We investigate projective varieties which are binary symmetric models of trivalent phylogenetic trees. We prove that they have Gorenstein terminal singularities and are Fano varieties of index 4 and dimension equal to the number of edges of the tree in question. Moreover any two such varieties which are of the same dimension are deformation equivalent, that is, they are in the same connected component of the Hilbert scheme of the projective space. As an application we provide a simple formula for computing their Hilbert-Ehrhart polynomial.

The aim of this paper is to describe the structure of Fano bundles in dimension > 4. Introduction. In this paper rank 2 vector bundles E on projective spaces Ψ n and quadrics Q n are investigated which enjoy the additional property that their projectized bundles Ψ(E) are Fano manifolds, i.e. have negative canonical bundles. Such bundles are shortly called Fano bundles. Up to dimension 3 Fano bundles are completely classified by [SW], [SW], [SW"], [SSW]. The aim of this paper is to describe the structure of Fano bundles in dimension > 4. Namely we prove the following MAIN THEOREM. Let E be a rank 2 Fano bundle on Ψ n or Q n , n > 4. Then up to some explicit exceptions on Q 4 and Q$ (see ex. (2.1), (2.2), (2.3)), E splits into a direct sum of line bundles. A rank 2 bundle E on Ψ n is Fano if and only if the "Q-vector bundle" E (det£*)/2 ® &{?ψ) is ample, i.e.

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As an application of a recent characterization of complete flag manifolds as Fano manifolds having only P 1 -bundles as elementary contractions, we consider here the case of a Fano manifold X of Picard number one supporting an unsplit family of rational curves whose subfamilies parametrizing curves through a fixed point are rational homogeneous, and we prove that X is homogeneous. In order to do this, we first study minimal sections on flag bundles over the projective line, and discuss how Grothendieck's theorem on principal bundles allows us to describe a flag bundle upon some special sections.

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