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**Rational connectedness and Galois covers of the projective line.**
*(English)*
Zbl 0990.12003

The author considers the regular inverse Galois problem over “large” fields \(k\). A field \(k\) is said to be large if for any smooth integral \(k\)-variety with a \(k\)-point the set of \(k\)-points is Zariski dense. Real closed fields, \(p\)-adic fields, pseudo algebraically closed fields are large. The main result of the paper is

Theorem 1: Let \(G\) be a finite group. Let \(k\) be a large field of characteristic zero. Let \(E=\text{Spec}(K)\) be a \(G\)-torsor over \(\text{Spec}(k)\). Then there exist an open set \(U\) of the affine line \({\mathbb A}_k^1\) containing a \(k\)-point \(O\) and a \(G\)-torsor \(V\to U\) such that the following two properties hold:

(i) The fibre of \(V\to U\) over \(O\) is isomorphic to \(E\) \((\)as a \(G\)-torsor over \(\text{Spec}(k))\);

(ii) The smooth \(k\)-curve \(V\) is geometrically connected.

When the \(G\)-torsor \(E/\text{Spec}(k)\) is trivial, this is a theorem of D. Harbater [Lect. Notes Math. 1240, 165-195 (1987; Zbl 0627.12015)] and F. Pop [Ann. Math. (2) 144, 1-34 (1996; Zbl 0862.12003)].

The method of proof is quite different from that of Harbater and Pop. The main tool is smoothing a tree of rational curves into a single rational curve. It was developed by J. Kollár, Y. Miyaoka and S. Mori [J. Algebr. Geom. 1, 429-448 (1992; Zbl 0780.14026)] in the case where \(k\) is algebraically closed, and then generalized by J. Kollár [Ann. Math. (2) 150, 357-367 (1999; Zbl 0976.14016)] to the case where \(k\) is large. The author applies this deformation technique to a natural versal model of a \(G\)-torsor.

Here is an interesting corollary: for any finite \(G\) there exists a finite set of number fields \(k_i\) such that the greatest common denominator of the degrees \([k_i:\mathbb Q]\) is equal to one, and such that \(G\) is a regular Galois group over each \(k_i(t)\), hence in particular a Galois group over each \(k_i\) (that is, for any group \(G\) the inverse Galois problem over \(\mathbb Q\) acquires a positive answer when passing from rational points to zero-cycles of degree one).

Note that Theorem 1 was generalized to the case where \(G\) is an arbitrary linear algebraic group by J. Kollár [Mich. Math. J. 48, Spec. Vol., 359-368 (2000; Zbl 1077.14520)].

Theorem 1: Let \(G\) be a finite group. Let \(k\) be a large field of characteristic zero. Let \(E=\text{Spec}(K)\) be a \(G\)-torsor over \(\text{Spec}(k)\). Then there exist an open set \(U\) of the affine line \({\mathbb A}_k^1\) containing a \(k\)-point \(O\) and a \(G\)-torsor \(V\to U\) such that the following two properties hold:

(i) The fibre of \(V\to U\) over \(O\) is isomorphic to \(E\) \((\)as a \(G\)-torsor over \(\text{Spec}(k))\);

(ii) The smooth \(k\)-curve \(V\) is geometrically connected.

When the \(G\)-torsor \(E/\text{Spec}(k)\) is trivial, this is a theorem of D. Harbater [Lect. Notes Math. 1240, 165-195 (1987; Zbl 0627.12015)] and F. Pop [Ann. Math. (2) 144, 1-34 (1996; Zbl 0862.12003)].

The method of proof is quite different from that of Harbater and Pop. The main tool is smoothing a tree of rational curves into a single rational curve. It was developed by J. Kollár, Y. Miyaoka and S. Mori [J. Algebr. Geom. 1, 429-448 (1992; Zbl 0780.14026)] in the case where \(k\) is algebraically closed, and then generalized by J. Kollár [Ann. Math. (2) 150, 357-367 (1999; Zbl 0976.14016)] to the case where \(k\) is large. The author applies this deformation technique to a natural versal model of a \(G\)-torsor.

Here is an interesting corollary: for any finite \(G\) there exists a finite set of number fields \(k_i\) such that the greatest common denominator of the degrees \([k_i:\mathbb Q]\) is equal to one, and such that \(G\) is a regular Galois group over each \(k_i(t)\), hence in particular a Galois group over each \(k_i\) (that is, for any group \(G\) the inverse Galois problem over \(\mathbb Q\) acquires a positive answer when passing from rational points to zero-cycles of degree one).

Note that Theorem 1 was generalized to the case where \(G\) is an arbitrary linear algebraic group by J. Kollár [Mich. Math. J. 48, Spec. Vol., 359-368 (2000; Zbl 1077.14520)].

Reviewer: B.Kunyavskii (Ramat Gan)