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On covering and quasi-unsplit families of curves. (English) Zbl 1107.14015
Summary: 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 $$\dim(X) \leq 4$$, every covering and quasi-unsplit family $$V$$ of rational curves generates a geometric extremal ray of the Mori cone $$\overline{\text{NE}}(X)$$ of classes of effective 1-cycles and that the associated Mori contraction yields a geometric quotient for $$V$$ provided $$X$$ has canonical singularities.

##### MSC:
 14E30 Minimal model program (Mori theory, extremal rays) 14J99 Surfaces and higher-dimensional varieties 14M99 Special varieties
##### Keywords:
Covering families of curves; extremal curves; quotient
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##### References:
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