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

Hacon, Christopher D.; McKernan, James

Duke Math. J. 138, No. 1, 119-136 (2007).

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.

Reviewer: Massimiliano Mella (Ferrara)

MSC:

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

Keywords:

rational connection; dlt; mmp; fibres of a resolution

Citations:

Zbl 0074.26301; Zbl 0423.14006; Zbl 1047.14006; Zbl 1121.14011

References:

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