On covering and quasi-unsplit families of curves

Abstract

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 $\mathbb{Q}$-factorial projective variety $X$ with $\dim (X)\le 4$, every covering and quasi-unsplit family $V$ of rational curves generates a geometric extremal ray of the Mori cone $\overline{\mathrm{NE}}(X)$ of classes of effective 1-cycles and that the associated Mori contraction yields a geometric quotient for $V$ provided $X$ has canonical singularities.