Related papers: Fano varieties of middle pseudoindex

We classify Sarkisov links from index 1 Fano 3-folds anticanonically embedded in codimension 4 that start from so-called Type I Tom centres. We apply this to compute the Picard rank of many such Fano 3-folds.

We prove that a Fano variety (with arbitrary singularities) of dimension $n$ in positive characteristic is isomorphic to $\mathbb{P}^n$ if the Seshadri constant of the anti-canonical divisor at some smooth point is greater than $n$ and…

We classify Fano threefolds with only Gorenstein terminal singularities and Picard number greater than 1 satisfying an additional assumption that the $G$-invariant part of the Weil divisor class group is of rank 1 with respect to an action…

In this paper we show that a smooth toric variety $X$ of Picard number $r\leq 3$ always admits a nef primitive collection supported on a hyperplane admitting non-trivial intersection with the cone $\Nef(X)$ of numerically effective divisors…

We classify $2$-Fano horospherical varieties with Picard number $1$. We also review all the known examples of $2$-Fano manifolds and investigate the relation between the $2$-Fano condition and different notions of stability. This paper was…

We prove that the deformations of a smooth complex Fano threefold X with Picard number 1, index 1, and degree 10, are unobstructed. The differential of the period map has two-dimensional kernel. We construct two two-dimensional components…

This paper obtains criteria for a Fano variety X with normal crossing singularities defined over an algebraically closed field of characteristic zero, to be smoothable. The difference with the original version is that the theory of…

We study the Picard variety of the Fano surface of nodal and mildly cuspidal cubic threefolds in arbitrary characteristic by relating divisors on the Fano surface to divisors on the symmetric product of a curve of genus 4.

A horospherical variety is a normal algebraic variety where a reductive algebraic group acts with an open orbit which is a torus bundle over a flag variety. For example, toric varieties and flag varieties are horospherical. In this paper,…

In this paper, we give the complete classification of full exceptional collections on smooth toric Fano threefolds and fourfolds with Picard rank two. To be precise, we give a partial answer to the conjecture in \cite{Kuz} and \cite{LYY}:…

The classification of toric Fano manifolds with large Picard number corresponds to the classification of smooth Fano polytopes with large number of vertices. A smooth Fano polytope is a polytope that contains the origin in its interior such…

For Fano varieties of various singularities such as canonical and terminal, we construct examples with large Fano index. By low-dimensional evidence, we conjecture that our examples have the largest Fano index for all dimensions.

Let X be a smooth, complex Fano variety, and delta(X) its Lefschetz defect. It is known that if delta(X) is at least 4, then X is isomorphic to a product SxT, where dim T=dim X-2. In this paper we prove a structure theorem for the case…

We provide a systematic method to classify all smooth weak Fano toric varieties of Picard rank $3$ in any dimension using Macaulay2, and describe the classification explicitly in dimensions $3$ and $4$. There are $28$ and $114$ isomorphism…

Let X be an $n$-dimensional Fano manifold of Picard number 1. We study how many different ways X can compactify the complex vector group C^n equivariantly. Hassett and Tschinkel showed that when X = P^n with n \geq 2, there are many…

Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. We show that if rho(X)>12, then X is a product of del Pezzo surfaces. The proof relies on a careful study of divisorial elementary contractions f: X->Y such that the…

In this note we consider the problem of determining which Fano manifolds can be realised as fibres of a Mori fibre space. In particular, we study the case of toric varieties, Fano manifolds with high index and some Fano manifolds with high…

In this short note we give a characterization of smooth projective varieties of Picard number one that are separably uniruled but not separably rationally connected. We also give a sufficient condition involving the torsion order and the…

In this paper we study smooth toric Fano varieties using primitive relations and toric Mori theory. We show that for any irreducible invariant divisor D in a toric Fano variety X, we have $0\leq\rho_X-\rho_D\leq 3$, for the difference of…

Let $X \subset \mathbb P(a_0,\ldots,a_n)$ be a quasi-smooth weighted Fano hypersurface of degree $d$ and index $I_X$ such that $a_i |d$ for all $i$, with $a_0 \le \ldots \le a_n$. If $I_X=1$, we show that, under a suitable condition, the…