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Manin’s conjecture for a singular sextic del Pezzo surface. (English. French summary) Zbl 1258.14029

The main object of the paper under review is a particular singular del Pezzo surface \(S\) of degree 6. It has one singular point of type \(A_ 2\), contains two rational lines, and can be obtained as an equivariant compactification of \(\mathbb G_ a^ 2\). The author’s goal is to check for \(S\) (in the case where the ground field is \(\mathbb Q\)) a conjecture of Manin concerning the asymptotic behaviour of the number of rational points of \(S\) (lying outside the rational lines) of bounded height. The main results of the paper confirm and sharpen this conjecture. In particular, the E. Peyre constant (see [Duke Math. J. 79, No. 1, 101–218 (1995; Zbl 0901.14025)]) is computed explicitly. Besides that, the author gives an explicit expression and meromorphic continuation for the associated height zeta function.

MSC:

14G05 Rational points
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11D45 Counting solutions of Diophantine equations
11G35 Varieties over global fields
11G50 Heights
14J26 Rational and ruled surfaces

Citations:

Zbl 0901.14025
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References:

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