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Quasi-lines and their degenerations

Laurent BonaveroAndreas Höring — 2007

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper we study the structure of manifolds that contain a quasi-line and give some evidence towards the fact that the irreducible components of degenerations of the quasi-line should determine the Mori cone. We show that the minimality with respect to a quasi-line yields strong restrictions on fibre space structures of the manifold.

On covering and quasi-unsplit families of curves

Laurent BonaveroCinzia CasagrandeStéphane Druel — 2007

Journal of the European Mathematical Society

Given a covering family V of effective 1-cycles on a complex projective variety X , we find conditions allowing one to construct a geometric quotient q : X Y , with q regular on the whole of X , such that every fiber of q is an equivalence class for the equivalence relation naturally defined by V . Among other results, we show that on a normal and -factorial projective variety X with canonical singularities and dim X 4 , every covering and quasi-unsplit family V of rational curves generates a geometric extremal...

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