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.

MSC:

 14M22 Rationally connected varieties 14J26 Rational and ruled surfaces
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References:

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