quantitative arithmetic of projective varieties - t. browning (birkhauser, 2009) ww

[...]... interesting objects Thus, in the setting of Diophantine geometry, analytic number theory is especially suited to questions concerning the “distribution” of integral and rational points on algebraic varieties Determining the arithmetic of affine varieties, both qualitatively and quantitatively, is much more complicated than for projective varieties Given the breadth of the domain and the inherent difficulties... with the arithmetic of curves being central in our understanding of Diophantine equations The terrain for equations in 4 or more variables remains relatively obscure, however, with only a scattering of results and conjectures available The focus of this book will be on quantitative aspects of the arithmetic of higher-dimensional projective varieties Thus our interest lies with the second and third questions... Skinner & Wooley [113] Browning [21] Browning & Heath-Brown [31] Salberger [106] Table 1.1: Estimates for νd (x) 1.3.2 Waring’s problem It is a famous result of Lagrange from 1770 that every positive integer can be written as the sum of 4 squares of non-negative integers This is the least number of squares with which one can hope to achieve such a representation, since we know from work of Legendre that... which is a del Pezzo surface of degree 8 in P8 (isomorphic to P1 × P1 ), or as the blow-up of P2 along 9 − d points in general 2.2 The conjectures 23 position, in which case the degree of S satisfies 3 d 9 Here 9 − d coplanar points are said to be in general position if no 3 of them are collinear and no 6 of them lie on a conic One of the remarkable features of del Pezzo surfaces of small degree is that... earlier draft of this book Over the years, the author has greatly benefited from discussing mathematics with Professors de la Bret`che, Colliot-Th´l`ne, Foue ee vry, Hooley, Salberger, Swinnerton-Dyer and Wooley A sincere debt of thanks is owed to them all Finally, it is essential to single out Professor Heath-Brown for special gratitude, both as a mathematical inspiration and for the generosity of his explanations... the Manin conjectures, a series of predictions concerning the density of rational points on suitable algebraic varieties It turns out that a number of successful attacks upon the conjecture have made essential use of divisor problems for binary forms of one sort or another The underlying problem here is one of pure analytic number theory and rests upon estimating sums of the shape D(X; ϕ, F ) := ϕ(F... primitive cube root of unity In general terms the blow-up of a point on a surface replaces the point in a particular way by an exceptional divisor, which is just a divisor with genus 0 20 Chapter 2 The Manin conjectures and negative self-intersection number (see [66, Section I.4]) We will refer to a (−k)-curve as an exceptional divisor on a surface with self-intersection −k As is well-known a non-singular cubic... surfaces in this analysis Nonetheless, a significant part of this book will be dedicated to cubic surfaces in P3 and intersections of 2 quadrics in P4 It is now time to give a formal definition of a del Pezzo surface Let us begin with a discussion of non-singular del Pezzo surfaces Let d 3 Then a del Pezzo surface of degree d is a non-singular surface S ⊂ Pd of degree d, with very ample anticanonical divisor... satisfactory to work with arbitrary projective algebraic varieties V ⊆ Pn−1 All of the varieties that we will work with are assumed to be cut out by a finite system of homogeneous equations defined over Q Moreover, whenever we speak of a variety as being irreducible we will henceforth take this to mean that the variety is geometrically reduced and irreducible In the case of varieties cut out by a single equation... construct examples of homogeneous polynomials in d2 variables that have no non-zero integer solutions The construction is purely local, relying upon showing that the 2 polynomial fails to have a non-zero solution in Qd It was conjectured by Artin p that Qp is a C2 field, so that f should have a non-trivial p-adic zero as soon as n > d2 The latter property is certainly true of forms of degree at most . with only a scattering of results and conjectures available. The focus of this book will be on quantitative aspects of the arithmetic of higher-dimensional projective varieties. Thus our interest. Z >1 . It is through this rˆole as a characteristic function that it figures in the quantitative study of Diophantine equations. We illustrate the procedure by showing how it allows us to relate. chlorine-free pulp. TCF ∞ Printed in Germany ISBN 97 8-3 -0 34 6-0 12 8-3 e-ISBN 97 8-3 -0 34 6-0 12 9-0 9 8 7 6 5 4 3 2 1 www.birkhauser.ch Author: Timothy D. Browning School of Mathematics University of Bristol Bristol