Related papers: Toric varieties whose blow-up at a point is Fano

We prove that the derived category of a smooth complete intersection variety is equivalent to a full subcategory of the derived category of a smooth projective Fano variety. This enables us to define some new invariants of smooth projective…

We find all K-polystable smooth Fano threefolds that can be obtained as blowup of projective space along the disjoint union of a twisted cubic curve and a line.

In this paper, we prove that for a threefold of Fano type $X$ and a movable $\mathbb{Q}$-Cartier Weil divisor $D$ on $X$, the number of smooth varieties that arise during the running of a $D$-MMP is bounded by $1 + h^1(X, 2D)$.…

We give a simple criterion for slope stability of Fano manifolds $X$ along divisors or smooth subvarieties. As an application, we show that $X$ is slope stable along an ample effective divisor $D\subset X$ unless $X$ is isomorphic to a…

We survey various notions of symmetry for toric varieties. These notions range from algebraic geometric, complex geometric, representation theoretic, combinatorial, convex geometric, to geometric stability. The main theorem gives the…

We show that Fano lattice polygons define a class of balanced quivers with interesting properties. The combinatorics of these quivers is related to singularities of the underlying toric Fano surface. This allows us to show that every Fano…

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 give conditions for the Mayer-Vietoris property to hold for the algebraic K-theory of blow-up squares of toric varieties in any characteristic, using the theory of monoid schemes. These conditions are used to relate algebraic K-theory to…

Let $f:Y\to X$ be a finite morphism between Fano manifolds $Y$ and $X$ such that the Fano index of $X$ is greater than 1. On the one hand, when both $X$ and $Y$ are fourfolds of Picard number 1, we show that the degree of $f$ is bounded in…

The main result of this paper is a structural theorem for projective Q-factorial toric varieties X in P^N, covered by lines. We prove that there exists a toric fibration f: X -> Z, locally trivial in the Zariski topology, with fiber a…

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 an open Richardson variety in the complete flag variety for $\mathrm{GL}_n$ is isomorphic to a torus if and only if the corresponding closed Richardson variety is toric. Such toric varieties can be classified in terms of the…

To any graph $G$ one can associate a toric variety $X(\mathcal{P}G)$, obtained as a blowup of projective space along coordinate subspaces corresponding to connected subgraphs of $G$. The polytope of this toric variety is the graph…

The fundamental property of Fano varieties with mild singularities is that they have a finite polyhedral Mori cone. Thus, it is very interesting to ask: What we can say about algebraic varieties with a finite polyhedral Mori cone? I give a…

Let $X$ be a complex smooth Fano variety of dimension $n$. Assume that $X$ admits a birational contraction of an extremal ray. In this paper, we give a classification of such $X$ when the pseudoindex is equal to $\frac{\dim X}{2}$.

We consider complete toric varieties $X$ such that a maximal unipotent subgroup $U$ of the automorphism group $\text{Aut}(X)$ acts on $X$ with an open orbit. It turns out that such varieties can be characterized by several remarkable…

We classify the d-dimensional simplicial, terminal, and reflexive polytopes with at least 3d-2 vertices. In particular, it turns out that these are all smooth Fano polytopes. This improves on previous results of Casagrande in 2006 and Oebro…

Let $X$ be a complete $\Q$-factorial toric variety of dimension $n$ and $\del$ the fan in a lattice $N$ associated to $X$. For each cone $\sigma$ of $\del$ there corresponds an orbit closure $V(\sigma)$ of the action of complex torus on…

Complete intersections inside rational homogeneous varieties provide interesting examples of Fano manifolds. For example, if $X = \cap_{i=1}^r D_i \subset G/P$ is a general complete intersection of $r$ ample divisors such that $K_{G/P}^*…

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…