JournalsjemsVol. 9 , No. 1DOI 10.4171/jems/71

On covering and quasi-unsplit families of curves

  • Laurent Bonavero

    Université Grenoble I, Saint-Martin-d'Hères, France
  • Stéphane Druel

    Université Grenoble I, Saint-Martin-d'Hères, France
  • Cinzia Casagrande

    Università di Pisa, Italy
On covering and quasi-unsplit families of curves cover

Abstract

Given a covering family VV of effective 1-cycles on a complex projective variety XX, we find conditions allowing to construct a geometric quotient q :XYq~: X \to Y, with qq regular on the whole of XX, such that every fiber of qq is an equivalence class for the equivalence relation naturally defined by VV. Among others, we show that on a normal and \Q\Q-factorial projective variety XX with dim(X)4\dim(X) \leq 4, every covering and quasi-unsplit family VV of rational curves generates a geometric extremal ray of the Mori cone NE(X)\overline{\rm NE}(X) of classes of effective 1-cycles and that the associated Mori contraction yields a geometric quotient for VV provided XX has canonical singularities.