Uniform families of minimal rational curves on Fano manifolds

It is a well-known fact that families of minimal rational curves on rational homogeneous manifolds of Picard number one are uniform, in the sense that the tangent bundle to the manifold has the same splitting type on each curve of the family. In this note we prove that certain --stronger-- uniformity conditions on a family of minimal rational curves on a Fano manifold of Picard number one allow to prove that the manifold is homogeneous

Nestings of rational homogeneous varieties

In this paper we study the existence of sections of universal bundles on rational homogeneous varieties -- called nestings -- classifying them completely in the case in which the Lie algebra of the automorphism group of the variety is simple of classical type. In particular we show that, under this hypothesis, nestings do not exist unless there exists a proper algebraic subgroup of the automorphism group acting transitively on the base variety.Comment: Major revision of the exposition. To appear in Tranformation Group

Rank two Fano bundles on G(1,4)

We classify rank two Fano bundles over the Grassmannian of lines \G(1,4). In particular we show that the only non-split rank two Fano bundle over \G(1,4) is, up to a twist, the universal quotient bundle \cQ. This completes the classification of rank two Fano bundles over Grassmannians of lines

Geometric realizations of birational transformations via C∗\mathbb{C}^*-actions

In this paper we study varieties admitting torus actions as geometric realizations of birational transformations. We present an explicit construction of these geometric realizations for a particular class of birational transformations, and study some of their geometric properties, such as their Mori, Nef and Movable cones

Rank two Fano bundles on G(1, 4)

Abstract We classify rank two Fano bundles over the Grassmannian of lines G(1, 4). In particular we show that the only non-split rank two Fano bundle over

Chow quotients of C∗\mathbb{C}^*-actions

Given an action of the one-dimensional torus on a projective variety, the associated Chow quotient arises as a natural parameter space of invariant $1$-cycles, which dominates the GIT quotients of the variety. In this paper we explore the relation between the Chow and the GIT quotients of a variety, showing how to construct explicitly the former upon the latter via successive blowups under suitable assumptions. We also discuss conditions for the smoothness of the Chow quotient, and present some examples in which it is singular.Comment: 29 pages, 4 figure