11,624 research outputs found

### On some Fano manifolds admitting a rational fibration

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.Comment: 31 pages. Revised version, minor changes. To appear in the Journal of
the London Mathematical Societ

### Centrally symmetric generators in toric Fano varieties

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

### Quasi elementary contractions of Fano manifolds

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 Mathematic

### Fano 4-folds, flips, and blow-ups of points

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.Comment: 44 pages. Minor changes, to appear in the Journal of Algebr

### On the birational geometry of Fano 4-folds

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.Comment: 43 pages. Final version, minor changes, to appear in Mathematische
Annale

### On Fano manifolds with a birational contraction sending a divisor to a curve

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 figure

### Fano 4-folds with rational fibrations

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.Comment: 25 pages. Minor changes. To appear in Algebra & Number Theor

### The number of vertices of a Fano polytope

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 Fourie

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