Let C be a compact Riemann surface. Let us consider a finite group acting on CxC, having some elements that exchange the factors, and assume that the subgroup of those elements that do not exchange the factors acts freely. We call the quotient a Semi-isogenous Mixed Surface. In this work we investigate these surfaces and we explain how their geometry is encoded in the group. Based on this, we present an algorithm to classify the Semi-isogenous Mixed Surfaces with given geometric genus, irregularity and self-intersection of the canonical class. In particular we give the classification of Semi-isogenous Mixed Surfaces with K^2>0 and holomorphic Euler-Poincaré characteristic equal to 1, where new examples of minimal surfaces of general type appear. Minimality of Semi-isogenous Mixed Surfaces is discussed using two different approaches. The first one involves the study of the bicanonical system of such surfaces: we prove that we can relate the dimension of its first cohomology group to the rank of a linear map that involves only curves. The second approach exploits Hodge index theorem to bound the number of exceptional curves that live on a Semi-isogenous Mixed Surface.

In this thesis I've explored the theory of quaternionic slice regular functions. More precisely I've studied some rigidity properties, some differential issues and an application in complex differential geometry. This application concerns the constructions and classifications of orthogonal complex structures on open domains of the four dimensional euclidean space.

In this thesis we define and study the mixed quasi-étale surfaces. In particularwe classify all the mixed quasi-étale surfaces whose minimal resolution of the singularities is a regular surface with p_g=0 and K^2>0. It is a well known fact that each Riemann surface with p_g=0 is isomorphic to P^1. At the end of XIX century M. Noether conjectured that an analogous statement holds for the surfaces: in modern words, he conjectured that every smooth projective surface with p_g=q=0 be rational. The first counterexample to this conjecture is due to F. Enriques (1869). He constructed the so called Enriques surfaces. The Enriques-Kodaira classification divides compact complex surfaces in four main classes according to their Kodaira dimension k: -oo, 0, 1, 2. A surface is said to be of general type if k=2. Nowadays this class is much less understood than the other three. The Enriques surfaces have k=0. The first examples of surfaces of general type with p_g=0 have been constructed in the 30's by L. Campedelli e L. Godeaux. The idea of Godeaux to construct surfaces was to consider the quotient of simpler surfaces by the free action of a finite group. In this spirit, Beauville proposed a simple construction of surfaces of general type, considering the quotient of a product of two curves C_1 and C_2 by the free action of a finite group G. Moreover he gave an explicit example with p_g=q=0 considering the quotient of two Fermat curves of degree 5 in P^2. There is no hope at the moment to achieve a classification of the whole class of the surfaces of general type. Since for a surface in this class the Euler characteristic of the structure sheaf \chi is strictly positive, one could hope that a classification of the boundary case \chi=1 is more affordable. Some progresses in this direction have been done in the last years through the work of many authors, but this (a priori small) case has proved to be very challenging, and we are still very far from a classification of it. At the same time, this class of surfaces, and in particular the subclass of the surfaces with p_g=0 contains some of the most interesting surfaces of general type. If S is a surface of general type with \chi=1, which means p_g=q, then p_g = q < 5, and if p_g=q=4, then S is birational to the product of curves of genus 2. The surfaces with p_g = q = 3 are completely classified. The cases p_g = q < 3 are still far from being classified. Generalizing the Beauville example, we can consider the quotient (C_1 x C_2)/G, where the C_i are Riemann surfaces of genus at least two, and G is a finite group. There are two cases: the mixed case where the action of G exchanges the two factors (and then C_1 = C_2); and the unmixed case where G acts diagonally. Many authors studied the surfaces birational to a quotient of a product of two curves, mainly in the case of surfaces of general type with \chi=1. In all these works the authors work either in the unmixed case or in the mixed case under the assumption that the group acts freely. The main purpose of this thesis is to extend the results and the strategies of the above mentioned papers in the non free mixed case. Let C be a Riemann surface of genus at least 2, let G be a finite group that acts on C x C with a mixed action, i.e. there exists an element in G that exchanges the two factors. Let G^0 be the index two subgroup of the elements that do not exchange the factors. We say that X=(C x C)/G is a mixed quasi-étale surface if the quotient map C x C -> (C x C)/G has finite branch locus. We present an algorithm to construct regular surfaces as the minimal resolution of the singularities of mixed quasi-étale surfaces. We give a complete classification of the regular surfaces with p_g=0 and K^2>0 that arise in this way. Moreover we show a way to compute the fundamental group of these surfaces and we apply it to the surfaces we construct. Some of our construction are more interesting than others. We have constructed two numerical Campedelli surfaces (K^2 = 2) with topological fundamental group Z/4Z. Two of our constructions realize surfaces whose topological type was not present in the literature before. We also have three examples of Q-homology projective planes, and two of them realize new examples of Q-homology projective planes.

ABSTRACT: Roughly speaking, the moduli space of higher spin curves parametrizes equivalence classes of pairs (C, L) where C is a smooth genus g algebraic curve and L is a line bundle on it whose r-th tensor power is isomorphic to the canonical bundle of the curve. The aim of the talk is to discuss important geometrical properties of these spaces under different points of view: one possible compactification together with the description of the rational Picard group, their birational geometry in some low genus cases and their relation with some special locus inside the classical moduli spaces of curves.

We consider smooth complex projective varieties X which are rationally connected by rational curves of degree d with respect to a fixed ample line bundle L on X, and we focus our attention on conic connected manifolds (d=2) and on rationally cubic connected manifolds (d=3). Conic connected manifolds were studied by Ionescu and Russo; they considered conic connected manifolds embedded in projective space (i.e. L is very ample) and they proved a classification theorem for these manifolds. We show that their classification result holds true assuming just the ampleness of L. Moreover we give a different proof of a theorem due to Kachi and Sato; this result characterizes a special subclass of conic connected manifolds. As already said before, we study also rationally cubic connected manifolds. We prove that if rationally cubic connected manifolds are covered by “lines†, i.e. by curves of degree 1 with respect to L, then the Picard number of X is equal to or less than 3; moreover we show that if the Picard number is equal to 3 then there is a covering family of “lines†whose numerical class spans a negative extremal ray of the Kleiman-Mori cone of X. Unfortunately, for rationally cubic connected manifolds which don't admit a covering family of “lines†there isn't an upper bound on the Picard number. However we prove that if we consider rationally cubic connected manifolds which are not covered by “lines†but are Fano then up to a few exceptions in dimension 2 also the Picard number of these manifolds is equal to or less than 3. In particular, supposing that the dimension of X is greater than 2, we show that either the Picard number is equal to or less than 2 or X is the blow up of projective space along two disjoint subvarieties that are linear subspaces or quadrics.

In this dissertation I face three main arguments. 1) Classification of Fano-Mori contractions. 2) Chern numbers on smooth threefolds. 3) Pluricanonical systems.

Galois spaces, that is affine and projective spaces of dimension N ≥ 2 defined over a finite (Galois) field F_q, are well known to be rich of nice geometric, combinatorial and group theoretic properties that have also found wide and relevant applications in several branches of Combinatorics, as well as in more practical areas, notably Coding Theory and Cryptography. The systematic study of Galois spaces was initiated in the late 1950’s by the pioneering work of B. Segre [59]. The trilogy [34, 36, 42] covers the general theory of Galois spaces including the study of objects which are linked to linear codes. Typical such objects are plane arcs and their generalizations - especially caps and arcs in higher dimensions - whose code theoretic counterparts are distinguished types of error-correcting and covering linear codes. Their investigation has received a great stimulus from Coding Theory, especially in the last decades; see the survey papers [40, 41]. An important issue in this context is to ask for explicit constructions of small complete arcs and small complete caps. A cap in a Galois space is a set of points no three of which are collinear. A cap is complete if its secants (lines through two points of the set) cover the whole space. An arc in a Galois space of dimension N is a set of points no N+1 of which lying on the same hyperplane. In analogy with caps, an arc which is maximal with respect to set-theoretical inclusion is said to be complete. Also, arcs coincide with caps in Galois planes. From these geometrical objects, there arise linear codes which turn out to have very good covering properties, provided that the size of the set is small with respect to the dimension N and the order q of the ambient space. For the size t(AG(N,q)) of the smallest complete caps in a Galois affine space AG(N,q) of dimension N over F_q, the trivial lower bound is √2q^{N−1/2}. General constructions of complete caps whose size is close to this lower bound are only known for q even and N odd, see [16, 19, 29, 52]. When N is even, complete caps of size of the same order of magnitude as cq^{N/2}, with c a constant independent of q, are known to exist for both the odd and the even order case, see [16, 18, 28, 29, 31] (see also Section 2.2 and the references therein). Whereas, few constructions of small complete arcs in Galois spaces of dimension N>2 are known. In [65, 66, 67], small complete arcs having many points in common with the normal rational curve are investigated (see Section 4.2.3 for comparisons with our results). In this thesis, new infinite families of complete arcs and caps in higher dimensional spaces are constructed from algebraic curves defined over a finite field. In most cases, no smallest complete caps/arcs were previously known in the literature. Although caps and arcs are rather combinatorial objects, constructions and proofs sometimes heavily rely on concepts and results from Algebraic Geometry in positive characteristic.

Este trabalho de mestrado teve o objetivo de calcular a dimensão de um modelo de gravitação quântica euclidiana baseado no formalismo da geometria não comutativa. Este cálculo é uma continuação do artigo [4]. Em [4], observa-se que utilizar uma tripla espectral comuttiva permite generalizar e tratar a geometria usual de maneira puramente algébrica. Do formalismo da Geometria Não Comutativa [2], observa-se que existe uma relação entre uma variedade e uma tripla espectral comutativa. Depois de definida a ação, as variáveis dinâmicas e o observável dimensão desta teoria em termo de tripla espectral, verificou-se que havia um analogia clara com a teoria das Matrizes Aleatórias [6], assim foi possível identificar uma maneira numerica de calcular a dimensão deste problema. O espaço estudado foi um conjunto infinito de pontos, cuja dimensão era uma variável estocástica, ou seja, partimos de um conjunto de pontos arbitrário que poderia ter qualquer dimensão real. Os resultados obtidos mostram que a dimensão do modelo é um número perto de 1. Neste trabalho de mestrado encontramos dois métodos distintos para calcular a dimensão, um baseando-se na definição de dimensão de dimensão oriunda da Geometria Não Comutativa e a outra baseada na lei de Weyl. Dadas estas duas alternativas, o trabalho consistiu em criar, construir, implementar e testar um algoritmo capaz de extrair a dimensão destas duas maneiras através de simulações de Monte Carlo. Este resultado é muito interessante devido as características muito gerais do espaço escolhido, dado que o limite superior da dimensão calculada em [4] foi 2, estávamos esperando qualquer valor no intervalo entre 0 e 2, e o que os resultados sugerem é que a dimensão é uma \"variável termodinâmica\", ou seja, uma distribuição delta com centro em 1 / This work had the intention of calculating the dimension of a Euclidian Quantum Gravity model which in based on the formalism of non commutative geometry. This work is a continuation of the article [4]. From [4] we know that using a commutative spectral triple allows us to generalize and treat ordinary geometry in a purely algebraic manner. From Non commutative Geometry [2] follows that manifolds and a spectral triples are related. After defining action, dynamical variables and observables in terms of the spectral triple, a relationship between Random Matrix Theory [6] came up. That relationship was used to evaluate the model dimension using numerical computation algorithms. We studied on particular manifold which is equivalent to an ordinary set of points. The dimension is a stochatic variable, for instance this model allow any possible value for dimension inside the interval [0;2] We found two distinct methods for calculating the dimension. One of them was based on the dimension defined by Alain Connes and the other is based on the dimension defined by Weyl. It is known thad both definitions are equivalent for the simple cases. Given these two methods we calculated the dimension through a Monte Carlo algorithm created and implemented during this work. The program simulated a finite approximation for the dimension and after running several simulations depending on the number of points we founds the asymptotic law for the dimension. We found that the dimension is a \"thermodynamical variable\" near the finite value 1, since we found that the estimator of this observable has variance thad goes to zero on the continuous case. This result is very interesting due to the broad characteristics of the space chosen. Given that the upper limit for the dimension calculated in [4] was 2, we were expecting any stochastic distribution in the interval between 0 and 2, and the results sugest that dimension is a delta

Orientadora: Renata Zotin Gomes de Oliveira / Banca: Suzinei Aparecida Siqueira Marconato / Banca: Maria Beatriz Ferreira Leite / Resumo: Este trabalho busca entender o que se espera do professor na sua prática, analisando os documentos o ciais e materiais de apoio (livros didáticos e outros), de modo a veri car o que eles oferecem na condução de aulas, com foco no estudo da circunfer ência. Junto com estes materiais, novas possibilidades são encontradas com o uso de tecnologias digitais, presentes e inseparáveis da vida diária da maioria dos estudantes nos níveis fundamental e médio. Assim, apresentamos uma proposta de utilização de um software educacional livre como motivação para o estudo da circunferência, especi camente o estudo de ângulos (em radianos), no segundo ano do Ensino Médio / Abstract: This paper seeks to understand what is expected of the teacher in his practice, analyzing o cial documents and material support (school and other books), checking what they can o er for lessons, focusing on the study of the circle. Along with these materials, new possibilities are found with the use of digital technologies, always present and inseparable from the daily lives of the majority of students in elementary and high school. Thus, we present a proposal to use a free educational software as motivation for the study of the circumference, speci cally the study of angles (in radians) in the second year of high school / Mestre

Orientador: Wladimir Seixas / Banca: Yuriko Yamomoto Baldin / Banca: João Peres Vieira / Resumo: Esta dissertação tem como objetivo introduzir os conceitos e os principais resultados da Geometria Hiperbólica, entre eles a não existência de retângulos. Verifica-se assim que as diferenças entre as geometrias euclidiana e hiperbólica se dá pela negação do Quinto Axioma de Euclides ou, como é conhecido, o Axioma das paralelas de Euclides. Na parte final deste trabalho abordaremos três principais modelos da Geometria Hiperb ólica: o Disco de Beltrami-Klein, o Disco de Poincaré e o Semiplano de Poincaré. Demonstraremos também que estes modelos são isomorfos / Abstract: The aim of this dissertation is to introduce the main concepts and results of hyperbolic geometry including the non-existence of rectangles. This statement is one of the many di erences between Euclidean geometry and Hyperbolic geometry from the negation of the Fifth Axiom of Euclid or as it is known, the Axiom of parallel of Euclid. In the nal part of this work we shall cover three main models of Hyperbolic Geometry: Beltrami-Klein, Poincaré Disk and the Poincaré Half Plane. We also demonstrate that these models are isomorphic / Mestre