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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
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