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Families of rationally connected varieties. (English) Zbl 1092.14063

A complex algebraic variety \(X\) is said to be rationally connected if two general points \(p,q\in X\) are contained in the image of a map \({\mathbb P^1} \to X\). In this beautiful paper the authors prove an important property of the rational connectivity (not shared by the rationality):
(T1) Let \(f: X\to Y\) be a dominant morphism of complex varieties, if \(Y\) and the general fiber of \(f\) are rationally connected, then \(X\) is rationally connected.
The proof is given in several steps.
1) (T1) is deduced from the following result:
(T2) Let \(\pi: X\to B\) a proper morphism of complex varieties, with \(B\) a smooth curve. If the general fiber of \(\pi\) is rationally connected, then \(\pi\) has a section.
2) The case \(B\simeq {\mathbb P^1}\) is considered. The notions of flexible and prefexible stable map \(f: C\to X\) are introduced and it is proved that:
(T3) if \(X\) admits a prefexible map, then \(\pi\) has a section.
3) A prefexible stable map is obtained, starting from a 1-dimensional linear section \(f:C \to X\) , by two ingenious constructions which consist in attaching rational curves to the fibres of \(\pi\) and in performing small deformations of the branch divisor of \(\pi f: C\to B\) (in the case of multiple fibers of \(\pi\)).
4) Finally the result (T2) is extended to the case of an arbitrary curve \(B\).

MSC:

14M20 Rational and unirational varieties
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
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