ALGEBRAIC GEOMETRY

Author: GRAU MONTAÑA M. TERESA.
Year: 2004.
University: AUTÓNOMA DE BARCELONA.
Place of defense: FACULTAD DE CIENCIAS.
Place of preparation: ESCUELA DE POSTGRADO.

Summary: This thesis is part of the qualitative theory of differential system in the plane. Caca chapter contains a different aspect. In the introduction, is a summary of the results known and presents the notation used for the rest of the thesis. In particular, it describes the problem of integrabilidad and some results concerning the determination of the stability of a unique point to a órbia periodically with the aim of introducing the final chapters. We define the problem of integrabilidad as the problem of finding a comprehensive system for a first differential plane and determine the functional class to which it should belong. Chapters 2 and 3 deal with the problem of integrabilidad. In Chapter 2, we get a result that allows to find explicit expression for a comprehensive first for a certain type of polynomial system. Through a rational change of variable, we match a homogeneous linear differential equation of second order: A2 (x) w''(x) + Al (x) + A0 (x) w (x) = 0, whose coefficients are polynomials, a polynomial differential system in the plane. We tried that system has an invariant for each solution is not arbitrary zero w (x) the edo second order, which, if it w (x) is a polynomial, resulting in a curve algebraic invariant. In addition, we explicit expression of a comprehensive system for the first constructed from two independent solutions of the second order edo. This is not the first comprehensive, in general, a funicón Liouvilliana. Finally, we verify that all known examples of families of systems with a quadratic curve algebraic invariant of arbitrarily high degree can be described by this construction (module transformations birracionales). In Chapter 3, invariant algebraic curves of a polynomial differential level play a key role. If an algebraic curve and irreducible invariant exists for a system level polynomial, then the values of its cofactor unique at each point are determined not degenerated. In fact, this value is a combination of linear coefficients of the natural values associated with the point singula rno degnerado. These coefficients can be determined completely natural depending on the nature of the singular point. We also can consider the points of infinity. Once the system is considered at the design complexity, the degree of an invariant algebraic curve becomes a parameter of its cofactor. If we consider a system of degree d, then is d ^ 2 + d + 1 singular points (counted with its multiplicity) and the cofactor an invariant algebraic curve is unpolinomio grade at most d-1. We proceed in the following manner: take a polynomial of degree d-1 with their d (d +1) / 2 coefficients arbitrary and assume that is the cofactor of a curve and irreducible invariant algebraic grade n. Then, impose all the conditions given by the singular points not degenerate. In the general case, we impose d ^ 2 + d + 1 conditions, and in conseucencia, completely determine the cofactor and the degree of the curve, whose existence can be determined by solving a system of linear equations, we show a condition of incompatibility. Therefore, we can determine the existence of all invariant algebraic curves for a comprehensive system. Chapter 4 focuses on the stability of a periodic orbit of a differential system level. We assume that f (x, y) = 0 is a curve and irreducible invariant with cofactor containing the periodic orbit. Pro 8 bamos qu a3b and integrals on the regular orbit of the division and the cofactor match. Hence, we can deduce the stability of a periodic orbit by integrating the cofactor on it. In Chapter 5, we describe uná implementation of the results given in Chapters 3 and 4. Believe systems with a quadratic algebraic limit cycle known until the wording of this thesis. These cycles are algebraic limit contained in invariant algebraic curves of grades, 2,4,5 and 6 and some of these families are birracionalmente equivalent quadratic systems. Applying the method described in Chapter 3, we show that there is no curve algebráica invariant except what ocntiene cycle limit. We use this result to show that these systems are not integrated first Liouvilliana. And, applying the formula given in Chapter 4, we tested these limits cycles are algebraic Hyperbolic. Chapter 6 focuses on the study of the properties of the function period asocaida a point with singular linear portion of type centro-foco. Given a cross section to flow with that singular point by extremes, we can define the application and Poincaré period and function associated with this section because this item is always monodrómico. We say that this point is synchronous if we can find a section that period associated with the role it is constant. This definition generalizes the definition usal given to centers anywhere with singular linear portion of type centro-foco. Caracterizamos this property through Lie symmetries and normal forms, widespread processes known for centers. In addition, is an example of a family of systems that depend on a real parameter, such that the origin is a unique conparte lienal type centro-foco and that is never a point isochrone.

Author: MENDOZA AGUILAR JUDIT.
Year: 2005.
University: LA LAGUNA.
Place of defense: FACULTAD DE MATEMATICAS.
Place of preparation: FACULTAD DE MATEMATICAS.

Author: MARTINEZ RAMIREZ CRISTINA.
Year: 2005.
University: AUTÓNOMA DE MADRID.
Place of defense: UNIVERSIDAD AUTONOMA DE MADRID.
Place of preparation: AUTONOMA DE MADRID.

Summary: The varieties of Severi curves flat irreducible degree d with Delta nodes were introduced by Enriques and Severi early in the last century. J. Harris showed that the varieties Sevri are irredubiles and was raised then calculate your degree. Since then, several authors have studied this problem, which is also attractive for its connection with the invariants of Gromov-Witten and cohomología quantum. In 1986 DF Coray and I. Vainsencher calculated the degree of certain strata of the variety to the family of parameterized regulated cubic surfaces. In 2001, R. Hernandez and MJ Vazquez calculated the degree of tiers of cubic unique in the space design parametrizando all surfaces grade d. One of the ways to bring problems enumerativos, is to find a space suitable parameters for objects that we enumerate and express the locus of objects satisfying conditions given as a certain cero-ciclo in the space of parameters. For the universal property of the Grassmanniana, we can identify an area regulated rational design space with a rational curve in the Grassmanniana. This allows us to use the variety of morfismos sound to Grassmanniana, as a space of parameters for areas regulated rational degree d. To apply techniques intersection to a space of parameters, it requires a compactificación it. The space morfismos is not compact, and uses two compactificaciones other than this space, compactificación of Grothendieck schema Quot ratios of a fibrado trivial rank 4 on the straight projective, and compactificación of Kontsevich Applications stable. Unfortunately, one type of dividers that intersecamos have a component in the border schema Quot. But the other dividers were intersected trasversalmente and apply the formula Atiyah-Bott plows calculate its autointersección. GEOMETRY ENUMERATIVA VIA APPLICATIONS STABLE One of the most powerful tools to solve problems enumerativos is through applications stable. The geometry of the moduli space of Kontsevich of the applications stable degree d of gender g curves with n points maracados space design, is well known. Ravi Vakil studying their connection with the geometry of enumerative sound and cruvas elliptic curves in the space design. R. Pandharipande studying theory dividers Q-Cartier in this space to the case rationally and test an algorithm to calculate all the numbers sound characteristic curves in the space design. It also calculates the degree of curves rather than sound cuspidales in linear system of curves degree d flat. There are many interesting examples in the literature of how techniques cohomología quantum stable and applications are used to solve problems enumerativos. L. GÃ ¶ ttsche and R. Pandharipande studying the ring cohomología quantum of blow-ups in the plan design, and A. Gathmann calculates invariatnes of blow-up space in a design point and examines their significance enumerative. T. Graber studies the cohomología quantum scheme Hilber plan and builds a recursive algorithm that counts the number of corners flat hiperelípticas grade dy gender g passing through 3d +1 general points. The geometry of surfaces enumerative regulated in space design is closely related to the geometry inherent moduli space of Kontsevich Applications stable curves sound with n points marked the Grassmanniana representing d times the generator positive group homology. As Grassmanniana is a variety homogeneous arguments trasversalidad imply a relationship between the invariants of Gromov-Witten and geometry enumerative. The compactificación given the moduli space of stable applications has the advantage over the scheme Quot, that the points of the border with torque ratios are not zero, which does not give morfismos. The ring cohomología quantum of a range is defined in terms of data 8 s to int ae0 ersección (invariants of Gromo-Witten) in the space of applications holomorfas curves marked gender 0 to the variety. The extent of the variety of surfaces regulated sound level is a fixed factor in the multiplication table ring cohomología quantum of Grassmanniana. The ring cohomología quantum of Grassmanniana was described by P. Di Francesco and C. Itzykson in 1994. Relationships of associativity proprocionan many equations between invariatnes of Gromov-Witten that often lead to the determination of all invariants in terms of a few numbers. In this paper, we have implemented the program farsta because Adrew Kresch, which uses relations associativity to calculate some of these quantum numbers that concern us. In the second part of this report we feel a certain stratification of the variety of morfismos and endorse an extension of the moduli space of Kontsevich that compactifica this variety. CONCLUSION AND ISSUES OPEN 1. The problem of extending these results to arbitrary gender is closely related to the problem of classifying bundles on a bend gender arbitrary. In the case elliptical, M. Atiyah has given a complete classification of the bundles on a bend gender 1. For gender greater than or equal to 2, are known only partial results. 2. We have given a description of the first group of cohomología space Kontsevich of the Grassmanniana, Â Is it possible to give a complete description of the ring Chow of this ring? There are two natural dividers that can be considered, for example, adivisor associated with the locus of applications stable whose image is tangent to a hiperplano fixed. Another locus of applications describing a codición of codimensión 1, is the dea plciaciones whose image is a curve in the Grassmanniana a pinnacle. One problem is natural to express these dividers in terms d elos generators given group Picard and calculate kinds of strata defined.

Author: PEREZ DEL POZO ANGEL.
Year: 2005.
University: COMPLUTENSE DE MADRID.
Place of defense: FACULTAD DE MATEMATICAS.
Place of preparation: FACULTAD DE MATEMATICAS.

Summary: This report deals with the study of certain subsets of Riemann surfaces and surfaces Klein that are invariant under the action of the group of automorphisms of them. Chapter 2 focuses on the overall points Weierstrass of a Riemann surface. It sets lower bounds for the weight of the fixed points of a automorfismo the surface. These levels depend on the order of automorfismo, the number of fixed points which owns and gender of the surface. In Chapter 3 extends the concepts of the theory of points Weierstrass context of the surfaces of Klein. It is associated with each point on the surface a succession of positive integers (formed by differences in dimensions of spaces functions meromorfas defined on the surface), which generalizes the concept of succession gaps at one point, we study some properties of this succession and determined for each point of a surface Klein hiperelíptica. In Chapter 4 are obtained heights higher order of a group of automorphisms of a surface of Klein's edge; these levels depend on the gender of algebraic surface and sub cardinal finite surface, invariant under the action of the group. Imposing conditions transitivity not in the group action on all related components at the edge of the area, it can apply the levels obtained to find others that only depend on the gender algebraic.

Author: CARAVANTES TORTAJADA JORGE.
Year: 2005.
University: COMPLUTENSE DE MADRID.
Place of defense: FACULTAD DE CIENCIAS MATEMÁTICAS.
Place of preparation: FACUTLAD DE CIENCIAS MATEMÁTICAS.

Summary: This paper presents a new method to check if a subvariety smooth codimensión small group inherits picard its variety environment (Except divisibility) we apply this method to subvariedades in grassmannianas of product lines and spaces proyectivos, so we extend the results of BARTH + LARSEN for space design and suavizamos restrictions that were obtained genetic results BARTH-Van VEN and SOMMESE.

Author: ÁLVAREZ SÁNCHEA AMELIA.
Year: 2005.
University: EXTREMADURA.
Place of defense: FACULTAD DE CIENCIAS.
Place of preparation: UNIVERSIDAD DE EXTREMADURA.

Author: RIERA BURGER CONSTANZA.
Year: 2005.
University: COMPLUTENSE DE MADRID.
Place of defense: FACULTAD DE MATEMÁTICAS.
Place of preparation: FACULTAD CIENCIAS MATEMÁTICAS.

Summary: Generalizamos property Bent for a Boolean function we spectral interpretation complementation local and pivot, the specter of a plane Boolean quadratic function with respect to certain unitary transformations relates to modified versions of its parent and associated adjacency. We estimate the number of flat spectra of some structures deduce an interpretation of the various spectral polynomials "interlacing" of a graph and relate one with a measure of quantum intertwining of quantum state partner. Caracterizamos values spectro a boolean function Quad. We give a formula for the "Weight Hierarchy" in terms of a polynomial "Interlace" modified generalizamos pivot to hipergráfos. Mostrmaos change the degree of a Boolean function through pivot. Finally, we show changes as the specter of a wide range of vector with respect to a series of significant changes.

Author: NAVARRO GARULO GABRIEL.
Year: 2006.
University: GRANADA.
Place of defense: FACULTA DE CIENCIAS.
Place of preparation: FACULTAD DE CIENCIAS.

Summary: The objective of the theory of representation Álgebras is to classify Álgebras, usually on a closed algebra body, depending on their level modules. Historically, efforts have focused on only considers the case finito-dimensional. Here highlights the work of Gabriel to translate the problem of context or quivers directed graphs and Auslander and Reiten providing tools is essential for the study of the modules of an algebra. However, this theory is not valid if the algebra is of infinite dimension. In this context arises the concept of coálgebra as a generalization of Álgebras finito-dimensionales and allows an approximation to the general case from the point of view classic. In this dissertation explores l possibility of a result for coálgebras analogous to known theorem Gabriel describing the Álgebras basic finite dimensional as ratios Álgebras road by an idea admissible. For this purpose using the notion of coálgebra road with a quiver relations defined by Simson. Since counterexamples are obtained in this regard, and even a criterion for deciding when a coálgebra admissible is the coálgebra road with a quiver relations, the class is considered low to coalgebras tame. To deal with this new problem is considered in the location categories comodulos, linking the property to be wild or tame a coálgebra and coálgebras localized. As a result of this analysis is obtained the following results: All subcoálgebra admissible tame a coálgebra road of a quiver aciclico is isomorfa a coálgebra road with a quiver relations.

Author: MARCO BUZUNÁRIZ MIGUEL ÁNGEL.
Year: 2006.
University: ZARAGOZA.
Place of defense: FACULTAD DE CIENCIAS.
Place of preparation: DEPARTAMENTO DE MATEMÁTICAS.

Summary: A configuration of straight is a finite set of lines at the complex design. You can isolate their combinatorial properties in the concept of combinatorial straight. One of the classic questions about these objects is the extent to which the combination of an upright configuration determines the topology of its socket. In this regard were known classics enough results that showed topological invariants that could be derived from the combinatorial information. The first result showed that the existence of different configurations combinatoriamente equivalent topology dates from 1994, when Rybnikov built two realizations of a combinatorial groups whose core is not isomorfos. Such groups can be distinguished except isomorfismo homológicamente trivial by Invariante Alexander. This report explores various combinatorial conditions that suggest that all isomorfismos among key groups are homológicamente trivial. Such combinatorial are called homológicamente rigid. To study the riorigidez homological a combinatorial, introduced in chapter 3 the concepts of class and allowable beam combinatorial, demonstrating equivalence. These objects, although they are combinatorial nature, allow for geometric information. Specifically, described beam curves embedded in the configuration. All isomorfismo among key groups swap these beams, which can be used to narrow the group of such isomorfismos. This marking is made possible by the existence of a structure subyaciente throughout combinatorial beams. We studied this structure through the concept of a triangle of eligible classes, which are classes of eligible subjects whose kernels are cut in a non-generic. Chapter 4 contains the description and justification of a method to establish the rigidity of a combinatorial homological. It uses the method above to give a criterion: any combinatorial strongly associated with sufficient triangles and dots that can be distinguished is homológicamente rigid. The power of this method is shown through several examples in chapter 5. In this chapter highlights some particularly interesting properties of some combinatorial and its achievements. The report includes an appendix with the code of the software that implements all over the same algorithms described, as well as some comments subre-su complexity.