On Shokurov’s rational connectedness conjecture. (English) Zbl 1128.14028

It is classically known that the exceptional locus of a birational morphism between smooth varieties is covered by rational curves, see [S. Abhyankar, Am. J. Math. 78, 321–348 (1956; Zbl 0074.26301)]. More recently, S. Mori proved that a smooth Fano variety is covered by rational curves [Ann. Math. (2) 110, 593–606 (1979; Zbl 0423.14006)]. The paper under review generalizes both results to special singular varieties, essentially dlt pairs, by solving a conjecture of V. V. Shokurov on rational connectedness [Math. Notes 68, No. 5, 652–660 (2000); translation from Mat. Zametki 68, No. 5, 771–782 (2000; Zbl 1047.14006)]. The statement of the main theorem, too technical to be stated in a review, implies the results afore mentioned for dlt pairs and much more. The rational connectedness of a fiber, say \(F\), can be desumed by studying rational maps starting from \(F\). To do this the authors produce, after a subtle analysis of Fano fibrations and a lifting theorem inherited from [C. D. Hacon and J. McKernan, Invent. Math. 166, No. 1, 1–25 (2006; Zbl 1121.14011)], a log pair \((F,\Delta)\) that shows by standard arguments the rational connectedness of \(F\). It has to be noted that the ideas of lifting and producing ad hoc log varieties from a starting one are central in MMP and has been, subsequently, used by the authors in proving the Minimal Model Conjecture.


14J45 Fano varieties
14E30 Minimal model program (Mori theory, extremal rays)
14E05 Rational and birational maps
14J17 Singularities of surfaces or higher-dimensional varieties
Full Text: DOI arXiv


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