Related papers: On the Picard number of almost Fano threefolds wit…

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

We give a characterization of Gorenstein toric Fano varieties of dimension $n$ with index $n$ among toric varieties. As an application, we give a strong version of Fujita's freeness conjecture and also give a simple proof of Fujita's very…

Let X be a smooth complex Fano 4-fold. We show that if X has a small elementary contraction, then the Picard number rho(X) of X is at most 12. This result is based on a careful study of the geometry of X, on which we give a lot of…

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…

We classify the cones of curves of Fano varieties of dimension greater or equal than five and (pseudo)index dim X -3, describing the number and type of their extremal rays.

Let $X$ be an $n$-dimensional normal $\mathbb{Q}$-factorial projective variety with canonical singularities and Picard number one such that $X$ is smooth in codimension two, $-K_X$ is ample and $n\geq 2$. We prove that $X$ satisfies the…

Let X be a complex Fano manifold of dimension n. Let s(X) be the sum of l(R)-1 for all the extremal rays of X, the edges of the cone NE(X) of curves of X, where l(R) denotes the minimum of (-K_X \cdot C) for all rational curves C whose…

For a log Fano manifold (X, D) with D\neq 0 and with the log Fano pseudoindex \geq 2, we prove that the restriction homomorphism Pic(X)\to Pic(D_1) of Picard groups is injective for any irreducible component D_1\subset D.The strategy of our…

In our previous work we conjectured - inspired by an algebro-geometric result of Fujita - that the height of an arithmetic Fano variety X of relative dimension $n$ is maximal when X is the projective space $\mathbb{P}^n_{\mathbb{Z}}$ over…

We construct an explicit complex smooth Fano threefold with Picard number 1, index 1, and degree 10 (also known as a Gushel-Mukai threefold) and prove that it is not rational by showing that its intermediate Jacobian has a faithful…

Let $X$ be a projective Fano manifold of Picard number one, different from the projective space. There is a folklore conjecture that any non-constant endomorphism of $X$ is an isomorphism. In the first half of this article, we will prove…

We classify the terminal Fano threefolds of Picard number one that come with an effective action of a two-torus. Our approach applies also to higher dimensions and generalizes the correspondence between toric Fano varieties and lattice…

We give the complete classification of Mukai pairs of dimension $4$ and rank $2$ with Picard number one, that is, pairs $(X,E)$ where $X$ is a Fano $4$-fold with Picard number one, and $E$ is an ample vector bundle of rank two on $X$ with…

A Fano-Enriques threefold is a three-dimensional non-Gorenstein Fano variety of index 1 with at most canonical singularities. We study the birational geometry of Fano-Enriques threefolds with terminal cyclic quotient singularities. We…

We show that some important classes of weak Fano $3$-folds of Picard rank $2$ do not satisfy Bott vanishing. Using this we show that any smooth projective $3$-fold $X$ of Picard rank $2$ with $-K_X$ nef which is the image of a projective…

We show that mixed-characteristic and equi-characteristic small deformations of 3-dimensional canonical (resp. terminal) singularities with perfect residue field of characteristic $p>5$ are canonical (resp. terminal). We discuss…

Let $X$, $Y$ be Fano threefolds of Picard number one and such that the ample generators of Picard groups are very ample. Let $X$ be of index one and $Y$ be of index two. It is shown that the only morphisms from $X$ to $Y$ are double…

We construct families of non-toric $\mathbb{Q}$-factorial terminal Fano ($\mathbb{Q}$-Fano) threefolds of codimension $\geq 20$ corresponding to 54 mutation classes of rigid maximally mutable Laurent polynomials. From the point of view of…

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.

Let $X$ be a Fano variety with at worst isolated quotient singularities. Our result asserts that if $C \cdot (-K_X) > max\{\frac{n}{2}+1,\frac{2n}{3}\}$ for every curve $C \subset X$, then $\rho_X=1$.