Manin’s conjecture for a singular sextic del Pezzo surface
Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 3, pp. 675-701.

We prove Manin’s conjecture for a del Pezzo surface of degree six which has one singularity of type A 2 . Moreover, we achieve a meromorphic continuation and explicit expression of the associated height zeta function.

On démontre la conjecture de Manin pour une surface de del Pezzo de degré six qui a une singularité de type A 2 . De plus, on établit un prolongement méromorphe et une expression explicite de la fonction zêta des hauteurs associées.

DOI: 10.5802/jtnb.739
Classification: 11D45, 14G05, 14G10
Loughran, Daniel 1

1 Department of Mathematics University Walk Bristol UK, BS8 1TW
@article{JTNB_2010__22_3_675_0,
     author = {Loughran, Daniel},
     title = {Manin{\textquoteright}s conjecture for a singular sextic del {Pezzo} surface},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {675--701},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
     number = {3},
     year = {2010},
     doi = {10.5802/jtnb.739},
     zbl = {1258.14029},
     mrnumber = {2769338},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jtnb.739/}
}
TY  - JOUR
AU  - Loughran, Daniel
TI  - Manin’s conjecture for a singular sextic del Pezzo surface
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2010
SP  - 675
EP  - 701
VL  - 22
IS  - 3
PB  - Université Bordeaux 1
UR  - http://archive.numdam.org/articles/10.5802/jtnb.739/
DO  - 10.5802/jtnb.739
LA  - en
ID  - JTNB_2010__22_3_675_0
ER  - 
%0 Journal Article
%A Loughran, Daniel
%T Manin’s conjecture for a singular sextic del Pezzo surface
%J Journal de théorie des nombres de Bordeaux
%D 2010
%P 675-701
%V 22
%N 3
%I Université Bordeaux 1
%U http://archive.numdam.org/articles/10.5802/jtnb.739/
%R 10.5802/jtnb.739
%G en
%F JTNB_2010__22_3_675_0
Loughran, Daniel. Manin’s conjecture for a singular sextic del Pezzo surface. Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 3, pp. 675-701. doi : 10.5802/jtnb.739. http://archive.numdam.org/articles/10.5802/jtnb.739/

[BB07] R. de la Bretèche and T. D. Browning, On Manin’s conjecture for singular del Pezzo surfaces of degree four, I. Michigan Mathematical Journal 55 (2007), 51–80. | MR | Zbl

[Bro07] T. D. Browning, An overview of Manin’s conjecture for del Pezzo surfaces. Analytic number theory - A tribute to Gauss and Dirichlet (Goettingen, 20th June - 24th June, 2005), Clay Mathematics Proceedings 7 (2007), 39–56. | MR | Zbl

[CT88] D. F. Coray and M. A. Tsfasman, Arithmetic on singular Del Pezzo surfces. Proc. London Math. Soc (3) 57(1) (1988), 25–87. | MR | Zbl

[CTS87] J.-L. Colliot-Thélène and J.-J. Sansuc, La descente sur les variétés rationnelles. II. Duke Math. J. 54(2) (1987), 375–492. | MR | Zbl

[CT02] A. Chambert-Loir and Y. Tschinkel, On the Distribution of points of bounded height on equivariant compactifications of vector groups. Invent. Math. 148 (2002), 421–452. | MR | Zbl

[Der06] U. Derenthal, Singular Del Pezzo surfaces whose universal torsors are hypersurfaces. arXiv:math.AG/0604194 (2006).

[Der07] U. Derenthal, On a constant arising in Manin’s Conjecture for Del Pezzo surfaces. Math. Res. Letters 14 (2007), 481–489. | MR | Zbl

[DL10] U. Derenthal and D. Loughran, Singular del Pezzo surfaces that are equivariant compactifications. Proceedings of Hausdorff Trimester on Diophantine equations in: Zapiski Nauchnykh Seminarov (POMI) 377 (2010), 26–43.

[DT07] U. Derenthal and Y. Tschinkel, Universal torsors over Del Pezzo surfaces and rational points. Equidistribution in Number theory, An Introduction, (A. Granville, Z. Rudnick eds.), NATO Science Series II, 237, Springer, (2007), 169–196. | MR | Zbl

[FMT89] J. Franke, Y. I. Manin and Y. Tschinkel, Rational Points of Bounded Height on Fano Varieties. Invent. Math 95 (1989), 421–435. | MR | Zbl

[Har77] R. Hartshorne, Algebraic Geometry. Springer-Verlag, New York, 1977. | MR | Zbl

[HB79] D. R. Heath-Brown, The fourth power moment of the Riemann zeta function. Proc. London Math. Soc. 38 (1979), 385–422. | MR | Zbl

[HK00] Y. Hu and S. Keel, Mori dream spaces and GIT. Michigan Math. J., dedicated to William Fulton on the occasion of his 60th birthday, 48 (2000) 331–348. | MR | Zbl

[Man86] Y. I. Manin, Cubic Forms. North-Holland Mathematical Library 4, North-Holland Publishing Co., 2nd ed. 1986. | MR | Zbl

[Pey95] E. Peyre, Hauteurs et measures de Tamagawa sur les variétiés de Fano. Duke Math. J., 79(1) (1995), 101–218. | MR | Zbl

[Sal98] P. Salberger, Tamagawa measures on universal torsors and points of bounded height on Fano varieties. Astérisque, Nombre et répartition de points de hauteur bornnée (Paris, 1996), 251 (1998), 91–258. | Numdam | MR | Zbl

[Sko01] A. Skorobogatov, Torsors and rational points. Cambridge University press, 2001. | MR | Zbl

[Ten95] G. Tenenbaum, Introduction to analytic and probabilistic number theory. Cambridge University press, 1995. | MR | Zbl

[Tit86] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function. Oxford University press, 2nd ed. edited by D.R.Heath-Brown, 1986. | MR | Zbl

Cited by Sources: