Given a covering family of effective 1-cycles on a complex projective variety , we find conditions allowing to construct a geometric quotient , with regular on the whole of , such that every fiber of is an equivalence class for the equivalence relation naturally defined by . Among others, we show that on a normal and \( \Q \)-factorial projective variety with , every covering and quasi-unsplit family of rational curves generates a geometric extremal ray of the Mori cone of classes of effective 1-cycles and that the associated Mori contraction yields a geometric quotient for provided has canonical singularities.
Cite this article
Laurent Bonavero, Stéphane Druel, Cinzia Casagrande, On covering and quasi-unsplit families of curves. J. Eur. Math. Soc. 9 (2007), no. 1, pp. 45–57DOI 10.4171/JEMS/71