Manin’s conjecture for a quartic del Pezzo surface with A 4 singularity
[La conjecture de Manin pour une surface quartique de del Pezzo avec singularité A 4 ]
Annales de l'Institut Fourier, Tome 59 (2009) no. 3, pp. 1231-1265.

Ce papier contient une preuve de la conjecture de Manin pour une surface quartique de del Pezzo, avec singularité A 4 .

The Manin conjecture is established for a split singular del Pezzo surface of degree four, with singularity type A 4 .

DOI : 10.5802/aif.2462
Classification : 11G35, 14G05, 14G10
Keywords: Del Pezzo surfaces, rational points, Manin’s conjecture
Mot clés : surfaces de del Pezzo, points rationnels, conjecture de Manin
Browning, Tim D. 1 ; Derenthal, Ulrich 2

1 University of Bristol School of Mathematics Bristol BS8 1TW (United Kingdom)
2 Universität Zürich Institut für Mathematik Winterthurerstrasse 190 8057 Zürich (Switzerland)
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Browning, Tim D.; Derenthal, Ulrich. Manin’s conjecture for a quartic  del Pezzo surface with $\mathbf{A}_4$ singularity. Annales de l'Institut Fourier, Tome 59 (2009) no. 3, pp. 1231-1265. doi : 10.5802/aif.2462. http://archive.numdam.org/articles/10.5802/aif.2462/

[1] Batyrev, V. V.; Tschinkel, Yu. Manin’s conjecture for toric varieties, J. Alg. Geom., Volume 7 (1998), pp. 15-53 | MR | Zbl

[2] de la Bretèche, R.; Browning, T. D. On Manin’s conjecture for singular del Pezzo surfaces of degree four, I, Michigan Math. J., Volume 55 (2007), pp. 51-80 | DOI | MR | Zbl

[3] Browning, T. D. An overview of Manin’s conjecture for del Pezzo surfaces, Analytic number theory — A tribute to Gauss and Dirichlet, Volume 7, Clay Math. Proceedings, 2007, pp. 39-56 | MR | Zbl

[4] Chambert-Loir, A.; Tschinkel, Yu.; Birkhäuser Fonctions zêta des hauteurs des espaces fibrés, Rational points on algebraic varieties, Volume 199, Progress in Math., 2001, pp. 71-115 | MR | Zbl

[5] Chambert-Loir, A.; Tschinkel, Yu. On the distribution of points of bounded height on equivariant compactifications of vector groups, Invent. Math., Volume 148 (2002), pp. 421-452 | DOI | MR | Zbl

[6] Coray, D. F.; Tsfasman, M. A. Arithmetic on singular del Pezzo surfaces, Proc. London Math. Soc., Volume 57 (1988), pp. 25-87 | DOI | MR | Zbl

[7] Derenthal, U. Singular del Pezzo surfaces whose universal torsors are hypersurfaces, 2006 (arXiv:math.AG/0604194)

[8] Derenthal, U. On a constant arising in Manin’s conjecture for del Pezzo surfaces, Math. Res. Letters, Volume 14 (2007), pp. 481-489 | MR | Zbl

[9] Derenthal, U.; Joyce, M.; Teitler, Z. The nef cone volume of generalized del Pezzo surfaces, Algebra & Number Theory, Volume 2 (2008), pp. 157-182 | DOI | MR

[10] Derenthal, U.; Tschinkel, Yu. Universal torsors over del Pezzo surfaces and rational points, Equidistribution in Number Theory, An Introduction (NATO Sci. Ser. II Math. Phys. Chem.), Volume 237, Springer, 2006 | MR | Zbl

[11] Franke, J.; Manin, I.; Tschinkel, Yu. Rational points of bounded height on Fano varieties, Invent. Math., Volume 95 (1989), pp. 421-435 | DOI | MR | Zbl

[12] Hassett, B.; Tschinkel, Yu. Geometry of equivariant compactifications of G a n ., Internat. Math. Res. Notices, Volume 22 (1999), pp. 1211-1230 | DOI | MR | Zbl

[13] Hodge, W. V. D.; Pedoe, D. Methods of algebraic geometry, 2, Cambridge University Press, 1952 | MR

[14] Peyre, E. Hauteurs et nombres de Tamagawa sur les variétés de Fano, Duke Math. J., Volume 79 (1995), pp. 101-218 | DOI | MR | Zbl

[15] Salberger, P. Tamagawa measures on universal torsors and points of bounded height on Fano varieties, Astérisque, Volume 251 (1998), pp. 91-258 | Numdam | MR | Zbl

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