Abstract
We give upper bounds for the number of rational points of bounded height on the complement of the lines on projective surfaces.
Similar content being viewed by others
References
Broberg N. (2004). A note on a paper by R. Heath-Brown: the density of rational points on curves and surfaces. J. Reine. Angew. Math. 571: 159–178
Broberg, N., Salberger, P.: Counting rational points on threefolds, in arithmetic of higher-dimensional varieties. In: Poonen, B., Tschinkel Y. (eds.) Progress in Mathematics vol. 206, Birkhäuser, Boston (2003)
Browning T.D. and Heath-Brown R. (2005). Plane curves in boxes and equal sums of two powers. Math. Zeit. 251: 233–247
Browning T.D. and Heath-Brown R. (2005). Counting rational points on hypersurfaces. J. Reine. Angew. Math. 584: 83–116
Browning T.D., Heath-Brown R. and Salberger P. (2006). Counting rational points on algebraic varieties. Duke Math. J. 132: 545–578
Coskun I. (2006). The enumerative geometry of del Pezzo surfaces via degenerations. Am. J. Math. 128: 751–786
Debarre O. (2001). Higher-Dimensional Algebraic Geometry. Springer, New York
Fischer P., Pontkowski J. (2001) Ruled Varieties. An Introduction to Differential Geometry, Advanced Lectures in Mathematics. Friedr.Viehvweg & Sohn, Braunschweig
Fulton W. (1984). Intersection Theory. Springer, Heidelberg
Gruson L., Lazarsfeld R. and Peskine C. (1983). On a theorem of Castelnuovo and the equations defining space curves. Invent. Math. 72: 491–506
Harris J. (1982). Curves in Projective Space. Montreal University Press, Canada
Heath-Brown R. (2002). The density of rational points on curves and surfaces. Ann. Math. 155: 553–595
Heath-Brown, R.: Counting rational points on algebraic varieties. I0n: Analytic Number Theory, Lectures given at the C.I.M.E. Summer School in Cetraro, 2002. Springer Lecture Notes in Mathematics, vol. 1891, pp. 51–96 (2006)
Hooley C. (1995). On another sieve method and the numbes that are a sum of two k-th powers. II.. J. Reine. Angew. Math. 475: 55–75
Iskovskikh V.A. and Prokhorov Yu.G. (1999). Algebraic Geometry V, Encyklopedia of Mathematical Sciences, vol. 47. Springer, Berlin
Kleiman, S.J.: Tangency and duality. In: Proceedings of the 1984 Vancouver Conference in Algebraic Geometry, CMS Conference Proceedings, vol. 6, pp. 163–225, Am. Math. Soc. (1986)
Salberger P. (2005). Counting rational points on hypersurfaces of low dimension. Ann. Sci. Éc. Norm. Sup. 38: 93–115
Salberger P. (2007). On the density of rational and integral points on algebraic varieties. J. Reine. Angew. Math. 606: 123–147
Semple J. and Roth L. (1949). Introduction to Algebraic Geometry. Oxford University Press, New York
Skinner C.M. and Wooley T.D. (1995). Sums of two kth powers. J. Reine. Angew. Math. 462: 57–68
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Salberger, P. Rational points of bounded height on projective surfaces. Math. Z. 258, 805–826 (2008). https://doi.org/10.1007/s00209-007-0199-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-007-0199-z