Neste projeto inicia-se o estudo de classificação de aplicações estáveis. Para isto usamos grafos que irão corresponder ao conjunto singular destas aplicações. Em um primeiro momento estudamos o caso de aplicações estáveis de superfícies no plano e depois estudamos aplicações estáveis de 3-variedades em \'R POT. 3\' / In this project we began the study of classification of stable maps. For this we use graphs that correspond to the singular set of these applications. At first we study the case of stable maps of surfaces in the plane and then we study stable maps of a 3-manifold in \'R POT. 3\'

Neste projeto inicia-se o estudo de classificação de aplicações estáveis. Para isto usamos grafos que irão corresponder ao conjunto singular destas aplicações. Em um primeiro momento estudamos o caso de aplicações estáveis de superfícies no plano e depois estudamos aplicações estáveis de 3-variedades em \'R POT. 3\' / In this project we began the study of classification of stable maps. For this we use graphs that correspond to the singular set of these applications. At first we study the case of stable maps of surfaces in the plane and then we study stable maps of a 3-manifold in \'R POT. 3\'

The moduli space of stable quotients, introduced by Marian, Oprea, and Pandharipande, provides a nonsingular compactification of the moduli space of degree d maps from smooth genus 1 curves into projective space ℙⁿ. This is done by allowing the domain curve to have nodal singularities and by admitting certain rational maps. The rational maps are introduced in the following way: A map to projective space can be defined by a quotient bundle of the trivial bundle on the domain curve; in the compactification, the quotient bundle is replaced by a sheaf which may not be locally free. The boundary is filtered by the degree of the torsion subsheaf of the quotient. Yijun Shao has defined a similar compactification of the moduli space of degree d maps from ℙ¹ into a Grassmannian. A blow-up process is carried out on the compactification in order to produce a boundary which is a simple normal crossings divisor: The closed subschemes in the filtration of the boundary are blown up in order of decreasing torsion. In this thesis, we carry out an analogous blow-up process on the moduli space of stable quotients. We show that the end result is a nonsingular compactification which has as its boundary a simple normal crossings divisor.

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Neste trabalho apresento a teoria de Gromov-Witten, cohomologia quÃntica e mapas estÃveis e uso estas ferramentas para obter alguns resultados enumerativos. Em particular, provo a fÃrmula de Kontsevich para curvas racionais projetivas planas de grau d. FaÃo um estudo introdutÃrio dos espaÃos de Mumford-Knudsen e construo os espaÃos de Kontsevich a fim de definir os invariantes de Gromov-Witten. Estes sÃo usados para definir o anel de cohomologia quÃntica. Em seguida, aplico a teoria geral para o caso do plano projetivo e, usando a associatividade do produto quÃntico, obtenho a fÃrmula de Kontsevich. TambÃm estudo a fronteira do espaÃo modulli de mapas estÃveis e descrevo o grupo de Picard destes. Com isso, seguindo as ideias de Pandharipand, especialmente o algoritmo por este desenvolvido, calculo alguns nÃmeros caracterÃsticos de curvas no espaÃo projetivo. / In this work, I present the Gromov-Witten theory, quantum cohomology and stable maps and use these tools to obtain some enumerative results. In particular, I proof the Kontsevich formula to projective rational plane curves of degree d. I do an introductory study of Mumford-Knudsen spaces and construct the Kontsevich spaces in order to define gromov-witten invariants. These are used to define the quantum cohomology ring. Next, I apply the general theory to the case of the projective plane and, using the the associativity of the quantum product, I obtain the Kontsevich formula. I also study the boundary of the modulli space of stable maps and describe its Picard group. Following the ideas of Pandharipand, especially the algorithm he developed, I calculate some characteristic numbers of curves in the projective space.

Made available in DSpace on 2015-03-26T13:45:32Z (GMT). No. of bitstreams: 1 texto completo.pdf: 636306 bytes, checksum: ff057bb39b98e76ae4a5997ce6271a4d (MD5) Previous issue date: 2010-03-12 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work, we study the graphs as invariants of stable maps from closed surface in the plane. We study the problem of realization of graphs by stable maps, emphasizing also the case of fold maps (without cusps). / Neste trabalho, estudamos os grafos como invariantes de aplicações estáveis de superfície fechada no plano. Abordamos o problema de realização de grafos por aplicações estáveis, enfatizando também o caso específico de aplicações de dobra (sem cúspides).

Made available in DSpace on 2015-03-26T13:45:34Z (GMT). No. of bitstreams: 1 texto completo.pdf: 833570 bytes, checksum: 24fbb8cfc82b6070bc53272b50a6af11 (MD5) Previous issue date: 2011-02-25 / This dissertation is devoted to the study of stable maps from closed orientable surfaces to the sphere. We study graphs as invariants of such maps and according to Hacon, Mendes de Jesus and Romero-Fuster [14], every bipartite graph is realized by a stable map with arbitrary degree. According to Demoto [2], we show that the minimal contour of a stable map f between two spheres has exactly 2deg(f) cusps and no self-intersections. / Esta dissertação é dedicada ao estudo de aplicações estáveis de superfícies fechadas e orientadas na esfera. Vamos estudar grafos como invariantes de tais aplicações estáveis e de acordo com Hacon, Mendes de Jesus e Romero-Fuster [14], todo grafo bipartido é realizado por aplicações estáveis desse tipo, com grau arbitrário. Segundo Demoto [2], vamos mostrar que o contorno minimal de uma aplicação estável f entre duas esferas possui exatamente 2deg(f) cúspides e nenhuma auto-interseção.

En géométrie algébrique, les invariants de Gromov—Witten sont des invariants énumératifs qui comptent le nombre de courbes complexes dans une variété projective lisse qui vérifient des conditions d’incidence. En 2001, A. Givental et Y.P. Lee ont défini de nouveaux invariants, dits de Gromov—Witten K-théoriques, en remplaçant les définitions cohomologiques dans la construction des invariants de Gromov—Witten par leurs analogues K-théoriques. Une question essentielle est de comprendre comment sont reliées ces deux théories. En 2013, Iritani- Givental-Milanov-Tonita démontrent que les invariants K-théoriques peuvent être encodés dans une fonction qui vérifie des équations aux q-différences. En général, ces équations fonctionnelles vérifient une propriété appelée “confluence”, selon laquelle on peut dégénérer ces équations pour obtenir une équationdifférentielle. Dans cette thèse, on propose de comparer les deux théories de Gromov— Witten à l’aide de la confluence des équations aux q-différences. On montre que, dans le cas des espaces projectifs complexes, que ce principe s’adapte et que les invariants Kthéoriques peuvent être dégénérés pour obtenir leurs analogues cohomologiques. Plus précisément, on montre que la confluence de la petite fonction J de Givental K-théorique permet de retrouver son analogue cohomologique après une transformation par le caractère de Chern. / In algebraic geometry, Gromov— Witten invariants are enumerative invariants that count the number of complex curves in a smooth projective variety satisfying some incidence conditions. In 2001, A. Givental and Y.P. Lee defined new invariants, called Ktheoretical Gromov—Witten invariants. These invariants are obtained by replacing cohomological objects used in the definition of the usual Gromov—Witten invariants by their Ktheoretical analogues. Then, an essential question is to understand how these two theories are related. In 2013, Iritani-Givental- Milanov-Tonita show that K-theoretical Gromov—Witten invariants can be embedded in a function which satisfies a q-difference equation. In general, these functional equations verify a property called “confluence”, which guarantees that we can degenerate these equations to obtain a differential equation. In this thesis, we propose to compare our two Gromov—Witten theories through the confluence of q-difference equations. We show that, in the case of complex projective spaces, this property can be adapted to degenerate Ktheoretical invariants into their cohomological analogues. More precisely, we show that theconfluence of Givental’s small K-theoretical Jfunction produces its cohomological analogue after applying the Chern character.