Related papers: Deformations of Smooth Toric Surfaces

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 investigate the class of degenerations of smooth cubic surfaces which are obtained from degenerating their Cox rings to toric algebras. More precisely, we work in the spirit of Sturmfels and Xu who use the theory of Khovanskii bases to…

A classical problem in algebraic geometry is to construct smooth algebraic varieties with prescribed properties. In the approach via smoothings, one first constructs a degenerate scheme with the prescribed properties, and then shows the…

It is shown that time-harmonic motions of spherical and toroidal surfaces can be deformed non-locally without loosing the existence of infinitely many constants of the motion.

The fundamental group of a smooth projective variety is fibered if it maps onto the fundamental group of smooth curve of genus 2 or more. The goal of this paper is to establish some strong restrictions on these groups, and in particular on…

We introduce toric $b$-divisors on complete smooth toric varieties and a notion of integrability of such divisors. We show that under some positivity assumptions toric $b$-divisors are integrable and that their degree is given as the volume…

There is a large number of different ways of constructing Calabi-Yau manifolds, as well as related non-geometric formulations, relevant in string compactifications. Showcasing this diversity, we discuss explicit deformation families 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 prove that a singular complex surface that admits a complete holomorphic vector field that has no invariant curve through a singular point of the surface is obtained from a Kato surface by contracting some divisor (in particular, it is…

We analize the semistable degeneration of the Fano surface F when the cubic threefold becomes the Segre primal. This gives an explicit topological decomposition for F. The decomposition is used to decide that the Fano surface is not an an…

We call a sheaf on an algebraic variety immaculate if it lacks any cohomology including the zero-th one, that is, if the derived version of the global section functor vanishes. Such sheaves are the basic tools when building exceptional…

It is shown that a smooth global deformation of quartic double solids, i.e. double covers of $\mathbb P^3$ branched along smooth quartics, is again a quartic double solid without assuming the projectivity of the global deformation. The…

Given a split $\mathbb{P}$-functor $F:\mathcal{D}^b(X) \to \mathcal{D}^b(Y)$ between smooth projective varieties, we provide necessary and sufficient conditions, in terms of the Hochschild cohomology of $X$, for it to become spherical on…

For $n\geq 3$, let $M$ be an $(n+r)$-dimensional irreducible Hermitian symmetric space of compact type and let $\mathcal{O}_M(1)$ be the ample generator of $Pic(M)$. Let $Y=H_1\cap\dots\cap H_r$ be a smooth complete intersection of…

In this paper, we give a description of holomorphic multi-vector fields on smooth compact toric varieties, which generalizes Demazure's result of holomorphic vector fields on toric varieties. Based on the result, we compute the Poisson…

Symmetric varieties are normal equivariant open embeddings of symmetric homogeneous spaces and they are interesting examples of spherical varieties. The principal goal of this article is to study the rigidity under K\"{a}hler deformations…

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…

Let $k$ be a perfect field and let $C_0:f=0$ be a smooth curve in the torus $\mathbb{G}_{m,k}^2$. Let $\mathbb{T}_\Delta$ be the toric variety associated to the Newton polygon of $f$. Extending the toric resolution of $C_0$ on…

We are interested in two classes of varieties with group action, namely toric varieties and spherical embeddings. They are classified by combinatorial objects, called fans in the toric setting, and colored fans in the spherical setting. We…

Let $S$ be a {\em Todorov surface}, {\it i.e.}, a minimal smooth surface of general type with $q=0$ and $p_g=1$ having an involution $i$ such that $S/i$ is birational to a $K3$ surface and such that the bicanonical map of $S$ is composed…