Related papers: Fano threefolds in positive characteristic IV

We construct 4 di erent families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of the form Z x A1, where Z is a quasiprojective variety. The affi ne cones over such a fourfold admit eff…

We prove that all smooth Fano threefolds in the families 2.1, 2.2, 2.3, 2.4, 2.6 and 2.7 are K-stable, and we also prove that smooth Fano threefolds in the family 2.5 that satisfy one very explicit generality condition are K-stable.

We classify toric Fano threefolds having at worst terminal singularities such that a rank of a $G$-invariant part of a class group equals one, where $G$ is a group acting on the variety by automorphisms.

We prove a characterization of Fano type varieties.

We construct some new deformation families of four-dimensional Fano manifolds of index $1$ in some known classes of Gorenstein formats. These families have explicit descriptions in terms of equations, defining their image under the…

We study the coregularity of smooth Fano threefolds. We prove that for 100 out of 105 families of smooth Fano threefolds, a general member in the family has coregularity 0; moreover, for 92 families out of these 100, any member in the…

We give a characterization of Fano type surfaces with large cyclic automorphisms.

We classify smooth Fano manifolds X with the Picard number $\rho_X \geq 3$ such that there exists an extremal ray which has a birational contraction that maps a divisor to a point.

We construct new families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of form $Z\times A^1$, where $Z$ is a quasiprojective variety. The affine cones over such a fourfold admit effective…

We give a necessary and sufficient condition for a generalized Bott manifold to be Fano or weak Fano. As a consequence we characterize Fano Bott manifolds.

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…

We prove that a prime Fano threefold of genus 8 over an algebraically closed field of positive characteristic is isomorphic to a linear section of the Grassmannian variety Gr(2, 6). As applications, it is shown that a prime Fano threefold…

Let $X$ be a smooth Fano threefold over an algebraically closed field of positive characteristic. Assume that $|-K_X|$ is very ample and each of the index and the Picard number is equal to one. We prove that $3 \leq g \leq 12$ and $g \neq…

Let $X$ be a smooth complex prime Fano fourfold having a semi-free action of $\mathbb{C}^*$, then $X$ is contained in one of the families $\mathbb{P}^4,Q^4,W_5, X^{m}_{8}$. All of the families contain members that have a semi-free…

We give some rationality constructions for Fano threefolds with canonical Gorenstein singularities.

We prove that every smooth Fano threefold from the family No 2.8 is K-stable. Such a Fano threefold is a double cover of the blow-up of $\mathbb{P}^3$ at one point branched along an anti-canonical divisor.

In this article, we prove that any $\Bbb Q$-factorial weak Fano 3-fold with only terminal singularities has a smoothing.

In this paper we prove that any smooth prime Fano threefold, different from the Mukai-Umemura threefold, contains a 1-dimensional family of intersecting lines. Combined with a result of the second author (see J. Algebr. Geom. 8:2 (1999),…

We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we show that the Fano varieties are always birational to…

In this paper, we investigate Fano manifolds whose Chern characters satisfy some positivity conditions. We prove that such manifolds admit long chains of higher order minimal families of rational curves and are covered by higher rational…