Gianluca Occhetta

Generalized Mukai conjecture for special Fano varieties

Let X be a Fano variety of dimension n, pseudoindex iX and Picard number ρX. A generalization of a conjecture of Mukai says that ρX(iX−1)≤n. We prove that the conjecture holds for a variety X of pseudoindex iX≥n+3/3 if X admits an unsplit covering family of rational curves; we also prove that this condition is satisfied if ρX> and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension five.

Special rays in the Mori cone of a projective variety

Let X be a smooth n -dimensional projective variety over an algebraically closed field k such that K X is not nef. We give a characterization of non nef extremal rays of X of maximal length (i.e of length n – 1); in the case of Char( k ) = 0 we also characterize non nef rays of length n – 2.

THE CONE OF CURVES OF FANO VARIETIES OF COINDEX FOUR

We classify the cones of curves of Fano varieties of dimension greater or equal than five and (pseudo)index dim X - 3, describing the number and type of their extremal rays.

Ample vector bundles with sections vanishing on special varieties

Let e be an ample vector bundle of rank r on a complex projective manifold X such that there exists a section s ∈ Γ(e) whose zero locus Z = (s = 0) is a smooth submanifold of the expected dimension dim X-r:= n - r. Assume that Z is not minimal; we investigate the hypothesis under which the extremal contractions of Z can be lifted to X. Finally we study in detail the cases in which Z is a scroll, a quadric bundle or a del Pezzo fibration.

Fano manifolds whose elementary contractions are smooth P^1-fibrations: a geometric characterization of flag varieties

The present paper provides a geometric characterization of complete flag varieties for semisimple algebraic groups. Namely, if

Rational curves, Dynkin diagrams and Fano manifolds with nef tangent bundle

X

On rank

is a Fano manifold whose all elementary contractions are

2

\mathbb P^1

vector bundles on Fano manifolds

-fibrations then

Projective manifolds containing a large linear subspace with nef normal bundle

X