A section of the total tangent space of a scheme X of finite type over a field k, i.e. a vector field on X, corresponds to an X-valued 1-jet on X. In the language of jets the notion of a vector field becomes functorial, and the total tangent space constitutes one of an infinite family of jet schemes Jm(X) for m ≥ 0. We prove that there exist families of “logarithmic” jet schemes JDm(X) for m ≥ 0, in the category of k-schemes of finite type, associated to any given X and its family of divisors D = (D₁, . . . ,Dr). The sections of JD₁(X) correspond to so-called vector fields on X with logarithmic poles along the family of divisors D = (D₁, . . . ,Dr). To prove this, we first introduce the categories of pairs (X,D) where D is as mentioned, an r-tuple of (effective Cartier) divisors on the scheme X. The categories of pairs provide a convenient framework for working with only those jets that pull back families of divisors.

A section of the total tangent space of a scheme X of finite type over a field k, i.e. a vector field on X, corresponds to an X-valued 1-jet on X. In the language of jets the notion of a vector field becomes functorial, and the total tangent space constitutes one of an infinite family of jet schemes Jm(X) for m ≥ 0. We prove that there exist families of “logarithmic” jet schemes JDm(X) for m ≥ 0, in the category of k-schemes of finite type, associated to any given X and its family of divisors D = (D₁, . . . ,Dr). The sections of JD₁(X) correspond to so-called vector fields on X with logarithmic poles along the family of divisors D = (D₁, . . . ,Dr). To prove this, we first introduce the categories of pairs (X,D) where D is as mentioned, an r-tuple of (effective Cartier) divisors on the scheme X. The categories of pairs provide a convenient framework for working with only those jets that pull back families of divisors.

The theorem of De Rham-Saito is a generalization of a lemma due to De Rham [3], which was announced and used in [7] by Kyoji Saito, as noproof of this theorem was available, Le Dung Trang encouraged to Saito to publish the proof that can be seen in [8], which indirectly encourages us to detail the proof in this article for the many applications it has,we highlight the Godbillon-Vey algorithm [4]; in the proof of Theorem classical Frobenius given in [2]; in [6] we see some interesting applications, in the proof of Frobenius theorem with singularities [5]. In [1] we givefull details of the proof given by Moussu and Rolin. / El teorema de De Rham-Saito es una generalización de un lema debido a De Rham [3], el cual fue enunciado y usado en [11] por Kyoji Saito, al no haber prueba de este teorema Le Dung Trang anima a Saito a publicar la prueba que puede ser vista en [12], lo cual indirectamente nos motiva a detallarla prueba en este articulo por las muchas aplicaciones que tiene, destacamos el algoritmo de Godbillon-Vey [5]; en la prueba del Teorema de Frobenius clásico dada en [2]; en [8] vemos unas aplicaciones interesantes; en la prueba del Teorema de Frobenius con singularidades [7]; en [1] se detalla la prueba realizada por Moussu y Rolin [10].

Submitted by Renata Lopes (renatasil82@gmail.com) on 2018-03-28T11:17:02Z No. of bitstreams: 1 andersoncorreaporto.pdf: 567494 bytes, checksum: e685a947374868ceaa838290c83bc61a (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-04-09T19:23:34Z (GMT) No. of bitstreams: 1 andersoncorreaporto.pdf: 567494 bytes, checksum: e685a947374868ceaa838290c83bc61a (MD5) / Made available in DSpace on 2018-04-09T19:23:34Z (GMT). No. of bitstreams: 1 andersoncorreaporto.pdf: 567494 bytes, checksum: e685a947374868ceaa838290c83bc61a (MD5) Previous issue date: 2018-02-08 / O objetivo desse trabalho é o estudo de conceitos básicos da Geometria Algébrica sob o ponto de vista clássico. O foco central do trabalho é o estudo do Teorema de Riemann- Roch e algumas de suas aplicações. Esse teorema constitui uma importante ferramenta no estudo da Geometria Algébrica clássica uma vez que possibilita, por exemplo, o cáculo do gênero de uma curva projetiva não singular no espaço projetivo de dimensão dois. Para o desenvolvimento do estudo do Teorema de Riemann-Roch e suas aplicações serão estudados conceitos tais como: variedades, dimensão, diferenciais de Weil, divisores, divisores sobre curvas e o anel topológico Adèle. / The goal of this work is the study of basic concepts of Algebraic Geometry from the classical point of view. The central focus of the paper is the study of Riemann-Roch Theorem and some of its applications. This theorem constitutes an important tool in the study of classical Algebraic Geometry since it allows, for example, the calculation of the genus of a non-singular projective curve in the projective space of dimension two. For the development of the study of the Riemann-Roch Theorem and its applications we will study concepts such as: varieties, dimension, Weil differentials, divisors, divisors on curves and the Adèle topological ring.

A section of the total tangent space of a scheme X of finite type over a field k, i.e. a vector field on X, corresponds to an X-valued 1-jet on X. In the language of jets the notion of a vector field becomes functorial, and the total tangent space constitutes one of an infinite family of jet schemes Jm(X) for m ≥ 0. We prove that there exist families of “logarithmic” jet schemes JDm(X) for m ≥ 0, in the category of k-schemes of finite type, associated to any given X and its family of divisors D = (D₁, . . . ,Dr). The sections of JD₁(X) correspond to so-called vector fields on X with logarithmic poles along the family of divisors D = (D₁, . . . ,Dr). To prove this, we first introduce the categories of pairs (X,D) where D is as mentioned, an r-tuple of (effective Cartier) divisors on the scheme X. The categories of pairs provide a convenient framework for working with only those jets that pull back families of divisors. / Science, Faculty of / Mathematics, Department of / Graduate

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Diese Arbeit untersucht verschiedene Fragen hinsichtlich der birationalen Geometrie der Modulräume $\Mbar_g$ und $\Mbar_{g,n}$, mit besonderem Augenmerk auf der Berechnung effektiver Divisorklassen. In Kapitel 2 definieren wir für jedes $n$-Tupel ganzer Zahlen $\d$, die sich zu $g-1$ summieren, einen geometrisch bedeutsamen Divisor auf $\Mbar_{g,n}$, der durch Zurückziehen des Thetadivisors einer universellen Jacobi-Varietät mittels einer Abel-Jacobi-Abbildung erhalten wird. Er ist eine Verallgemeinerung verschiedener in der Literatur verwendeten Arten von Divisoren. Wir berechnen die Klasse dieses Divisors und zeigen, dass er für bestimmte $\d$ irreduzibel und extremal im effektiven Kegel von $\Mbar_{g,n}$ ist. Kapitel 3 beschäftigt sich mit einem birationalen Modell $X_6$ von $\Mbar_6$, das durch quadrische Hyperebenenschnitte auf der del-Pezzo-Fläche vom Grad $5$ erhalten wird. Wir berechnen die Klasse des großen Divisors, der die birationale Abbildung $\Mbar_6 \dashrightarrow X_6$ induziert, und erhalten so eine obere Schranke an die bewegliche Steigung von $\Mbar_6$. Wir zeigen, dass $X_6$ der letzte nicht-triviale Raum im log-minimalen Modellprogramm für $\Mbar_6$ ist. Weiterhin geben wir einige Resultate bezüglich der Unirationalität der Weierstraßorte auf $\Mbar_{g,1}$. Für $g = 6$ hängen diese mit der del-Pezzo-Konstruktion zusammen, die benutzt wurde, um das Modell $X_6$ zu konstruieren. Kapitel 4 konzentriert sich auf den Fall $g = 0$. Castravet and Tevelev führten auf $\Mbar_{0,n}$ kombinatorisch definierte Hyperbaumdivisoren ein, die für $n = 6$ zusammen mit den Randdivisoren den effektiven Kegel erzeugen. Wir berechnen die Klasse des Hyperbaumdivisors auf $\Mbar_{0,7}$, der bis auf Permutation der markierten Punkte eindeutig ist. Wir geben eine geometrische Charakterisierung für ihn an, die zu der von Keel und Vermeire für den Fall $n = 6$ gegebenen analog ist. / This thesis investigates various questions concerning the birational geometry of the moduli spaces $\Mbar_g$ and $\Mbar_{g,n}$, with a focus on the computation of effective divisor classes. In Chapter 2 we define, for any $n$-tuple $\d$ of integers summing up to $g-1$, a geometrically meaningful divisor on $\Mbar_{g,n}$ that is essentially the pullback of the theta divisor on a universal Jacobian variety under an Abel-Jacobi map. It is a generalization of various kinds of divisors used in the literature, for example by Logan to show that $\Mbar_{g,n}$ is of general type for all $g \geq 4$ as soon as $n$ is big enough. We compute the class of this divisor and show that for certain choices of $\d$ it is irreducible and extremal in the effective cone of $\Mbar_{g,n}$. Chapter 3 deals with a birational model $X_6$ of $\Mbar_6$ that is obtained by taking quadric hyperplane sections of the degree $5$ del Pezzo surface. We compute the class of the big divisor inducing the birational map $\Mbar_6 \dashrightarrow X_6$ and use it to derive an upper bound on the moving slope of $\Mbar_6$. Furthermore we show that $X_6$ is the final non-trivial space in the log minimal model program for $\Mbar_6$. We also give a few results on the unirationality of Weierstraß loci on $\Mbar_{g,1}$, which for $g = 6$ are related to the del Pezzo construction used to construct the model $X_6$. Finally, Chapter 4 focuses on the case $g = 0$. Castravet and Tevelev introduced combinatorially defined hypertree divisors on $\Mbar_{0,n}$ that for $n = 6$ generate the effective cone together with boundary divisors. We compute the class of the hypertree divisor on $\Mbar_{0,7}$, which is unique up to permutation of the marked points. We also give a geometric characterization of it that is analogous to the one given by Keel and Vermeire in the $n = 6$ case.

Ce travail se divise en deux parties. Dans la première partie, nous généralisons le travail de Khuri-Makdisi qui décrit des algorithmes pour l'arithmétique des diviseurs sur une courbe sur un corps. Nous montrons que les analogues naturelles de ses résultats se vérifient pour les diviseurs de Cartier relatifs effectifs sur un schéma projectif, lisse et de dimension relative un sur un schéma affine noetherien quelconque, et que les analogues naturelles de ses algorithmes se vérifient pour une certaine classe d'anneaux de base. Nous présentons un formalisme pour tels anneaux qui sont caractérisés par l'existence d'un certain sous-ensemble des opérations standards de l'algèbre linéaire pour les modules projectifs sur ces anneaux.Dans la deuxième partie de ce travail, nous considérons un type de problème de Riemann-Roch pour les diviseurs sur certaines surfaces algébriques. Plus précisément, nous analysons les surfaces algébriques qui émanent d'un produit ou d'un produit symétrique d'une courbe hyperelliptique de genre supérieur à un sur un corps (presque) arbitraire. Les résultats principaux sont une décomposition des espaces de sections globales de certains diviseurs sur telles surfaces et des formules explicites qui décrivent les dimensions des espaces de sections de ces diviseurs. Dans le dernier chapitre, nous présentons un algorithme qui produit une base pour l'espace de sections globales d'un tel diviseur. / The contribution of this thesis is divided naturally into two parts. In Part I we generalise the work of Khuri-Makdisi (2004) on algorithms for divisor arithmetic on curves over fields to more general bases. We prove that the natural analogues of the results of Khuri-Makdisi continue to hold for relative effective Cartier divisors on projective schemes which are smooth of relative dimension one over an arbitrary affine Noetherian base scheme and that natural analogues of the algorithms remain valid in this context for a certain class of base rings. We introduce a formalism for such rings,which are characterised by the existence of a certain subset of the usual linear algebra operations for projective modules over these rings.Part II of this thesis is concerned with a type of Riemann-Roch problem for divisors on certain algebraic surfaces. Specifically we consider algebraic surfaces arising as the square or the symmetric square of a hyperelliptic curve of genus at least two over an (almost) arbitrary field. The main results are a decomposition of the spaces of global sections of certain divisors on such surfaces and explicit formulæ for the dimensions of the spaces of sections of these divisors. In the final chapter we present an algorithm which generates a basis for the space of global sections of such a divisor.

Tropical geometry is algebraic geometry over the tropical semiring, or min-plus algebra. In this thesis, I discuss the basic geometry of plane tropical curves. By introducing the notion of abstract tropical curves, I am able to pass to a more abstract metric-topological setting. In this setting, I discuss divisors on tropical curves. I begin a study of $G$-invariant divisors and divisor classes.