Arithmetic on singular del Pezzo surfaces.

*(English)*Zbl 0653.14018For the general context of their results, the authors refer the reader to the survey by Yu. I. Manin and M. A. Tsfasman [“Rational varieties: Algebra, geometry, arithmetic”, Russ. Math. Surv. 41, No.2, 51-116 (1986), translation from Usp. mat. Nauk 41, No.2 (248), 43-94 (1986; Zbl 0621.14029)]. In the paper under review, the authors study the arithmetic of certain singular surfaces: cubic surfaces in part I, intersections of quadrics in part II, and, in part III, generalizations of these to higher degrees which they call singular del Pezzo surfaces.

Each part starts with a discussion of possible singularities on the surfaces. For cubic surfaces, this leads to a classification into birational equivalence classes depending on the number of singularities; furthermore, explicit birational maps are given. - In the other two parts the authors prove theorems relating to the k-rationality of the surfaces. It turns out that with the exception of a certain class of surfaces of degree \( 4\), singular del Pezzo surfaces of degree \( 4\leq d\leq 8\) satisfy the Hasse principle and are k-rational whenever they have a k-rational point (they are always k-rational if the degree is 5 or 7).

The exceptional class is the class of Iskovskih surfaces. These are arithmetically very interesting; they fail the Hasse principle and are not necessarily rational when they have a k-rational point, although they are unirational.

Each part of the paper also contains numerous examples and applications. For example, explicit equations are given for five members of a birational equivalence class, any smooth member Z of which satisfies the following striking properties, among others: (i) Z is \({\mathbb{Q}}\)- unirational; (ii) for every completion \({\mathbb{Q}}_ v\) of \({\mathbb{Q}}\), the surface \(Z_ v\) is \({\mathbb{Q}}_ v\)-rational; however, Z is not \({\mathbb{Q}}\)- rational; (iii) the Chow group \(A_ 0(Z)\) does not embed in \(\oplus_{v}A_ 0(Z_ v).\) The Brauer group of Z also has some interesting properties.

Each part starts with a discussion of possible singularities on the surfaces. For cubic surfaces, this leads to a classification into birational equivalence classes depending on the number of singularities; furthermore, explicit birational maps are given. - In the other two parts the authors prove theorems relating to the k-rationality of the surfaces. It turns out that with the exception of a certain class of surfaces of degree \( 4\), singular del Pezzo surfaces of degree \( 4\leq d\leq 8\) satisfy the Hasse principle and are k-rational whenever they have a k-rational point (they are always k-rational if the degree is 5 or 7).

The exceptional class is the class of Iskovskih surfaces. These are arithmetically very interesting; they fail the Hasse principle and are not necessarily rational when they have a k-rational point, although they are unirational.

Each part of the paper also contains numerous examples and applications. For example, explicit equations are given for five members of a birational equivalence class, any smooth member Z of which satisfies the following striking properties, among others: (i) Z is \({\mathbb{Q}}\)- unirational; (ii) for every completion \({\mathbb{Q}}_ v\) of \({\mathbb{Q}}\), the surface \(Z_ v\) is \({\mathbb{Q}}_ v\)-rational; however, Z is not \({\mathbb{Q}}\)- rational; (iii) the Chow group \(A_ 0(Z)\) does not embed in \(\oplus_{v}A_ 0(Z_ v).\) The Brauer group of Z also has some interesting properties.

Reviewer: W.McCallum

##### MSC:

14J25 | Special surfaces |

14M20 | Rational and unirational varieties |

14J17 | Singularities of surfaces or higher-dimensional varieties |

14G25 | Global ground fields in algebraic geometry |

14C05 | Parametrization (Chow and Hilbert schemes) |