On Manin’s conjecture for singular del Pezzo surfaces of degree 4. I. (English) Zbl 1132.14019

This paper concerns Manin’s conjecture for a certain singular del Pezzo surface \(X\) of degree 4, given by the equations \[ x_0x_1- x^2_2= x_0x_4- x_1x_2+ x^2_3= 0. \] This has just one singular point, which is of type \(D_5\). Let \(N(B)\) be the standard counting function for rational points of height at most \(B\), for the open subset of \(X\) for which one excludes points on the line \(x_0= x_2= x_3= 0\).
It is shown that Manin’s conjecture holds for this surface, in the strong form \[ N(B)= BP(\log B)+ O(B^\theta), \] for any \(\theta> 11/12\). Here \(P\) is a certain polynomial of degree 5, whose leading term is explicitly calculated, and shown to agree with the prediction by Peyre. Moreover, it is shown that the height zeta-function has a meromorphic continuation to \(\sigma> 5/6\), and is holomorphic for \(\sigma> 9/10\) apart from a sixth-order pole at \(s= 1\).
For the proof, the authors establish a bijection between the points under consideration and the integral points on the universal torsor associated to \(X\). These latter points are then handled by analytic techniques, including an estimate for the average of the fractional part of \((a- bx^2)/q\), as the integer \(x\) varies.


14G05 Rational points
14J26 Rational and ruled surfaces
11D45 Counting solutions of Diophantine equations
Full Text: DOI arXiv Euclid


[1] V. Batyrev and Y. Tschinkel, Manin’s conjecture for toric varieties, J. Algebraic Geom. 7 (1998), 15–53. · Zbl 0946.14009
[2] ——, Tamagawa numbers of polarized algebraic varieties, Astérisque 251 (1998), 299–340. · Zbl 0926.11045
[3] R. de la Bretèche and T. D. Browning, On Manin’s conjecture for singular del Pezzo surfaces of degree four, II, Math. Proc. Cambridge Philos. Soc., · Zbl 1132.14020
[4] R. de la Bretèche, T. D. Browning, and U. Derenthal, On Manin’s conjecture for a certain singular cubic surface, Ann. Sci. École Norm. Sup. (5), · Zbl 1125.14008
[5] 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. · Zbl 1067.11036
[6] J. Franke, Y. I. Manin, and Y. Tschinkel, Rational points of bounded height on Fano varieties, Invent. Math. 95 (1989), 421–435. · Zbl 0674.14012
[7] D. R. Heath-Brown, Mean values of the zeta-function and divisor problems, Recent progress in analytic number theory, vol. I (Durham, 1979), pp. 115–119, Academic Press, New York, 1981. · Zbl 0457.10019
[8] W. V. D. Hodge and D. Pedoe, Methods of algebraic geometry, vol. 2, Cambridge Univ. Press, 1952. · Zbl 0048.14502
[9] E. Peyre, Hauteurs et measures de Tamagawa sur les variétés de Fano, Duke Math. J. 79 (1995), 101–218. · Zbl 0901.14025
[10] E. Peyre and Y. Tschinkel, Tamagawa numbers of diagonal cubic surfaces of higher rank, Rational points on algebraic varieties, Progr. Math., 199, pp. 275–305, Birkhäuser, Basel, 2001. · Zbl 1079.11034
[11] P. Salberger, Tamagawa measures on universal torsors and points of bounded height on Fano varieties, Astérisque 251 (1998), 91–258. · Zbl 0959.14007
[12] P. Swinnerton-Dyer, Counting points on cubic surfaces, II, Geometric methods in algebra and number theory, Progr. Math., 235, pp. 303–310, Birkhäuser, Basel, 2005. · Zbl 1127.11043
[13] E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed. (D. R. Heath-Brown, ed.), Oxford Univ. Press, 1986. · Zbl 0601.10026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.