Extending extremal contractions from an ample section.
Raggi estremali di varietà proiettive lisce
Generalized Mukai conjecture for special Fano varieties
Let X be a Fano variety of dimension n, pseudoindex i X and Picard number ρX. A generalization of a conjecture of Mukai says that ρX(i X−1)≤n. We prove that the conjecture holds for a variety X of pseudoindex i X≥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.
A classification theorem on Fano bundles
In this paper we classify rank two Fano bundles on Fano manifolds satisfying . The classification is obtained via the computation of the nef and pseudoeffective cones of the projectivization , that allows us to obtain the cohomological invariants of and . As a by-product we discuss Fano bundles associated to congruences of lines, showing that their varieties of minimal rational tangents may have several linear components.
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