On covering and quasi-unsplit families of curves

• Autores: Laurent Bonavero, Cinzia Casagrande, Stéphane Druel
• Localización: Journal of the European Mathematical Society, ISSN 1435-9855, Vol. 9, Nº 1, 2007, págs. 45-57
• Idioma: inglés
• DOI: 10.4171/jems/71
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• Resumen

  • Given a covering family $V$ of effective 1-cycles on a complex projective variety $X$, we find conditions allowing to construct a geometric quotient $q~: X \to Y$, 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 others, we show that on a normal and $\Q$-factorial projective variety $X$ with $\dim(X) \leq 4$, every covering and quasi-unsplit family $V$ of rational curves generates a geometric extremal ray of the Mori cone $\overline{\rm NE}(X)$ of classes of effective 1-cycles and that the associated Mori contraction yields a geometric quotient for $V$ provided $X$ has canonical singularities.