Intersections of two quadrics and pencils of curves of genus 1.
(Intersections de deux quadriques et pinceaux de courbes de genre 1.)

*(French)*Zbl 1122.14001
Lecture Notes in Mathematics 1901. Berlin: Springer (ISBN 978-3-540-69137-2/pbk). viii, 218 p. (2007).

The monograph under review grew out of the author’s Ph.D thesis. Its main objects, smooth intersections of two quadrics in a projective space defined over a number field (which include an important particular case of del Pezzo surfaces of degree 4), have been extensively studied during the past decades. The focus has been made on the Hasse principle (in particular, on the problem whether the Brauer-Manin obstruction to this principle to hold is the only one). The author proves two important general results. Assuming two standard hypotheses to hold (Schinzel’s hypothesis (H) and finiteness of the Tate–Shafarevich group of the elliptic curves defined over number fields) he proves that the Hasse principle holds

a) for all smooth intersections of two quadrics in \({\mathbb P}^n\), \(n\geq 5\) (Theorem 3.3);

b) for several classes of del Pezzo surfaces \(X\) of degree 4 (Theorem 3.2).

More precisely, if in case b) \(X\) is defined by the system of homogeneous quadratic equations in five variables \(q_1=q_2=0\) over \(k\) and \(G\) stands for the Galois group of the extension \(k'/k\) generated by the roots of the homogeneous polynomial \(f(\lambda ,\mu ):=\det (\lambda q_1+\mu q_2)\), the Hasse principle holds in each of the following cases:

b1) the group \(G\) is 3-transitive (i.e. \(G=S_5\) or \(G=A_5\));

b2) the polynomial \(f\) has exactly two roots in \(k\) and \(\text{{Br}}(X)/\text{{Br}}(k)=0\);

b3) the polynomial \(f\) splits in \(k\) and \(\text{{Br}} (X)/\text{{Br}}(k)=0\).

Note that case b1) shows that the Hasse principle holds for “general” del Pezzo surfaces of degree 4.

These two theorems are proved in the third chapter of the monograph. Theorem 3.3 is deduced from Theorem 3.2 by induction, using the techniques of the fibration method mainly developed by J.-L. Colliot-Thélène, J.-J. Sansuc and P. Swinnerton-Dyer [J. Reine Angew. Math. 373, 37–107 (1987; Zbl 0622.14029)] and a theorem of A. N. Skorobogatov [Prog. Math. 91, 205–219 (1990; Zbl 0748.14002)]. The proof of Theorem 3.3 requires a lot of new ingredients (partly presented in the first two chapters) which are interesting by their own. In particular, Theorem 1.1 proved in the first chapter generalizes the main results of two important earlier papers: J.-L. Colliot-Thélène, A. N. Skorobogatov and P. Swinnerton-Dyer [Invent. Math. 134, No. 3, 579–650 (1998; Zbl 0924.14011)] and P. Swinnerton-Dyer [Proc. Lond. Math. Soc., III. Ser. 80, No. 3, 513–544 (2000); corrigenda ibid. 85, No. 3, 564 (2002; Zbl 1066.11029)]. The second chapter develops the ideas of the papers by A. O. Bender and P. Swinnerton-Dyer [Proc. Lond. Math. Soc., III. Ser. 83, No. 2, 299–329 (2001; Zbl 1018.11031)] and J.-L. Colliot-Thélène [Prog. Math. 199, 117–161 (2001; Zbl 1079.14510)]. One of the critical innovations introduced by the author, consists in systematic use of Néron models of elliptic curves appearing in the fibrations under consideration rather than their Weierstrass equations.

The monograph is written in extremely careful manner. Although the exposition inevitably contains a lot of technical details, the style can be characterized as reader-friendly. There are two introductions: the English one makes emphasis on the general context, whereas the French one contains more technical descriptions and references.

To sum up, the monograph can be recommended to everyone interested in the state of the art of methods and results concerning arithmetic of rational varieties.

a) for all smooth intersections of two quadrics in \({\mathbb P}^n\), \(n\geq 5\) (Theorem 3.3);

b) for several classes of del Pezzo surfaces \(X\) of degree 4 (Theorem 3.2).

More precisely, if in case b) \(X\) is defined by the system of homogeneous quadratic equations in five variables \(q_1=q_2=0\) over \(k\) and \(G\) stands for the Galois group of the extension \(k'/k\) generated by the roots of the homogeneous polynomial \(f(\lambda ,\mu ):=\det (\lambda q_1+\mu q_2)\), the Hasse principle holds in each of the following cases:

b1) the group \(G\) is 3-transitive (i.e. \(G=S_5\) or \(G=A_5\));

b2) the polynomial \(f\) has exactly two roots in \(k\) and \(\text{{Br}}(X)/\text{{Br}}(k)=0\);

b3) the polynomial \(f\) splits in \(k\) and \(\text{{Br}} (X)/\text{{Br}}(k)=0\).

Note that case b1) shows that the Hasse principle holds for “general” del Pezzo surfaces of degree 4.

These two theorems are proved in the third chapter of the monograph. Theorem 3.3 is deduced from Theorem 3.2 by induction, using the techniques of the fibration method mainly developed by J.-L. Colliot-Thélène, J.-J. Sansuc and P. Swinnerton-Dyer [J. Reine Angew. Math. 373, 37–107 (1987; Zbl 0622.14029)] and a theorem of A. N. Skorobogatov [Prog. Math. 91, 205–219 (1990; Zbl 0748.14002)]. The proof of Theorem 3.3 requires a lot of new ingredients (partly presented in the first two chapters) which are interesting by their own. In particular, Theorem 1.1 proved in the first chapter generalizes the main results of two important earlier papers: J.-L. Colliot-Thélène, A. N. Skorobogatov and P. Swinnerton-Dyer [Invent. Math. 134, No. 3, 579–650 (1998; Zbl 0924.14011)] and P. Swinnerton-Dyer [Proc. Lond. Math. Soc., III. Ser. 80, No. 3, 513–544 (2000); corrigenda ibid. 85, No. 3, 564 (2002; Zbl 1066.11029)]. The second chapter develops the ideas of the papers by A. O. Bender and P. Swinnerton-Dyer [Proc. Lond. Math. Soc., III. Ser. 83, No. 2, 299–329 (2001; Zbl 1018.11031)] and J.-L. Colliot-Thélène [Prog. Math. 199, 117–161 (2001; Zbl 1079.14510)]. One of the critical innovations introduced by the author, consists in systematic use of Néron models of elliptic curves appearing in the fibrations under consideration rather than their Weierstrass equations.

The monograph is written in extremely careful manner. Although the exposition inevitably contains a lot of technical details, the style can be characterized as reader-friendly. There are two introductions: the English one makes emphasis on the general context, whereas the French one contains more technical descriptions and references.

To sum up, the monograph can be recommended to everyone interested in the state of the art of methods and results concerning arithmetic of rational varieties.

Reviewer: Boris Kunyavskii (Ramat Gan)

##### MSC:

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

11G35 | Varieties over global fields |

14J20 | Arithmetic ground fields for surfaces or higher-dimensional varieties |

14J27 | Elliptic surfaces, elliptic or Calabi-Yau fibrations |

14J26 | Rational and ruled surfaces |

11D25 | Cubic and quartic Diophantine equations |

14G25 | Global ground fields in algebraic geometry |

14D10 | Arithmetic ground fields (finite, local, global) and families or fibrations |