Related papers: On algebraic varieties with finite polyhedral Mori…

Suppose that X is a Fano manifold that corresponds under Mirror Symmetry to a Laurent polynomial f, and that P is the Newton polytope of f. In this setting it is expected that there is a family of algebraic varieties over the unit disc with…

We study finiteness properties, especially the noetherian property, the Krull dimension and a variation of finite presentation, in categories of polynomial functors from a small symmetric monoidal category whose unit is an initial object to…

In this note we prove that any affine algebraic monoid can be obtained as the endomorphisms' monoid of a finite-dimensional (nonassociative) algebra.

This paper explores the Fano variety of lines in hypersurfaces, particularly focusing on those with mild singularities. Our first result explores the irreducibility of the variety $\Sigma$ of lines passing through a singular point $y$ on a…

We show that every Mori dream space of globally $F$-regular type is of Fano type. As an application, we give a characterization of varieties of Fano type in terms of the singularities of their Cox rings.

We describe recent progress in a program to understand the classification of three-dimensional Fano varieties with $\mathbb{Q}$-factorial terminal singularities using mirror symmetry. As part of this we give an improved and more conceptual…

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.

Perfect quadratic forms give a toroidal compactification of the moduli space of principally polarized abelian g-folds that is Q-factorial and whose ample classes are characterized, over any base. In characteristic zero it has canonical…

I extend the definitions of schemes relative to monoids with zero - and therefore, toric geometry - to the world of formal schemes. This expands the usual framework to include, for instance, models for Mumford's degenerating Abelian…

We study Fano threefolds with~terminal singularities admitting a "minimal" action of a finite group. We prove that under certain additional assumptions such a variety does not contain planes. We also obtain an upper bounds of the number of…

Let $X$ be a cubic fourfold that has only simple singularities and does not contain a plane. We prove that the Fano variety of lines on $X$ has the same analytic type of singularity as the Hilbert scheme of two points on a surface with only…

We survey some recent progress in the study of algebraic varieties X with log terminal singularities, especially, the uni-ruledness of the smooth locus X^0 of X, the fundamental group of X^0 and the automorphisms group on (smooth or…

Motivated by the study of Fano type varieties we define a new class of log pairs that we call asymptotically log Fano varieties and strongly asymptotically log Fano varieties. We study their properties in dimension two under an additional…

We classify smooth toric Fano varieties of dimension $n\geq 3$ containing a toric divisor isomorphic to $\PP^{n-1}$. As a consequence of this classification, we show that any smooth complete toric variety $X$ of dimension $n\geq 3$ with a…

We prove that the genus of a finite-dimensional division algebra is finite whenever the center is a finitely generated field of any characteristic. We also discuss potential applications of our method to other problems, including the…

Cone spherical surfaces are orientable Riemannian surfaces with constant curvature one and a finite set of conical singularities. A subset of these surfaces, referred to as dihedral surfaces, is characterized by their monodromy groups,…

Let X be a complex, Gorenstein, Q-factorial, toric Fano variety. We prove two conjectures on the maximal Picard number of X in terms of its dimension and its pseudo-index, and characterize the boundary cases. Equivalently, we determine the…

Let $k$ be \emph{any} algebraically closed field in any characteristic, let $R$ be any regular local ring such that $R$ contains $k$ as a subring, the residue field of $R$ is isomorphic to $k$ as $k$-algebras and $\dim R\geq 1$, let $P$ be…

In this paper we generalize the algebraic density property to not necessarily smooth affine varieties relative to some closed subvariety containing the singular locus. This property implies the remarkable approximation results for…

We prove that for any e>0, there exists only finitely many e-log terminal spherical Fano varieties of fixed dimension. We also introduce an invariant of a spherical subgroup H in a reductive group G which measures how nice an equivariant…