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Fundamental groups of rationally connected varieties. (English) Zbl 1077.14520
The notion of a rationally connected variety was introduced by the author, Y. Miyaoka and S. Mori [J. Algebr. Geom. 1, 429–448 (1992; Zbl 0780.14026)] and says that any two points of a proper variety $$X$$ over a field of characteristic $$0$$ are in the image of an appropriate morphism $$f:\mathbb{P}^1\to X$$.
By a result of J.-P. Serre [J. Lond. Math. Soc. 34, 481–484 (1959; Zbl 0097.36301)], it turns out that rationally connected varieties are simply connected. In the paper under review, the author investigates the more sophisticated problem of the fundamental groups of open subsets in a rationally connected variety. The main result is the following
Theorem. Let $$X$$ be a smooth projective rationally connected variety $$X$$ over an algebraically closed field of characteristic $$0$$, let $$U\subset X$$ be an open subset, and let $$x_0\in U$$ be a point. Then there is an open subset $$0\in V\subset {\mathbb{A}}^1$$ and a morphism $$f: V\to U$$ such that $$f(0)=x_0$$, and $$\pi_1(V,0)\to\pi_1(U,x_0)$$ is surjective.
Several applications to $$p$$-adic fields and the image of the monodromy representations for the moduli space of genus $$g$$ curves (which is known to be unirational for $$2\leq g \leq 13$$) are given.

##### MSC:
 14F35 Homotopy theory and fundamental groups in algebraic geometry 14E08 Rationality questions in algebraic geometry 14J45 Fano varieties
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