Geometry of Universal Torsors

Ulrich Derenthal Geometry of universal torsors Dissertation Geometry of universal torsors Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakultäten der Georg-August-Universität zu Göttingen vorgelegt von Ulrich Derenthal aus Brakel Göttingen, 2006 D7 Referent: Prof. Dr. Yuri Tschinkel Korreferent: Prof. Dr. Victor Batyrev Tag der mundlichen¨ Prufung:¨ 13. Oktober 2006 Preface Many thanks are due to my advisor, Prof. Dr. Yuri Tschinkel, for introducing me to the problems considered here and for his guidance. I have benefitted from many discussions, especially with V. Batyrev, H.-C. Graf v. Bothmer, R. de la Bretèche, T. D. Browning, B. Hassett, J. Hausen, D. R. Heath-Brown, J. Heinloth, M. Joyce, E. Peyre, and P. Salberger. I am grateful for their suggestions and advice. Parts of this work were done while visiting the University of Oxford (March 2004), Rice University (February 2005), the CRM at Universitéde Montréal(July 2005) during the special year on Analysis in Number Theory, the Courant Institute of Mathematical Sciences at New York University (December 2005), and the MSRI in Berkeley (March–May 2006) during the program Rational and Integral Points on Higher-Dimensional Varieties.I thank these institutions for their support and ideal working conditions. Calculations leading to some of my results were carried out on computers of the Gauss-Labor (Universität Göttingen). Financial support was provided by Studienstiftung des deutschen Volkes and DFG (Graduiertenkolleg Gruppen und Geometrie). For proofreading, thanks go to Kristin Stroth and Robert Strich. I am grateful to my parents Anne-Marie and Josef Derenthal and my sisters Birgitta and Kirstin for their support and encouragement. Finally, I thank Inga Kenter for the wonderful time during the last years. Ulrich Derenthal v Contents Preface v Introduction 1 Part 1. Universal torsors of Del Pezzo surfaces 5 Chapter 1. Del Pezzo surfaces 7 1.1. Introduction 7 1.2. Blow-ups of smooth surfaces 8 1.3. Smooth Del Pezzo surfaces 11 1.4. Weyl groups and root systems 12 1.5. Singular Del Pezzo surfaces 15 1.6. Classification of singular Del Pezzo surfaces 17 1.7. Contracting (−1)-curves 20 1.8. Toric Del Pezzo surfaces 22 Chapter 2. Universal torsors and Cox rings 25 2.1. Introduction 25 2.2. Universal torsors 25 2.3. Cox rings 27 2.4. Generators and relations 27 Chapter 3. Cox rings of smooth Del Pezzo surfaces 31 3.1. Introduction 31 3.2. Relations in the Cox ring 31 3.3. Degree 3 34 3.4. Degree 2 38 3.5. Degree 1 40 Chapter 4. Universal torsors and homogeneous spaces 45 4.1. Introduction 45 4.2. Homogeneous spaces 47 4.3. Rescalings 50 4.4. Degree 3 53 4.5. Degree 2 55 Chapter 5. Universal torsors which are hypersurfaces 57 5.1. Introduction 57 5.2. Strategy of the proofs 58 5.3. Degree ≥ 6 61 5.4. Degree 5 63 5.5. Degree 4 66 vii viii CONTENTS 5.6. Degree 3 73 Chapter 6. Cox rings of generalized Del Pezzo surfaces 81 6.1. Introduction 81 6.2. Generators 82 6.3. Relations 86 6.4. Degree 4 89 6.5. Degree 3 93 6.6. Families of Del Pezzo surfaces 96 Part 2. Rational points on Del Pezzo surfaces 99 Chapter 7. Manin’s conjecture 101 7.1. Introduction 101 7.2. Peyre’s constant 105 7.3. Height zeta functions 106 Chapter 8. On a constant arising in Manin’s conjecture 109 8.1. Introduction 109 8.2. Smooth Del Pezzo surfaces 111 8.3. Singular Del Pezzo surfaces 113 Chapter 9. Manin’s conjecture for a singular cubic surface 117 9.1. Introduction 117 9.2. Manin’s conjecture 118 9.3. The universal torsor 119 9.4. Congruences 123 9.5. Summations 124 9.6. Completion of the proof 129 Chapter 10. Manin’s conjecture for a singular quartic surface 133 10.1. Introduction 133 10.2. Geometric background 134 10.3. Manin’s conjecture 136 10.4. The universal torsor 136 10.5. Summations 140 10.6. Completion of the proof 144 Bibliography 145 Index 149 Lebenslauf 151 Introduction The topic of this thesis is the geometry and arithmetic of Del Pezzo surfaces. Prime examples are cubic surfaces, which were already studied by Cayley, Schläfli, Steiner, Clebsch, and Cremona in the 19th century. Many books and articles followed, for example by Segre [Seg42] and Manin [Man86]. On the geometric side, our main goal is to understand the geometry of universal torsors over Del Pezzo surfaces. On the arithmetic side, we apply universal torsors to questions about rational points on Del Pezzo surfaces over number fields. We are concerned with the number of rational points of bounded height in the context of Manin’s conjecture [FMT89]. More precisely, a smooth cubic surface S in three-dimensional projective 3 space P over the field Q of rational numbers is defined by the vanishing of a non-singular cubic form f ∈ Z[x0, x1, x2, x3]. Its rational points are 3 S(Q) := {x = (x0 : x1 : x2 : x3) ∈ P (Q) | f(x) = 0}. If S(Q) 6= ∅, then S(Q) is dense with respect to the Zariski topology, i.e., there is no finite set of curves on S containing all rational points. One natural approach to understand S(Q) is to ask how many rational points of bounded height there are on S. Here, the height of a point x ∈ S(Q), represented by coprime integral coordinates x0, x1, x2, x3, is H(x) := max{|x0|, |x1|, |x2|, |x3|}. The number of rational points on S whose height is bounded by a positive number B is NS,H (B) := #{x ∈ S(Q) | H(x) ≤ B}. As it is difficult to determine the exact number NS,H (B), our question is: How does NS,H (B) behave asymptotically as B tends to infinity? It is classically known that S contains 27 lines defined over the algebraic closure of Q. If a line ` ⊂ S is rational, i.e., defined over Q, the number of rational points on ` bounded by B behaves asymptotically as a non-zero 2 constant multiple of B . The behavior of NS,H (B) is dominated by the rational points lying on rational lines, so we modify our question as follows: Question. Let U be the complement of the lines on a smooth cubic surface S. How does NU,H (B) := #{x ∈ U(Q) | H(x) ≤ B} behave asymptotically, as B → ∞? 1 2 INTRODUCTION The answer to this question depends on the geometry of S. Suppose that S is split, i.e., all 27 lines on S are defined over Q. Then Manin’s conjectural answer is: Conjecture. There is a positive constant c such that 6 NU,H (B) ∼ c · B · (log B) , as B → ∞. This conjecture is open for smooth cubic surfaces. Analogous state- ments have been proved for some smooth Del Pezzo surfaces of degrees ≥ 5 ([BT98], [Bre02]). An approach which is expected to lead to a proof of Manin’s conjecture for Del Pezzo surfaces is the use of universal torsors. For the projective plane, this method is used as follows: In order to estimate the number of 2 rational points x ∈ P (Q) whose height H(x) is bounded by B, we observe that these points are in bijection to the integral points 3 y = (y0, y1, y2) ∈ A (Z) \{(0, 0, 0)} which satisfy the coprimality condition gcd(y0, y1, y2) = 1 and the height condition max{|y0|, |y1|, |y2|} ≤ B, up to identification of y and −y. The number of these points y can be estimated using standard methods of analytic number theory. There is a similar bijection between rational points of bounded height on a cubic surface S and integral points on a certain affine variety TS, which is called the universal torsor (see Chapter 2 for the definition), subject to certain coprimality and height conditions. We can establish such a bijection in two ways. On the one hand, this can be done by elementary transformations of the form f defining S such as introducing new variables which are common divisors of previous coordinates (see Chapter 9 for an example). This way of passing to the universal torsor has been used in the proof of Manin’s conjecture for some Del Pezzo surfaces of other degrees, e.g., [BB04]. We will see that these transformations are motivated by the geometric structure of S. On the other hand, we can compute universal torsors via Cox rings. The Picard group Pic(S) of isomorphy classes of line bundles on S is an abelian group which is free of rank 7 if S is a split cubic surface. Effective line bundles have global sections, and the global sections of all isomorphy classes of line bundles on S can be given the structure of a ring, resulting in the Cox ring Cox(S) (see Chapter 2). Its generators and relations correspond to the coordinates and equations defining an affine variety A(S) which contains the universal torsor TS as an open subset. Batyrev and Popov [BP04] have determined the Cox ring of smooth Del Pezzo surfaces of degree ≥ 3. In the case of smooth cubic surfaces, this realizes the universal torsor as an open subset of a 9-dimensional variety, defined by 81 equations in 27-dimensional affine space (see Section 3.3). INTRODUCTION 3 However, estimating the number of points on the universal torsor seems to be very hard in this case. Schläfli [Sch63] and Cayley [Cay69] classified singular cubic surfaces.

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