric…
Using the technique of categorical absorption of singularities we prove that the nontrivial components of the derived categories of del Pezzo threefolds of degree $d \in \{2,3,4,5\}$ and crepant categorical resolutions of the nontrivial…
For every complete toric variety, there exists a projective toric variety which is isomorphic to it in codimension one. In this paper, we show that every smooth non-projective complete toric threefold of Picard number at most five becomes…
In this paper, we study derived categories of certain toric varieties with Picard number three that are blowing-up another toric varieties along their torus invariant loci of codimension at most three. We construct strong full exceptional…
Let X be a complete toric variety and let Y be a smooth projective variety with Picard number one. We prove that, if there exists a surjective morphism from X to Y, then Y is a projective space.
The space of torus translations and degenerations of a projective toric variety forms a toric variety associated to the secondary fan of the integer points in the polytope corresponding to the toric variety. This is used to identify a…
In this paper we construct a full exceptional collection of sheaves on some family of log Del Pezzo surfaces, viewed as a smooth stack. These surfaces can be embedded as a quasi-smooth hypersurfaces into a certain weighted projective space,…
We develop a theory of Prym varieties and cubic threefolds over fields of characteristic $2$. As an application, we prove that smooth cubic threefolds are non-rational over an arbitrary field and solve a conjecture of Deligne regarding…
The existence of a full strong exceptional sequence in the derived category of a smooth quadric hypersurface was proved by Kapranov. In this paper, we present a skew generalization of this result. Namely, we show that if $S$ is a standard…
Generalising toric geometry we study compact varieties admitting lower dimensional torus actions. In particular we describe divisors on them in terms of convex geometry and give a criterion for their ampleness. These results may be used to…
A dissection of a polygon is obtained by drawing diagonals such that no two diagonals intersect in their interiors. In this paper, we define a toric variety of Schr\"{o}der type as a smooth toric variety associated with a polygon…
In this paper, we introduce the notion of "extension" of a toric variety and study its fundamental properties. This gives rise to infinitely many toric varieties with a special property, such as being set theoretic complete intersection or…
We show that a weight variety, which is a quotient of a flag variety by the maximal torus, admits a flat degeneration to a toric variety. In particular, we show that the moduli spaces of spatial polygons degenerate to polarized toric…
Real Legendrian subvarieties are classical objects of differential geometry and classical mechanics and they have been studied since antiquity. However, complex Legendrian subvarieties are much more rigid and have more exceptional…