相关论文: The number of vertices of a Fano polytope

We show that the optimal upper bound for the anticanonical degrees of non-Gorenstein $\mathbb{Q}$-factorial canonical Fano threefolds with Picard number one is 200/3.

We solve the infinitesimal Torelli problem for $3$-dimensional quasi-smooth ${\mathbb{Q}}$-Fano hypersurfaces with at worst terminal singularities. We also find infinite chains of double coverings of increasing dimension which alternatively…

We determine the maximum number of edges that a planar graph can have as a function of its maximum degree and matching number.

We show that the complex projective space has maximal degree (volume) among all n-dimensional Kahler-Einstein Fano manifolds admitting a holomorphic C^*-action with a finite number of fixed points. The toric version of this result,…

Twinned chain polytopes form a broad class of non-centrally symmetric reflexive polytopes and exhibit intriguing structures. In the present paper, we show that the number of facets of $d$-dimensional twinned chain polytopes is at most…

We consider some families of smooth Fano hypersurfaces $X_{n+2}$ in ${\bf P}^{n+2} \times {\bf P}^3$ given by a homogeneous polynomial of bidegree $(1,3)$. For these varieties we obtain lower bounds for the number of $F$-rational points of…

In this paper, we classify smooth toric Fano 5-folds of index 2. There exist exactly 10 smooth toric Fano 5-folds of index 2 up to isomorphisms.

We show that a non-toric $\mathbb{Q}$-factorial terminal Fano threefold of Picard rank $1$ and Fano index $13$ is a weighted hypersurface of degree $12$ in $\mathbb{P}(3,4,5,6,7)$.

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.

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…

In this paper, we consider terminal reflexive polytopes arising from finite directed graphs and study the problem of deciding which directed graphs yield smooth Fano polytopes. We show that any centrally symmetric or pseudo-symmetric smooth…

We show that finitely generated Cox rings are Gorenstein. This leads to a refined characterization of varieties of Fano type: they are exactly those projective varieties with Gorenstein canonical quasicone Cox ring. We then show that for…

We prove rationality criteria over algebraically non-closed fields of characteristic $0$ for five out of six types of geometrically rational Fano threefolds of Picard number $1$ and geometric Picard number bigger than $1$. For the last type…

In this paper we classify mildly singular Fano varieties with maximal Picard number whose effective divisors are numerically eventually free.

We classify all 1-nodal degenerations of smooth Fano threefolds with Picard number 1 (both nonfactorial and factorial) and describe their geometry. In particular, we describe a relation between such degenerations and smooth Fano threefolds…

We prove the Shafarevich conjecture for Fano threefolds of Picard rank 1, index 1 and degree 4.

It is pretty well-known that toric Fano varieties of dimension k with terminal singularities correspond to convex lattice polytopes P in R^k of positive finite volume, such that intersection of P and Z^k consists of the point 0 and vertices…

We show that for a $\mathbb Q$-factorial canonical Fano $3$-fold $X$ of Picard number $1$, $(-K_X)^3\leq 72$. The main tool is a Kawamata--Miyaoka type inequality which relates $(-K_X)^3$ with $\hat{c}_2(X)\cdot c_1(X)$, where…

We classify some special classes of non-rational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found.

We classify Fano 3-folds with canonical Gorenstein singularities whose anticanonical linear system has no base points but does not give an embedding, and we classify anticanonically embedded Fano 3-folds with canonical Gorenstein…