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Mordell Orbifold Conjecture and Special Varieties Benozzo Marta A

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Mordell Orbifold Conjecture and Special Varieties Benozzo Marta A Density of integral points on varieties: Mordell orbifold conjecture and special varieties Benozzo Marta A Thesis In the Department of Mathematics and Statistics Presented in Partial Fulfillment of the Requirements For the Degree of Master of art (Mathematics) at Concordia university Montréal, Québec, Canada July 2020 c Benozzo Marta, 2020 Concordia university School of Graduate Studies This is to certify that the thesis prepared By: Benozzo Marta Entitled: Density of integral points on varieties: Mordell orbifold conjecture and special varieties and submitted in partial fulfillment of the requirements for the degree of Master of Art (Mathematics) complies with the regulations of this University and meets the accepted standards with respect to originality and quality. Signed by the Final Examining Commitee: Approved by Dr. Cody Hyndam, Chair of Department of Mathematics and Statistics 2020 Dr. Pascale Sicotte, Dean of the Faculty of Arts and Science iii ABSTRACT Density of integral points on varieties: Mordell orbifold conjecture and special varieties Marta Benozzo The objective of this thesis is to understand F.Campana’s conjectures about density of integral points on varieties defined over a number field. In the case of curves this is solved by Siegel and Faltings’ theorems: the degree of the canonical divisor of a curve determines whether it has finitely or infinitely many rational points (possibly after a finite extension of the base field). In higher dimension, varieties can be neither potentially dense (the set of rational points becomes dense after at most a finite extension of the base field), nor mordellic (rational points are always finitely many in an open dense subset). The conjectures studied in the thesis address the problems of characterizing potential density and mordellicity of a variety and of constructing a (unique) fibration which splits each variety in its mordellic part, the base of the fibration, and potentially dense part, the fibers. To deal with this problem, F.Campana introduces the notion of orbifold pair: to each variety, one can attach an orbifold divisor, which allows keeping track of multiple fibers of fibrations. Using this tool, he was able to construct a fibration, the core map, which has (orbifold) special general fiber, conjectured to be the potentially dense varieties, and base of (orbifold) general type, conjectured to be the mordellic ones. Contents Introduction 1 Notations 5 1 Orbifold integral points: curves and surfaces 7 1.1 Orbifold integral points . .8 1.2 Orbifold integral points on curves . 12 1.3 Siegel’s theorem . 14 1.4 Classical orbifold Mordell conjecture . 17 1.4.1 Tools . 17 1.4.2 Proof . 24 1.5 Rational and integral points on surfaces . 30 1.5.1 Numerical properties . 31 1.5.2 An analogue of Siegel’s theorem for surfaces . 34 2 Kodaira dimension 40 2.1 Preliminaries . 40 2.1.1 Divisors and embeddings . 40 2.1.2 Canonical sheaf . 45 2.2 Iitaka and Kodaira dimension . 47 2.3 Easy additivity . 53 2.4 Iitaka–Moishezon fibration . 55 2.5 The maximal rationally connected quotient . 61 3 The core map 69 3.1 Decompositions . 69 3.1.1 Some Chow space theory . 69 3.1.2 The C-quotient . 72 3.2 The weak core map . 75 3.3 Orbifold case . 79 3.4 The core map . 84 iv CONTENTS v I wandered lonely as a cloud I wandered lonely as a cloud That floats on high o’er vales and hills, When all at once I saw a crowd, A host, of golden daffodils; Beside the lake, beneath the trees, Fluttering and dancing in the breeze. Continuous as the stars that shine And twinkle on the milky way, They stretched in never-ending line Along the margin of a bay: Ten thousand saw I at a glance, Tossing their heads in sprightly dance. The waves beside them danced; but they Out-did the sparkling waves in glee: A poet could not but be gay, In such a jocund company: I gazed—and gazed—but little thought What wealth the show to me had brought: For oft, when on my couch I lie In vacant or in pensive mood, They flash upon that inward eye Which is the bliss of solitude; And then my heart with pleasure fills, And dances with the daffodils. ∼ William Wordsworth Introduction A problem that arises in Diophantine geometry is whether rational and integral points on a variety defined over a number field are dense or not. In particular, the goal would be to find geometric conditions that can characterize potential density (i.e. the closed rational points become Zariski dense after at most a finite extension of the base field) and mordellicity (i.e. there exists an open subset of the variety that contains only finitely many rational points, even after finite extensions of the base field). Integral points can be interpreted as points that do not become points at infinity of the variety reduced modulo every prime of the ring of integers. This concept can be generalized introducing the notion of an orbifold pair, a variety together with a simple Ps normal crossing divisor in the form ∆ = j=1(1 − 1/mj)Dj, where Dj are prime (Weil) divisors. This is not completely new, it generalizes log-pairs that arise naturally in arith- metic geometry and play an important role in the minimal model program and in the study of moduli spaces of higher dimensional varieties. The integral points of the orbifold pair are then the ones that, modulo all but finitely many primes, if they belong to any of the divisors Dj’s, their intersection number with them is "higher than mj", in a sense that is made precise in the first chapter. In partic- ular, if all multiplicities are infinite, these are the integral points on the quasi-projective variety complement of the support of the divisor ∆; on the other hand, if ∆ is the zero divisor, we are considering exactly the rational points. According to Campana, the prob- lem of potential density of rational and integral points cannot be solved without taking into account also the "intermediate" cases. The density problem was first formulated by Mordell in the case of projective curves (1922) and solved by Siegel (1929) in the affine case. Later, Faltings proved the projective case (1983). They understood that the geometrical invariant that characterizes density is the genus of a curve, or, equivalently, the degree of its canonical divisor: curves are potentially dense if and only if the degree of the canonical divisor is non-positive and they are mordellic elsewhere (see theorems 1.2.1 and 1.2.2). In the orbifold case, the problem is still open, but there are many results towards the proof. Also in this case, it is conjectured that positivity of the degree of the canonical divisor of the orbifold pair (defined as the canonical divisor of the curve plus the divisor ∆) characterizes potential density as above. In particular, this is proven in the "classical version" 1.4 by reducing to Faltings theorem by means of suitable ramified covers. To finish the proof of the "general version" it is possible, using the previous results, to reduce 1 2 INTRODUCTION the situation to the study of just a few orbifold pairs. In this cases it is even possible to show that abc conjecture implies the result [4, section 3.6]. Thanks to the birational classification of surfaces made by the Italian school in the 19th century, also in the case of surfaces a lot of progress has been made. The situation about rational points is almost understood and, as for integral and orbifold integral points, there are many results and examples that agree with the conjectures made for higher dimensional varieties. Anyway, the work is not completed yet, in many cases it is still open how to solve the conjectures. In higher dimension the problem is pretty much open. The guiding ideas to approach it come from the fact that we expect that density properties are preserved by birational morphisms, Chevalley–Weil theorem 1.3.2 that says that they are preserved also by étale covers and Bombieri, Lang and Vojta’s conjectures that say that varieties of general type are mordellic. Also in the higher dimensional case, the canonical divisor plays a central role. Indeed, it is conjectured that the Kodaira dimension of a variety can characterize density prop- erties. Thus, tools from birational geometry come into play. Harris and Tschinkel recently introduced the notion of weakly special varieties, varieties X such that, if X0 → X is an étale cover, then X0 does not admit any dominant map towards a variety of general type. They conjecture that these are exactly the potentially dense varieties. Campana was able to construct a map that divides the weakly special part of a variety and the part of general type: the weak core map. It is a fibration from the variety X to a variety of general type, the weak core, with the property that the fibers are weakly special. The main flaw of this approach is that the weak core is not preserved under étale covers. However, this problem can be solved by taking into account possible multiple fibers of fibrations and this is possible by considering orbifold pairs and "orbifold morphisms". This is also the main reason why we need to introduce orbifolds in the discussion. The subsequent modification of the notion of weakly special varieties is the notion of special varieties (and special orbifold pairs), conjectured by Campana to be the poten- tially dense ones. The weak core map is modified into the core map, a fibration with special fibers and base of (orbifold) general type.
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