Related papers: On the Picard number of singular Fano varieties

We prove the following main result: Let X be a Fano 3-fold with terminal Q-factorial singularities and X does not have a small extremal ray and a face of Kodaira dimension 1 or 2 for Mori polyhedron of X. Then the Picard number \rho (X) <…

We study Gorenstein almost Fano threefolds X with canonical singularities and pseudoindex > 1. We show that the maximal Picard number of X is 10 in general, 3 if X is Fano, and 8 if X is toric. Moreover, we characterize the boundary cases.…

Let X be a complex Fano manifold of arbitrary dimension, and D a prime divisor in X. We consider the image H of N_1(D) in N_1(X) under the natural push-forward of 1-cycles. We show that the codimension c of H in N_1(X) is at most 8.…

In this paper we study smooth, complex Fano 4-folds X with large Picard number rho(X), with techniques from birational geometry. Our main result is that if X is isomorphic in codimension one to the blow-up of a smooth projective 4-fold Y at…

Let X be a complex, Gorenstein, Q-factorial, toric Fano variety. We prove two conjectures on the maximal Picard number of X in terms of its dimension and its pseudo-index, and characterize the boundary cases. Equivalently, we determine the…

We classify mildly singular Fano varieties $X$ such that $\mathrm{Nef}(X)=\mathrm{Psef}(X)$ and that the Picard number of $X$ is equal to the dimension of $X$ minus $1$.

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…

Using a construction due to C. Casagrande and further developed by the author, we prove that the Picard number of a non-smooth Fano 3-fold with isolated factorial canonical singularities, is at most 6.

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…

We study a particular kind of fiber type contractions between complex, projective, smooth varieties f:X->Y, called Fano conic bundles. This means that X is a Fano variety, and every fiber of f is isomorphic to a plane conic. Denoting by…

We give examples of Fano varieties $X$ with Picard number 1, which have terminal singularities and admit endomorphisms with degree larger than 1.

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 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.

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…

Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. If X contains a prime divisor D with rho(X)-rho(D)>2, then either X is a product of del Pezzo surfaces, or rho(X)=5 or 6. In this setting, we completely classify the case…

We study (smooth, complex) Fano 4-folds X having a rational contraction of fiber type, that is, a rational map X-->Y that factors as a sequence of flips followed by a contraction of fiber type. The existence of such a map is equivalent to…

Let X be a Fano manifold of pseudoindex i_X whose Picard number is at least two and let R be an extremal ray of X with exceptional locus Exc(R). We prove an inequality which bounds the length of R in terms of i_X and of the dimension of…

In this paper we study the geometry of mildly singular Fano varieties on which there is an effective prime divisor of Picard number one. Afterwards, we address the case of toric varieties. Finally, we treat the lifting of extremal…

We study Fano 3-folds with Fano index 2: that is, 3-folds X with rank Pic(X) = 1, Q-factorial terminal singularities and -K_X = 2A for an ample Weil divisor A. We give a first classification of all possible Hilbert series of such polarised…

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…