Related papers: Fano varieties of middle pseudoindex

We study the deformation theory of a Fano variety X with normal crossing singularities of dimension at most three. We obtain a formula for the sheaf T^1(X) of first order deformations of X in a suitable log resolution of X and its singular…

We study (smooth, complex) Fano 4-folds X with Picard number rho(X)>6. We show that if rho(X)>9, then X is a product of del Pezzo surfaces, thus improving recent results by the author and by the author and S.A. Secci; the statement is now…

We prove that a smooth well formed Picard rank one Fano complete intersection of dimension at least 2 in a toric variety is a weighted complete intersection.

Let $\mathcal{X}$ be a smooth Fano threefold over the complex numbers of Picard rank $1$ with finite automorphism group. We give numerical restrictions on the order of the automorphism group $\mathrm{Aut}(\mathcal{X})$ provided the genus…

Let X be a (smooth, complex) Fano 4-fold. For any prime divisor D in X, consider the image of N_1(D) in N_1(X) under the push-forward of 1-cycles, and let c_D be its codimension in N_1(X). We define an integral invariant c_X of X as the…

This article settles the question of existence of smooth weak Fano threefolds of Picard number two with small anti-canonical map and previously classified numerical invariants obtained by blowing up certain curves on smooth Fano threefolds…

We give a complete classification of smooth, complex projective Fano 4-folds of Picard number 3 having a prime divisor of Picard number 1. They form 28 distinct families, and we compute the main numerical invariants, study the base locus of…

We show that, for a Q-Fano threefold X of Fano index 2, the inequality dim |-1/2K_X| <= 4 holds with a single well understood family of varieties having dim |-1/2K_X| = 4.

Determining when the birational automorphism group of a Fano variety is finite is an interesting and difficult problem. The main technique for studying this problem is by the Noether-Fano method. This method has been effective in studying…

Consider a family of Fano varieties $\pi: X \longrightarrow B\ni o$ over a curve germ with a smooth total space $X$. Assume that the generic fiber is smooth and the special fiber $F=\pi^{-1}(o)$ has simple normal crossings. Then $F$ is…

In this paper, we study the structure of Fano fibrations of varieties admitting an int-amplified endomorphism. We prove that if a normal $\mathbb{Q}$-factorial klt projective variety $X$ has an int-amplified endomorphism, then there exists…

We show that Fano 4-folds with Picard number 5 have Lefschetz defect 3 if and only if they are toric of combinatorial type K. We also find a characterization for such varieties in terms of Picard number of prime divisors. Moreover, we…

We study Fano threefolds with Picard number one equipped with a holomorphic section in $\Omega_V^1(1)$.

In this paper, an update on the classification of smooth weak Fano threefolds with Picard number two and small anti-canonical maps is given. Geometric constructions are provided for previously open numerical cases by blowing up certain…

Q-factorial Gorenstein toric Fano varieties X of dimension d with Picard number rho(X) correspond to simplicial reflexive d-polytopes with rho(X)+d vertices. Casagrande showed that any simplicial reflexive d-polytope has at most 3d…

We classify primitive Fano threefolds in positive characteristic whose Picard numbers are at least two. We also classify Fano theefolds of Picard rank two.

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 V_{12} Fano threefold is a smooth Fano threefold X of index 1 with Pic X = Z and (-K_X)^3=12. We show that the bounded derived category of coherent sheaves on any V_{12} threefold X admits a semiorthogonal decomposition consisting of two…

We give some bounds on the anticanonical degrees of Fano varieties with Picard number 1 and mild singularities, extending results of Koll\'ar et al. from the early 90's and improving them even in the smooth case. The proof is based on a…

This paper considers Q-Fano 3-folds X with \rho=1. The aim is to determine the maximal Fano index f of X. We prove that f<= 19, and that in case of equality, the Hilbert series of X equals that of weighted projective space PP(3,4,5,7). From…