Thesis (Ph. D.)--University of Washington, 2006. / Vita. Includes bibliographical references (p. 54-56).

In this dissertation we prove a version of the tropical Bezout's theorem which is applicable to all tropical projective plane curves. There is a version of tropical Bezout's theorem presented in other works which applies in special cases, but we provide a proof of the theorem for all tropical projective plane curves. We provide several different definitions of intersection multiplicity and show that they all agree. Finally, we will use a tropical resultant to determine the intersection multiplicity of points of intersection at infinite distance. Using these new definitions of intersection multiplicity we prove the complete tropical Bezout's theorem.

Thesis (Ph. D.)--University of Washington, 1998. / Vita. Includes bibliographical references (leaf [106]).

Thesis (Ph. D.)--University of Chicago, Department of Mathematics, June 2000. / Includes bibliographical references. Also available on the Internet.

Thesis advisor: Dawei Chen / We investigate the count of meromorphic differentials on the Riemann sphere pos-sessing a single zero, multiple poles with prescribed orders, and fixed residues at each pole. Gendron and Tahar previously examined this problem with respect to general residues using flat geometry, while Sugiyama approached it from the perspective of fixed-point multipliers of polynomial maps in the case of simple poles. In our study, we employ intersection theory on compactified moduli spaces of differentials, enabling us to handle arbitrary residues and pole orders, which provides a complete solution to this problem. We also determine interesting combinatorial properties of the solution formula. This thesis is organized as follows: In Chapter 1 we give an introduction to the problem and summarize the main results obtained. In Chapter 2 we review the compactification of moduli spaces of differentials and introduce various divisor classes. In Section 2.3 we explain how to identify the universal line bundle class with the divisor class of the locus of differentials satisfying a general given residue tuple and prove Theorem 1.0.1 (i). In Section 2.4 we impose exactly one independent partial sum vanishing condition to the residues and prove Theorem 1.0.1 (ii). In Section 2.5 we give a polynomial expression in terms of the zero order for the degree of mixed products between powers of the dual tautological class and the psi-class of the zero. Finally in Chapter 3 we prove Theorem 1.0.2 for arbitrary residues and investigate combinatorial properties of the solution formula. We have also verified our formula numerically for a number of cases by using the software package [CMZ2]. / Thesis (PhD) — Boston College, 2024. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

With the publication of the famous Erdős-Ko-Rado Theorem in 1961, intersection problems became a popular area of combinatorics. A family of combinatorial objects is t-intersecting if any two of its elements mutually t-intersect, where the latter concept needs to be specified separately in each instance. This thesis is split into two parts; the first is concerned with intersecting injections while the second investigates intersecting posets. We classify maximum 1-intersecting families of injections from {1, ..., k} to {1, ..., n}, a generalisation of the corresponding result on permutations from the early 2000s. Moreover, we obtain classifications in the general t>1 case for different parameter limits: if n is large in terms of k and t, then the so-called fix-families, consisting of all injections which map some fixed set of t points to the same image points, are the only t-intersecting injection families of maximal size. By way of contrast, fixing the differences k-t and n-k while increasing k leads to optimal families which are equivalent to one of the so-called saturation families, consisting of all injections fixing at least r+t of the first 2r+t points, where r=|_ (k-t)/2 _|. Furthermore we demonstrate that, among injection families with t-intersecting and left-compressed fixed point sets, for some value of r the saturation family has maximal size . The concept that two posets intersect if they share a comparison is new. We begin by classifying maximum intersecting families in several isomorphism classes of posets which are linear, or almost linear. Then we study the union of the almost linear classes, and derive a bound for an intersecting family by adapting Katona's elegant cycle method to posets. The thesis ends with an investigation of the intersection structure of poset classes whose elements are close to the antichain. The overarching theme of this thesis is fixing versus saturation: we compare the sizes and structures of intersecting families obtained from these two distinct principles in the context of various classes of combinatorial objects.

Nous construisons l'espace des différentielles stables : un espace des modules de différentielles méromorphes avec des pôles d'ordres fixés. Cet espace est un cône au dessus de l'espace Mg,n des courbes stables. Si l'ensemble de poles est vide, il s'agit du fibré de Hodge. Nous introduisons l'anneau tautologique du projectivisé de l'espace des différentielles stables par analogie avec Mg,n. L'espace des différentielles stables est stratifié en fonction des ordres des zéros de la différentielle. Nous montrons que la classe de cohomologie Poincaré-duale de chaque strate est tautologique et peut être calculée explicitement, ce qui constitue le résultat principal de la thèse. Nous appliquons ces résultats pour calculer des nombres de Hurwitz et pour prouver plusieurs identités dans le groupe de Picard des strates. Ensuite, nous nous intéressons aux espaces des modules des différentielles d'ordre supérieur. Une courbe munie d'une k-différentielle holomorphe possède un revêtement naturel de groupe de Galois Z/kZ. Le fibré de Hodge sur la courbe revêtante se décompose en une somme directe de sous-fibrés en fonction du car- actère de Z/kZ. Nous calculons la première classe de Chern de chacun de ces sous-fibrés. Un dernier chapitre sera consacré à l'exposé des liens conjecturaux entre les classes des strates de différentielles, les espaces de courbes r-spin et les cycles de double ramification. / We construct the space of stable differentials: a moduli space of meromorphic differentials with poles of fixed order. This space is a cone over the moduli space Mg,n of stable curves. If the set of poles is empty, then this cone is the Hodge bundle. We introduce the tautological ring of the projectivized space of stable differentials by analogy with Mg,n. The space of stable differentials is stratified according to the orders of zeros of the differential. We show that the Poincaré-dual cohomology classes of these strata are tautological and can be explicitly computed, this constitutes the main result of this thesis. We apply this result to compute Hurwitz numbers and to show several identities in the Picard group of the strata. Then, we interest ourselves to moduli spaces of differentials of superior order. A curve endowed with a k-differential carry a natural ramified covering of Galois group Z/kZ. The Hodge bundle over the covering curve is decomposed into a direct sum of sub-vector bundles according to the character of Z/kZ. We compute the first Chern class of each of these sub-bundles. A last chapter will be dedicated to the presentation of conjectural relations between classes of strata of differentials, moduli of r-spin structures and double ramification cycles.

Es sei A_2 der toroidal kompaktifizierte Modulraum prinzipal polarisierter komplexer abelscher Flächen, und M_k(Sp_4(Z)) das Geradenbündel Siegel'scher Modulformen von Gewicht k auf A_2, versehen mit der Petersson-Metrik. Betrachtet man A_2 als komplexe Faser einer arithmetischen Varietät über Spec(Z), und M_k(Sp_4(Z)) als das von einem Geradenbündel auf dieser arithmetischen Varietät induzierte Geradenbündel, so kann man die Frage nach dem arithmetischen Grad dieses Geradenbündels stellen. Wir stellen nachfolgend den Grad als Ausdruck in speziellen Werten der logarithmischen Ableitung der Riemann'schen Zeta-Funktion dar. Der arithmetische Grad setzt sich aus einem Beitrag vom Schnitt über den endlichen Fasern und einem Integral von Green'schen Formen über die komplexe Faser zusammen. Die Berechnung des von der komplexen Faser A_2 induzierten Anteils am arithmetischen Grad erfolgt durch eine spezifische Wahl von Schnitten von M_k(Sp_4(Z)), deren Eigenschaften bekannt oder durch ihre Darstellung als Polynome in Theta-Funktionen ableitbar sind. Mittels eines induktiven Arguments werden wir das Integral über das Stern-Produkt der zugehörigen Green'schen Formen auf eine Summe von Integralen über spezielle Zykel zurückführen, die beim sukzessiven Schneiden der zu den Schnitten gehörigen Divisoren auftauchen. Bei diesem Prozess entstehen Randterme in Form von Integralen um den toroidalen Rand. Wir werden zeigen, dass diese verschwinden, indem wir Minkowski-Theorie anwenden und eine bestimmte Wahl der Teilung der Eins treffen, die in der arithmetischen Schnitttheorie für logarithmisch singuläre Metriken auftaucht. Die Integrale über die speziellen Zykel berechnen wir durch Zurückführen auf ein Resultat von Kudla sowie auf eine modulare Version der Jensen-Formel. / Let A_2 be the toroidally compactified moduli stack of principally polarized complex abelian surfaces, and let M_k(Sp_4(Z)) be the line bundle of Siegel modular forms of weight k on A_2, equipped with the Petersson metric. Viewing A_2 as the complex fibre of an arithmetic variety over Spec(Z), and M_k(Sp_4(Z)) as the complex line bundle induced by a line bundle on this arithmetic variety, we can ask for the arithmetic degree of this line bundle. We will state a formula for the arithmetic degree in terms of special values of the logarithmic derivative of the Riemann zeta-function. The arithmetic degree consists of a contribution from intersection over Spec(Z), and from an integral of Green forms over the complex fibre. The computation of the summand of the arithmetic degree coming from the complex fibre A_2 will be approached by making a specific choice of sections of M_k(Sp_4(Z)), whose behaviour is well-known or can be worked out by their representation via theta-functions. With an induction argument, we will trace back the integral over the star-product of the corresponding Green forms to a sum of integrals over particular cycles on A_2 coming from the successive intersection of the divisors of these sections, as well as some boundary terms in the form of integrals around the toroidal boundary. We will prove that the boundary terms vanish, using Minkowski theory and a specific choice of the partition of unity that appears in arithmetic intersection theory for logarithmically singular metrics. The integrals over the special cycles will be traced back to results of Kudla and an application of a modular version of Jensen's formula.

Dans cette thèse nous étudions deux problèmes dans le domaine de la géométrie arithmétique, concernant respectivement les points de torsion des variétés abéliennes et le polylogarithme motivique sur les schémas abéliens. La conjecture de Manin-Mumford (démontrée par Raynaud en 1983) affirme que si A est une variété abélienne et X est une sous-variété de A ne contenant aucune translatée d'une sous-variété abélienne de A, alors X ne contient qu'un nombre fini de points de torsion de A. En 1996, Buium présenta une forme effective de la conjecture dans le cas des courbes. Dans cette thèse, nous montrons que l'argument de Buium peut être utilisé aussi en dimension supérieure pour prouver une version quantitative de la conjecture pour une classe de sous-variétés avec fibré cotangent ample étudiée par Debarre. Nous généralisons aussi à toute dimension un résultat sur la dispersion des relèvements p-divisibles non ramifiés obtenu par Raynaud dans le cas des courbes. En 2014, Kings and Roessler ont montré que la réalisation en cohomologie de Deligne analytique de la part de degré zéro du polylogarithme motivique sur les schémas abéliens peut être reliée aux formes de torsion analytique de Bismut-Koehler du fibré de Poincaré. Dans cette thèse, nous utilisons la théorie de l'intersection arithmétique dans la version de Burgos pour raffiner ce résultat dans le cas où la base du schéma abélien est propre. / In this thesis we approach two independent problems in the field of arithmetic geometry, one regarding the torsion points of abelian varieties and the other the motivic polylogarithm on abelian schemes. The Manin-Mumford conjecture (proved by Raynaud in 1983) states that if A is an abelian variety and X is a subvariety of A not containing any translate of an abelian subvariety of A, then X can only have a finite number of points that are of finite order in A. In 1996, Buium presented an effective form of the conjecture in the case of curves. In this thesis, we show that Buium's argument can be made applicable in higher dimensions to prove a quantitative version of the conjecture for a class of subvarieties with ample cotangent studied by Debarre. Our proof also generalizes to any dimension a result on the sparsity of p-divisible unramified liftings obtained by Raynaud in the case of curves. In 2014, Kings and Roessler showed that the realisation in analytic Deligne cohomology of the degree zero part of the motivic polylogarithm on abelian schemes can be described in terms of the Bismut-Koehler higher analytic torsion form of the Poincaré bundle. In this thesis, using the arithmetic intersection theory in the sense of Burgos, we give a refinement of Kings and Roessler's result in the case in which the base of the abelian scheme is proper.

The study of algebraic varieties originates from the study of smooth manifolds. One of the focal points is the theory of differential forms and de Rham cohomology. It’s algebraic counterparts are given by algebraic cycles and Chow groups. Linearizing and taking the pseudo-abelian envelope of the category of smooth projective varieties, one obtains the category of pure motives. In this thesis, we concentrate on studying the pure Chow motives of Severi-Brauer varieties. This has been a subject of intensive investigation for the past twenty years, with major contributions done by Karpenko, [Kar1], [Kar2], [Kar3], [Kar4]; Panin, [Pan1], [Pan2]; Brosnan, [Bro1], [Bro2]; Chernousov, Merkurjev, [Che1], [Che2]; Petrov, Semenov, Zainoulline, [Pet]; Calmès, [Cal]; Nikolenko, [Nik]; Nenashev, [Nen]; Smirnov, [Smi]; Auel, [Aue]; Krashen, [Kra]; and others. The main theorem of the thesis, presented in sections 4.3 and 4.4, extends the result of Zainoulline et al. in the paper [Cal] by providing new examples of motivic decompositions of generalized Severi-Brauer varieties.