##
**Strong rational connectedness of surfaces.**
*(English)*
Zbl 1246.14064

Let \(X\) be a complex projective variety, then \(X\) is rationally connected if for any two very general points \(x,y\in X\) there is a rational curve \(f:\mathbb P ^1\to X\) such that \(x,y\in f(\mathbb P ^1)\). By a result of Campana and Kollár-Miyaoka-Mori, it is known that all smooth Fano varieties are rationally connected and by a result of Zhang, the same is true for log Fano varieties i.e. for projective klt pairs \((X,\Delta )\) such that \(X\) is normal and \(-(K_X+\Delta)\) is ample. It is also known that if \(X\) is smooth then \(X\) is rationally connected if and only if it is strongly rationally connected so that for any \(x\in X\) there is a rational curve \(f:\mathbb P ^1\to X\) such that \(x\in f(\mathbb P ^1)\) and \(f^*T_X\) is ample. One can also ask if the smooth locus of a log Fano variety is rationally connected. In general this is a very hard question, which in dimension \(2\) was affirmatively answered by Keel and M{c}Kernan. In the paper under review it is shown that the smooth locus of a two dimensional log Fano variety is strongly rationally connected and that if \(S\) is a projective surface with at worst Du Val singularities such that the smooth locus \(S^{\text{sm}}\) is rationally connected, then \(S^{\text{sm}}\) is strongly rationally connected.

Reviewer: Christopher Hacon (Salt Lake City)

### References:

[1] | DOI: 10.1090/S0894-0347-01-00380-0 · Zbl 0991.14007 |

[2] | DOI: 10.1007/BF01390174 · Zbl 0317.14001 |

[3] | Campana F., Ann. Sci. E’c. Norm. 25 pp 539– (1992) |

[4] | DOI: 10.1007/BF02684599 · Zbl 0181.48803 |

[5] | DOI: 10.1090/S0894-0347-02-00402-2 · Zbl 1092.14063 |

[6] | DOI: 10.1215/S0012-7094-07-13813-4 · Zbl 1128.14028 |

[7] | Hassett B., Pure Appl. Math. Quart. 4 pp 743– (2008) |

[8] | Keel S., Mem. Amer. Math. Soc. 140 pp 669– (1999) |

[9] | DOI: 10.2307/2152888 · Zbl 0839.14031 |

[10] | Kollár J., J. Alg. Geom. 1 pp 429– (1992) |

[11] | Kollár J., J. Di{\currency}. Geom. 36 pp 765– (1992) |

[12] | DOI: 10.1007/s00209-005-0875-9 · Zbl 1096.14007 |

[13] | DOI: 10.1215/S0012-7094-06-13414-2 · Zbl 1114.14002 |

[14] | Olsson M., Comp. Math. 143 pp 476– (2007) |

[15] | DOI: 10.1007/BF01388892 · Zbl 0694.14001 |

[16] | Zhang Q., Math. 590 pp 131– (2006) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.