Strong rational connectedness of surfaces. (English) Zbl 1246.14064

Xu, Chenyang

J. Reine Angew. Math. 665, 189-205 (2012).

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)

MSC:

14M22	Rationally connected varieties
14J26	Rational and ruled surfaces

Keywords:

strong rational connectedness; log del Pezzo surface

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