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Quasi-lines and their degenerations. (English) Zbl 1139.14017
Let \(X\) be a Fano variety. Mori Theory tells us that the cone of effective curves of \(X\) is polyhedral and generated by rational curves. It is quite hard in general to understand the generators of the cone. The paper under review studies this question under the assumption that the variety \(X\) contains a quasi line. Namely a smooth rational curve with normal bundle isomorphic to the normal bundle of a line in projective space, see L. Bădescu et al. [in: Complex analysis and algebraic geometry. A volume in memory of Michael Schneider. Berlin: Walter de Gruyter. 1–27 (2000; Zbl 1078.14010)]. For an interesting relation between this notion and rational connectedness, see P. Ionescu and D. Naie [Int. J. Math. 14, No. 10, 1053–1080 (2003; Zbl 1080.14512)].
The authors expect that under this hypothesis the cone is generated by, irreducible components appearing in, deformations of the quasi line. The main results are related to Fano varieties admitting fiber type contractions. The presence of quasi lines in either \(X\) or the fiber is then used to study the various possibility. The main techniques are Mori extremal ray and bend and break. The paper is enriched by many examples.
MSC:
14E30 Minimal model program (Mori theory, extremal rays)
14J10 Families, moduli, classification: algebraic theory
14J30 \(3\)-folds
14J40 \(n\)-folds (\(n>4\))
14J45 Fano varieties
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