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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 where delta(X)=3. We show that there exists a smooth Fano variety T with dim T=dim X-2 such that X is obtained from T with two possible explicit constructions; in both cases there is a P^2-bundle Z over T such that X is the blow-up of Z along three pairwise disjoint smooth, irreducible, codimension 2 subvarieties. Then we apply the structure theorem to Fano 4-folds, to the case where X has Picard number 5, and to Fano varieties having an elementary divisorial contraction sending a divisor to a curve. In particular we complete the classification of Fano 4-folds with delta(X)=3., Comment: 30 pages, 2 figures. This version of the paper incorporates the published article with its corrigendum [CRS22], where a missing case in Prop. 7.1 is added

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 information. We also show that in the boundary case rho(X)=12 an open subset of X has a smooth fibration with fiber the projective line. Together with previous results, this implies if X is a Fano 4-fold with rho(X)>12, then every elementary contraction of X is divisorial and sends a divisor to a surface. The proof is based on birational geometry and the study of families of rational curves. More precisely the main tools are: the study of families of lines in Fano 4-folds and the construction of divisors covered by lines, a detailed study of fixed prime divisors, the properties of the faces of the effective cone, and a detailed study of rational contractions of fiber type., Comment: 46 pages, minor revision, to appear in Advances in Mathematics

Building on the work of Mukai, we explore the birational geometry of the moduli spaces M_{S,L} of semistable rank two torsion-free sheaves, with c_1=-K_S and c_2=2, on a polarized degree one del Pezzo surface (S,L); this is related to the birational geometry of the blow-up X of P^4 in 8 points. Our analysis is explicit and is obtained by looking at the variation of stability conditions. Then we provide a careful investigation of the blow-up X and of the moduli space Y=M_{S,-K_S}, which is a remarkable family of smooth Fano 4-folds. In particular we describe the relevant cones of divisors of Y, the group of automorphisms, and the base loci of the anticanonical and bianticanonical linear systems., Comment: Revised version, 46 pages, to appear in Revista Matematica Complutense

Let N be the moduli space of stable rank 2 quasiparabolic vector bundles of fixed degree on the projective line with 2g+1 marked points, where g>1, and stability is with respect to the weights {0,1/2} at each marked point. In this note we show that N is isomorphic to the variety of (g-2)-dimensional linear subspaces of P^{2g}, contained in the intersection of two quadrics. The proof relies on the work of Bhosle on the relation among quasiparabolic vector bundles on P^1 and invariant vector bundles on hyperelliptic curves, and the description by Bhosle and Ramanan of the moduli space of stable rank 2 vector bundles on a hyperelliptic curve, with fixed determinant of odd degree., Comment: 9 pages

Let X be a smooth, complex Fano variety. For every prime divisor D in X, we set c(D):=dim ker(r:H^2(X,R)->H^2(D,R)), where r is the natural restriction map, and we define an invariant of X as c_X:=max{c(D)|D is a prime divisor in X}. In a previous paper we showed that c_X<9, and that if c_X>2, then either X is a product, or X has a flat fibration in Del Pezzo surfaces. In this paper we study the case c_X=2. We show that up to a birational modification given by a sequence of flips, X has a conic bundle structure, or an equidimensional fibration in Del Pezzo surfaces. We also show a weaker property of X when c_X=1., Comment: 31 pages. Revised version, minor changes. To appear in the Journal of the London Mathematical Society

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 maximal c_D, where D varies among all prime divisors in X. One easily sees that c_X is at most rho_X-1 (where rho is the Picard number), and that c_X is greater or equal than rho_X-rho_D, for any prime divisor D in X. We know from previous works that if c_X > 2, then either X is a product of Del Pezzo surfaces and rho_X is at most 18, or c_X=3 and rho_X is at most 6. In this paper we show that if c_X=2, then rho_X is at most 12., Comment: 11 pages

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. Moreover if c>2, then either X=SxY where S is a Del Pezzo surface, or c=3 and X has a flat fibration in Del Pezzo surfaces onto a Fano manifold Y, such that the difference of the Picard numbers of X and Y is 4. We give applications to Fano 4-folds, to Fano varieties with pseudo-index >1, and to surjective morphisms whose source is Fano, having some high-dimensional fibers or low-dimensional target., Comment: Final version, to appear in the Annales Scientifiques de l'Ecole Normale Superieure

Let X be a smooth Fano variety of dimension at least 4. We show that if X has an elementary birational contraction sending a divisor to a curve, then the Picard number of X is smaller or equal to 5., Comment: 24 pages, 6 figures

Let X be a smooth complex Fano variety. We define and study 'quasi elementary' contractions of fiber type f: X -> Y. These have the property that rho(X) is at most rho(Y)+rho(F), where rho is the Picard number and F is a general fiber of f. In particular any elementary extremal contraction of fiber type is quasi elementary. We show that if Y has dimension at most 3 and Picard number at least 4, then Y is smooth and Fano; if moreover rho(Y) is at least 6, then X is a product. This yields sharp bounds on rho(X) when dim(X)=4 and X has a quasi elementary contraction, and other applications in higher dimensions., Comment: Final version, minor changes, to appear in Compositio Mathematica

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. In the Fano case, we prove that the generalized Mukai conjecture holds., Comment: 18 pages

The main result of this paper is a structural theorem for projective Q-factorial toric varieties X in P^N, covered by lines. We prove that there exists a toric fibration f: X -> Z, locally trivial in the Zariski topology, with fiber a product of projective joins. All lines in X intersecting the open subset isomorphic to the torus, are contained in some fiber of f. This characterization has a geometrical application to dual defective toric varieties, and a combinatorial application to discriminants of lattice subsets. We prove that X has positive dual defect if and only if it has an elementary extremal contraction of fiber type, whose general fiber is a projective join with dual defect bigger than its codimension in X. Turning to combinatorics, we characterize lattice subsets A with discriminant D_A equal to one, under suitable assumptions on the polytope Conv(A)., Comment: 24 pages. Refereed version, to appear in CCM (Communications in Contemporary Mathematics)

We study extremality properties of covering families of rational curves on projective varieties. Among others, we show that on a normal and Q-factorial projective variety of dimension at most 4, every covering and quasi-unsplit family of rational curves generates a geometric extremal ray of the Mori cone., Comment: 13 pages

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 maximal number of vertices of a simplicial reflexive polytope., Comment: Final version, to appear in Annales de l'Institut Fourier

Generalizing a question of Mukai, we conjecture that a Fano manifold $X$ with Picard number $\rho_X$ and pseudo-index $\iota_X$ satisfies $\rho_X (\iota_X-1) \le \dim(X)$. We prove this inequality in several situations: $X$ is a Fano manifold of dimension $\le 4$, $X$ is a toric Fano manifold of dimension $\le 7$ or $X$ is a toric Fano manifold of arbitrary dimension with $\iota_X \ge \dim(X)/3+1$. Finally, we offer a new approach to the general case., Comment: 21 pages, in french