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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 the existence of a non-zero, non-big movable divisor on X. Our main result is that if Y is not P^1 or P^2, then the Picard number rho(X) of X is at most 18, with equality only if X is a product of surfaces. We also show that if a Fano 4-fold X has a dominant rational map X-->Z, regular and proper on an open subset of X, with dim(Z)=3, then either X is a product of surfaces, or rho(X) is at most 12. These results are part of a program to study Fano 4-folds with large Picard number via birational geometry., 25 pages. Minor changes. To appear in Algebra & Number Theory

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 where rho(X)=5; there are 6 families, among which one is new. We also deduce the classification of Fano 4-folds with rho(X)>4 with an elementary divisorial contraction sending a divisor to a curve., 16 pages

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., 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 $$\mathbb {P}^4$$ 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 fourfolds. 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.

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.

In this paper we address Fano manifolds X with a locally unsplit dominating family of rational curves of anticanonical degree equal to the dimension of X. We first observe that their Picard number is at most 3, and then we provide a classification of all cases with maximal Picard number. We also give examples of locally unsplit dominating families of rational curves whose varieties of minimal tangents at a general point is singular., Comment: 34 pages, 2 figures, to appear in IMRN

In this paper we describe the geometry of the 2m-dimensional Fano manifold G parametrizing (m-1)-planes in a smooth complete intersection Z of two quadric hypersurfaces in the complex projective space P^{2m+2}, for m>0. We show that there are exactly 2^{2m+2} distinct isomorphisms in codimension one between G and the blow-up of P^{2m} at 2m+3 general points, parametrized by the 2^{2m+2} distinct m-planes contained in Z, and describe these rational maps explicitly. We also describe the cones of nef, movable and effective divisors of G, as well as their dual cones of curves. Finally, we determine the automorphism group of G. These results generalize to arbitrary even dimension the classical description of quartic del Pezzo surfaces (m=1)., 31 pages. Minor revision, to appear in Geometry & Topology

Let X be a Fano 4-fold. For any prime divisor D in X, consider where i* is the push-forward of one-cycles, and let cD be the codimension of in . We define an integral invariant cX of X as the maximal cD, where D varies among all prime divisors in X. We know from 3 that if cX > 2, then either X is a product of Del Pezzo surfaces and ρX ≤ 18, or cX = 3 and ρX ≤ 6. In this paper we show that if cX = 2, then ρX ≤ 12.

We study the birational geometry of a Fano 4-fold X from the point of view of Mori dream spaces; more precisely, we study rational contractions of X. Here a rational contraction is a rational map f: X-->Y, where Y is normal and projective, which factors as a finite sequence of flips, followed by a surjective morphism with connected fibers. Such f is called elementary if the difference of the Picard numbers of X and Y is 1. We first give a characterization of non-movable prime divisors in X, when X has Picard number at least 6; this is related to the study of birational and divisorial elementary rational contractions of X. Then we study the rational contractions of fiber type on X which are elementary or, more generally, quasi-elementary. The main result is that the Picard number of X is at most 11 if X has an elementary rational contraction of fiber type, and 18 if X has a quasi-elementary rational contraction of fiber type., 43 pages. Final version, minor changes, to appear in Mathematische Annalen

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., Final version, to appear in the Annales Scientifiques de l'Ecole Normale Superieure

Let X be a smooth complex Fano variety. We study ‘quasi-elementary’ contractions of fiber type of X, which are a natural generalization of elementary contractions of fiber type. If f:X→Y is such a contraction, then the Picard numbers satisfy ρX≤ρY+ρF, where F is a general fiber of f. We show that, if dim Y ≤3 and ρY≥4, then Y is smooth and Fano; if moreover ρY≥6, then X is a product. This yields sharp bounds on ρX when dim X=4 and X has a quasi-elementary contraction of fiber type, and other applications in higher dimensions.

We give a structural theorem for ℚ-factorial toric varieties covered by lines in ℙN, and compute their dual defect. This yields a characterization of defective ℚ-factorial toric varieties in ℙN. The combinatorial description of such varieties is used to characterize some finite sets of monomials with discriminant equal to one.

Given a covering family $V$ of effective 1-cycles on a complex projective variety $X$, we find conditions allowing to construct a geometric quotient $q~: X \to Y$, with $q$ regular on the whole of $X$, such that every fiber of $q$ is an equivalence class for the equivalence relation naturally defined by $V$. Among others, we show that on a normal and $\Q$-factorial projective variety $X$ with $\dim(X) \leq 4$, every covering and quasi-unsplit family $V$ of rational curves generates a geometric extremal ray of the Mori cone $\overline{\rm NE}(X)$ of classes of effective 1-cycles and that the associated Mori contraction yields a geometric quotient for $V$ provided $X$ has canonical singularities.

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 a point, then rho(X) is at most 12. We give examples of such X with Picard number up to 9. The main theme (and tool) is the study of fixed prime divisors in Fano 4-folds, especially in the case rho(X)>6, in which we give some general results of independent interest., 44 pages. Minor changes, to appear in the Journal of Algebra

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_X2, 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., 31 pages. Revised version, minor changes. To appear in the Journal of the London Mathematical Society

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., 24 pages, 6 figures

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., Final version, to appear in Annales de l'Institut Fourier

The geometry of the moduli space of stable spin curves is studied, with emphasis on its combinatorial properties. In this context, the standard graph theoretic framework is not just a book-keeping device: some purely combinatorial results are proved, having moduli theoretic applications. In particular, certain strata of the moduli space of stable curves are characterized by a (finite) set of integers, measuring the non-reducedness of the scheme of spin curves, and definable in purely graph-theoretical terms., LaTeX, 16 pages, 30 figures

Let X be a smooth, complete, toric variety. We study those curves C in X that are contractible, in the sense that there exists an equivariant morphism with connected fibers, with source X, that contracts exactly the irreducible curves that are numerically equivalent to a multiple of C. When X is projective, we compare contractible and extremal curves, and we show that every curve in X is numerically equivalent to a linear combination with positive integral coefficients of contractible curves., LaTeX, 27 pages, 5 figures; revised version, to appear in Mathematische Zeitschrift

We give a structure theorem for n-dimensional smooth toric Fano varieties whose associated polytope has "many" pairs of centrally symmetric vertices., LaTeX, 15 pages

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

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 the Picard numbers of X and D. Moreover, if $\rho_X-\rho_D>0$ (with some additional hypotheses if $\rho_X-\rho_D=1$), we give an explicit birational description of X. Using this result, we show that when dim X=5, we have $\rho_X\leq 9$. In the second part of the paper, we study equivariant birational morphisms f whose source is Fano. We give some general results, and in dimension 4 we show that f is always a composite of smooth equivariant blow-ups. Finally, we study under which hypotheses a non-projective toric variety can become Fano after a smooth equivariant blow-up.

Here we focus on the compactification of the moduli space of curves of genus g together with an unramified double cover, constructed by Arnaud Beauville in order to compactify the Prym mapping. We present an alternative description of it, inspired by the moduli space of spin curves of Maurizio Cornalba, and we discuss in detail its main features, both from a geometrical and a combinatorial point of view., Revised version, exposition improved, modified statement of Theorem 6