## Rational points on some pencils of conics with 6 singular fibres.(English)Zbl 0976.14014

The author writes: “Let $${\mathcal Y}\to\mathbb{P}^1$$ be a pencil of conics defined over an algebraic number field $$k$$. It is conjectured that the only obstruction to the Hasse principle on $${\mathcal Y}$$, and also to weak approximation, is the Brauer-Manin obstruction; and it was shown by J.-L. Colliot-Thélène and Sir P. Swinnerton-Dyer [J. Reine Angew. Math. 453, 49-112 (1994; Zbl 0805.14010)] that this follows from Schinzel’s hypothesis.$$\dots$$ If one does not assume Schinzel’s hypothesis, little is known. The only promising-looking line of attack is through the geometry of the universal torseurs on $${\mathcal Y}$$; and these are much easier to study when $${\mathcal Y}$$ has the special form $$U^2-cV^2=P(W)$$, where $$c$$ is a non-square in $$k$$ and $$P(W)$$ is a separable polynomial in $$k[W]$$. By writing $$W=X/Y$$ we can take the solubility of this equation into the equivalent (though ungeometric) problem of the solubility of $$U^2-cV^2=f(X,Y)$$ in $$k$$, where $$f$$ is homogeneous of even degree; here $$\deg f$$ is $$1+\deg P$$ or $$\deg P$$. The simplest non-trivial case is that of Châtelet surfaces, when $$P(W)$$ has degree 3 or 4; in this case the conjecture was proved by J.-L. Colliot-Thélène, J.-J. Sansuc, and P. Swinnerton-Dyer [J. Reine Angew. Math. 373, 37-107 (1987; Zbl 0622.14029)] and 374, 71-168 (1987; Zbl 0622.14030)]. The object of this paper is to prove the conjecture when $$\deg f=6$$ and $$f=f_4f_2$$ over $$k$$, where $$\deg f_4=4$$ and $$\deg f_2=2$$.”
The author solves this problem by explicitly constructing a finite set $$S$$ of projective algebraic varieties defined over $$k$$, with the following properties:
(i) if $$V\in S$$, then $$V(k)\neq \emptyset$$ as soon as $$V(k_v)\neq \emptyset$$ for every completion $$k_v$$ of $$k$$;
(ii) if $$V(k)\neq \emptyset$$ for some $$V$$ in $$S$$, then the variety $$W:U^2-cV^2 =f(X,Y)$$ contains a Zariski dense set of points defined over $$k$$;
(iii) if the equation $$U^2-cV^2= f(X,Y)$$ has a non-trivial solution in $$k$$, then $$V(k)\neq\emptyset$$ for some $$V$$ in $$S$$.
Moreover, each of the varieties in $$S$$ can be proved to be isomorphic to a factor of an universal torsor associated to the corresponding pencil of conics $${\mathcal Y}:U^2-cV^2=P(W)$$.
Reviewer: B.Z.Moroz (Bonn)

### MSC:

 14G05 Rational points 14C21 Pencils, nets, webs in algebraic geometry 11D72 Diophantine equations in many variables 11G35 Varieties over global fields

### Citations:

Zbl 0805.14010; Zbl 0622.14029; Zbl 0622.14030
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### References:

 [1] Colliot-Thélène, J.-L.and Sansuc, J.-J., La descente sur les variétés rationnelles II, Duke Math. J.54 (1987), 375-492. · Zbl 0659.14028 [2] Colliot-Thélène, J.-L., Sansuc, J.-J. and Swinnerton-Dyer, Sir Peter, Intersections of two quadrics and Châtelet surfaces, J. reine angew. Math.373 (1987), 37-107 and 374 (1987), 72-168. · Zbl 0622.14030 [3] Colliot-Thélène, J.-L. and Swinnerton-Dyer, Sir Peter, Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties, J. reine angew. Math.453 (1994), 49-112. · Zbl 0805.14010 [4] Manin, Y.I., Cubic Forms, algebra, geometry, arithmetic. (2nd edition, North-Holland, 1986) · Zbl 0582.14010 [5] Swinnerton-Dyer, Sir Peter, The Brauer group of cubic surfaces, Math. Proc. Camb. Phil. Soc.113 (1993), 449-460. · Zbl 0804.14018
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