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

Non-elementary Fano conic bundles

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 rho_{X} the Picard number of X, we investigate such contractions when rho_{X}-rho_{Y} is greater than 1, called non-elementary. We prove that rho_{X}-rho_{Y} is at most 8, and we deduce new geometric information about our varieties, depending on rho_{X}-rho_{Y}. Moreover, when X is locally factorial with canonical singularities and with at most finitely many non-terminal points, we consider fiber type K_{X}-negative contractions f:X->Y with one-dimensional fibers, and we show that rho_{X}-rho_{Y} is at most 9.Comment: Final version, to appear in Collectanea Mathematic