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Arrangements of Curves and Algebraic Surfaces Arrangements of Curves and Algebraic Surfaces by Giancarlo A. Urz¶ua A dissertation submitted in partial ful¯llment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2008 Doctoral Committee: Professor Igor Dolgachev, Chair Professor Robert K. Lazarsfeld Associate Professor Mircea I. Mustat»¸a Assistant Professor Radu M. Laza Assistant Professor Vilma M. Mesa °c Giancarlo A. Urz¶ua 2008 All Rights Reserved To Coni and Max ii ACKNOWLEDGEMENTS Firstly, I would like to thank my adviser for motivation, inspiration, and great support. I am indebted to him for sharing with me his knowledge and insight. I would like to thank the Fulbright-CONICYT 1 Fellowship, the Rackham Grad- uate School, and the Department of Mathematics of the University of Michigan for supporting me during my Ph.D. studies. Finally, I am grateful to the Dissertation Committee for their helpful comments. 1CONICYT stands for Comisi¶onNacional de Investigaci¶onen Ciencia y Tecnolog¶³a. This is the government institution in charge of sciences in my country Chile. iii TABLE OF CONTENTS DEDICATION :::::::::::::::::::::::::::::::::::::::::: ii ACKNOWLEDGEMENTS :::::::::::::::::::::::::::::::::: iii LIST OF FIGURES :::::::::::::::::::::::::::::::::::::: vi CHAPTER I. Introduction ....................................... 1 1.1 Complex projective surfaces. 1 1.2 Moduli and geography. 4 1.3 Fibrations. 9 1.4 Logarithmic surfaces. 14 1.5 Main results. 18 II. Arrangements of curves ................................ 25 2.1 De¯nitions. 25 2.2 Line arrangements in P2.............................. 28 2.3 (p; q)-nets. 32 2.4 Arrangements of sections. 36 2.5 More examples of arrangements of curves. 40 2.5.1 Plane curves. 40 2.5.2 Lines on hypersurfaces. 41 2.5.3 Platonic arrangements. 42 2.5.4 Modular arrangements. 43 2.5.5 Hirzebruch elliptic arrangements. 45 III. Arrangements as single curves and applications . 46 3.1 Moduli space of marked rational curves. 46 3.2 Two proofs of the one-to-one correspondence for line arrangements. 53 3.3 General one-to-one correspondence. 60 3.4 Examples applying the one-to-one correspondence. 62 3.5 Applications to (p; q)-nets. 64 3.5.1 (4; q)-nets. 64 3.5.2 (3; q)-nets for 2 · q · 6. 67 3.5.3 The Quaternion nets. 77 3.5.4 Realizable Latin squares. 79 IV. n-th root covers ..................................... 81 4.1 General n-th root covers. 81 iv 4.2 n-th root covers for curves. 85 4.3 n-th root covers for surfaces. 86 4.3.1 A formula for Dedekind sums. 92 4.3.2 Pull-back of branch divisors. 93 4.4 (¡1)- and (¡2)-curves on X............................ 95 4.4.1 Along line arrangements. 96 V. Projective surfaces vs. logarithmic surfaces ................... 99 5.1 Divisible arrangements. 101 5.1.1 Example: Rational surfaces from p-th root covers. 103 5.2 The theorem relating Chern and log Chern invariants. 103 5.3 Simply connected surfaces with large Chern numbers ratio. 110 5.4 More examples. 115 VI. Deforming p-th root covers ..............................122 6.1 Deformations of surfaces of general type. 122 6.2 Some general formulas for n-th root covers. 126 6.3 The case of surfaces. 127 VII. Further directions ....................................135 7.1 Minimality and rigidity. 135 7.2 3-nets and characteristic varieties. 137 7.3 p-th root covers over algebraically closed ¯elds. 140 7.4 Coverings and geometric normalizations. 145 7.5 Upper bounds for log Chern ratios of divisible arrangements. 148 APPENDICES :::::::::::::::::::::::::::::::::::::::::: 150 BIBLIOGRAPHY :::::::::::::::::::::::::::::::::::::::: 161 v LIST OF FIGURES Figure 2.1 Con¯guration in example II.15. 39 3.1 Singular ¯ber type. 50 3.2 Some K(¸) sets for (complete quadrilateral;P ). 57 3.3 The ¯ve types of m-Veronese (conics) for d = 4. 63 4.1 Resolution over a singular point of Di \ Dj....................... 89 5.1 Fibration with sections and simply connected ¯ber. 111 5.2 Some possible singularities for general arrangements. 120 vi CHAPTER I Introduction 1.1 Complex projective surfaces. De¯nition I.1. A variety is an integral separated scheme of ¯nite type over C. We call it a curve (a surface) if its dimension is one (two). A projective variety is a variety which has a closed embedding into Pn for some positive integer n. The main objects of this work are complex smooth projective surfaces. These Pr objects are studied by means of divisors lying on them. We write D = i=1 ºiDi for a divisor D on a smooth projective surface X, where the Di are projective curves on X and ºi 2 Z. An e®ective divisor has ºi > 0, and it describes a one dimensional closed subscheme in X. The line bundle de¯ned by a divisor D is denoted by OX (D), and the corresponding Picard group by Pic(X). If D and D0 are linearly equivalent, we write D » D0. A divisor D is called nef if for every curve ¡, we have ¡:D ¸ 0. A distinguished divisor class in a smooth projective variety X is the canonical dim(X) class, which is de¯ned by any divisor KX satisfying OX (KX ) ' ­X . The sheaf i of di®erential i-forms on a smooth variety X is denoted by ­X . We often use the dim(X) notation !X := ­X . For any two divisors D; D0 in a smooth projective surface X, we write D ´ D0 if they are numerically equivalent. The Neron-Severi group of X is denoted by 1 2 NS(X) = Pic(X)= ´. Some numerical invariants of X are: ½(X) = rank(NS(X)) 2 0 1 (Picard number), pg(X) = h (X; OX ) = h (X; KX ) (genus), q(X) = h (X; OX ) 0 (irregularity), Pm(X) = h (X; mKX ), m > 0 (m-th plurigenus). Any smooth projective surface X falls in one of the following classes. ¡1) Pm(X) = 0 for all m; X is a ruled surface. 0) Pm(X) are either 0 or 1 for all m; X is birational to either a K3 surface, or an Enriques surface, or an elliptic surface, or a bi-elliptic surface. 1) Pm(X) grows linearly in m >> 0; then, X has an elliptic ¯bration. 2) Pm(X) grows quadratically in m >> 0; X is called a surface of general type. This is Enriques' classi¯cation for algebraic surfaces. The Kodaira dimension ·(X) of X is the maximum dimension of the image of jmKX j for m > 0, or ¡1 if jmKX j = ; for all m > 0. This explains the indices of the previous list. We are mainly interested on surfaces X having ·(X) = 2. Important are the Chern classes of X, de¯ned as c1(X) := c1(TX ) and c2(X) := 1 _ c2(TX ), where TX = ­X is the tangent bundle of X. These classes are related via the Noether's formula 2 12Â(X; OX ) = c1(X) + c2(X) which is an instance of the Hirzebruch-Riemann-Roch theorem, so independent of the characteristic of the ground ¯eld [39, p. 433]. The invariant Â(X; OX ) = 1 ¡ q(X) + pg(X) is called the Euler characteristic of X. For any canonical divisor KX , 2 2 we have the equality KX = c1(X):c1(X) = c1(X). We now use the complex topology of X to de¯ne the Betti numbers i bi(X) = rankHi(X; Z) = dimCH (X; C). 3 Because of the Poincar¶eduality, they satisfy bi(X) = b4¡i(X). Hence, the topological Euler characteristic of X can be written as e(X) = 2 ¡ 2b1(X) + b2(X). We also i P p;q p;q have the Hodge decomposition H (X; C) = p+q=i H (X), where H (X) := q p p;q q;p 0 1 H (X; ­X ) and h (X) = h (X). Thus we have q(X) = h (X; ­X ), b1(X) = 1;1 2q(X), and b2(X) = 2pg(X) + h (X). Finally, it is well-known that c2(X) = e(X) (Gauss-Bonnet formula [35, p. 435], or Hirzebruch-Riemann-Roch Theorem and Hodge decomposition). More subtle topological invariants of X are the intersection form on H2(X; Z) and + ¡ the topological fundamental group ¼1(X). Let b and b be the numbers of positive and negative entries in the diagonalized version of the intersection form over R (so + ¡ + ¡ b2(X) = b + b ). Its signature sign(X) = b ¡ b can be expressed as (see [35]) 1¡ ¢ sign(X) = 2 + 2p (X) ¡ h1;1(X) = c2(X) ¡ 2c (X) : g 3 1 2 In particular, the Chern numbers of X are topological invariants. However, contrary to the case of curves, the Chern numbers do not determinate the 4 P4 25 P4 50 topology of X. For example, consider X in P de¯ned by i=0 xi = i=0 xi = 0. By the Lefschetz theorem ¼1(X) = f1g. There are two natural distinct free actions on X given by the groups Z=25Z and Z=5Z©Z=5Z. Therefore, the two corresponding quotients are smooth projective surfaces with the same Chern numbers but di®erent fundamental groups 1. It is known that any simply connected compact Riemann surface is isomorphic to the complex projective line. In the case of smooth projective surfaces, simply connectedness occurs for all Kodaira dimensions. There are several e®ective ways to compute fundamental groups when · < 2, but it becomes more di±cult for surfaces of general type. We are going to describe in Section 1.3 one well-known general tool 1This example due to Tankeev. 4 for ¯brations, which will be used later in this work. We will show how this tool ¯ts very well in p-th root covers to produce, in particular, several simply connected surfaces of general type.
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