This work concerns the construction of Calabi-Yau threefolds in codimension 4. Based on a study of Hilbert series, we give a list of families of Calabi-Yau threefolds which may exist in codimension 3 and codimension 4. Using birational methods, we construct Calabi-Yau threefolds that realize several of the listed families. The main result is that the cases we consider in codimension 4 lie in two different deformation components.

In the present thesis we analyze different types of additive decompositions of homogeneous polynomials. These problems are usually called Waring-type problems and their story go back to the mid-19th century and, recently, they received the attention of a large community of mathematicians and engineers due to several applications. At the same time, they are related to branches of Commutative Algebra and Algebraic Geometry. The classical Waring problem investigates decompositions of homogeneous polynomials as sums of powers of linear forms. Via Apolarity Theory, the study of these decompositions for a given polynomial F is related to the study of configuration of points apolar to F, namely, configurations of points whose defining ideal is contained in the ``perp'' ideal associated to F. In particular, we analyze which kind of minimal set of points can be apolar to some given polynomial in cases with small degrees and small number of variables. This let us introduce the concept of Waring loci of homogeneous polynomials. From a geometric point of view, questions about additive decompositions of polynomials can be described in terms of secant varieties of projective varieties. In particular, we are interested in the dimensions of such varieties. By using an old result due to Terracini, we can compute these dimensions by looking at the Hilbert series of homogeneous ideal. Hilbert series are very important algebraic invariants associated to homogeneous ideals. In the case of classical Waring problem, we have to look at power ideals, i.e., ideals generated by powers of linear forms. Via Apolarity Theory, their Hilbert series are related to Hilbert series of ideals of fat points, i.e., ideals of configurations of points with some multiplicity. In this thesis, we consider some special configuration of fat points. In general, Hilbert series of ideals of fat points is a very active field of research. We explain how it is related to the famous Fröberg's conjecture about Hilbert series of generic ideals. Moreover, we use Fröberg's conjecture to deduce the dimensions of several secant varieties of particular projective varieties and, then, to deduce results regarding some particular Waring-type problems for polynomials. In this thesis, we mostly work over the complex numbers. However, we also analyze the case of classical Waring decompositions for monomials over the real numbers. In particular, we classify for which monomials the minimal length of a decomposition in sum of powers of linear forms is independent from choosing the ground field as the field of complex or real numbers.

This thesis presents parallel algorithms for rational cones and affine monoids which pursue two main computational goals: finding the Hilbert basis, a minimal generating system of the monoid of lattice points of a cone; and counting elements degree-wise in a generating function, the Hilbert series.

No description available.

Polyominoes are two-dimensional objects obtained by joining edge by edge squares of same size. Originally, polyominoes appeared in mathematical recreations, but it turned out that they have applications in various fields, for example, theoretical physics and bio-informatics. Among the most popular topics in combinatorics related to polyominoes one finds enumerating polyominoes of given size, including the asymptotic growth of the numbers of polyominoes, tiling problems, and reconstruction of polyominoes. Recently Qureshi introduced a binomial ideal induced by the geometry of a given polyomino, called polyomino ideal, and its related algebra. From that moment different authors studied algebraic properties and invariants related to this ideal, such as primality, Gröbner bases, Gorensteinnes and Castelnuovo-Mumford regularity. In this thesis, we provide an overview on the results that we obtained about polyomino ideals and its related algebra. In the first part of the thesis, we discuss questions about the primality and the Gröbner bases of the polyomino ideal. In the second part of the thesis, we talk over the Castelnuovo-Mumford regularity, Hilbert series, and Gorensteinnes of the polyomino ideal and its coordinate ring.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2015-05-19T11:39:34Z No. of bitstreams: 2 Dissertação - Marcelo Bezerra Barboza - 2015.pdf: 1031294 bytes, checksum: 1a2c64373fbcf29d38e433509a38f1ab (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-05-19T11:45:05Z (GMT) No. of bitstreams: 2 Dissertação - Marcelo Bezerra Barboza - 2015.pdf: 1031294 bytes, checksum: 1a2c64373fbcf29d38e433509a38f1ab (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) / Made available in DSpace on 2015-05-19T11:45:05Z (GMT). No. of bitstreams: 2 Dissertação - Marcelo Bezerra Barboza - 2015.pdf: 1031294 bytes, checksum: 1a2c64373fbcf29d38e433509a38f1ab (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) Previous issue date: 2015-03-02 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Given a directed layered graph 􀀀, we present the algebra A(􀀀) as a quotient of the free associative or tensor algebra (with unit, over an arbitrarily fixed field of scalars), freely generated by the set of edges in 􀀀. We calculate the Hilbert series associated with the grading on A(􀀀) coming from degree in the tensor algebra. We also calculate the group of automorphisms of A(􀀀) that preserve the (ascending) filtration associated with the grading mentioned above. Despite the fact the main results within this notes remain true for a relatively large class of directed graphs, we stay close to the ones 􀀀Dn and Ln, n 3, that is, those consisting, respectively, on the Hasse diagram of the partially ordered sets of faces in a regular polygon containing n edges and the power set of {1, . . . , n}. The work teaching us all of the above is [1], by Colleen Duffy. / Dado um grafo 􀀀 orientado em níveis, apresentamos a álgebra A(􀀀) como um quociente da álgebra associativa livre ou tensorial (com unidade, sobre um corpo de escalares arbitrariamente fixado), livremente gerada pelo conjunto de arestas em 􀀀. Calculamos a série de Hilbert associada à graduação em A(􀀀) proveniente do grau na álgebra tensorial. Também calculamos o grupo dos automorfismos de A(􀀀) que preservam a filtração (crescente) associada à graduação acima mencionada. Apesar de os resultados principais permanecerem verdadeiros para uma classe relativamente ampla de grafos orientados, permanecemos próximos a 􀀀Dn e Ln, n 3, isto é, aqueles que consistem, respectivamente, no diagrama de Hasse dos conjuntos parcialmente ordenados das faces de um polígono regular de n lados e no conjunto das partes de {1, . . . , n}. O trabalho do qual aprendemos todo o acima é [1], por Collen Duffy.

Made available in DSpace on 2014-07-29T16:02:19Z (GMT). No. of bitstreams: 1 Dissertacao Bruno Trindade Reis.pdf: 1549283 bytes, checksum: 850cae1de80dba723aabf95e990ddd6a (MD5) Previous issue date: 2011-08-31 / We begin with a definition of the algebras Qn, who originated the study of algebra associated to directed graphs. Then, we define key concepts such as Hilbert series, graded and filtered algebras. Among the quadratic algebras, we introduce the Koszul algebras. The Hilbert series is a useful tool to study the Koszulity of a quadratic algebra. The homological interpretation of the coefficients of the Hilbert series of algebras associated with direct graphs allowed us to give conditions Koszulity these algebras in terms of the homological properties of the graph. We use this interpretation to construct algebras with Hilbert series prescribed. / Começamos definindo as álgebras Qn, que originaram o estudo das álgebras associadas a grafos orientados em níveis. Em seguida, definimos conceitos importantes, tais como séries de Hilbert , álgebras graduadas e álgebras filtradas. Entre as álgebras quadráticas, introduzimos as álgebras de Koszul. As séries de Hilbert são instrumentos úteis para estudar a Koszulidade de álgebras quadráticas. A interpretação homológica dos coeficientes da série de Hilbert de álgebras associadas a grafos em níveis nos permite dar condições de Koszulidade dessas álgebras em termos das propriedades homológicas do grafo. Usamos essa interpretação para construir álgebras com séries de Hilbert préestabelecidas.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2017-09-22T11:27:19Z No. of bitstreams: 2 Dissertação - Kariny de Andrade Dirino - 2017.pdf: 1986993 bytes, checksum: fd843aaf3aee361fd3b4dd512828d8f0 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-09-22T11:27:47Z (GMT) No. of bitstreams: 2 Dissertação - Kariny de Andrade Dirino - 2017.pdf: 1986993 bytes, checksum: fd843aaf3aee361fd3b4dd512828d8f0 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-09-22T11:27:47Z (GMT). No. of bitstreams: 2 Dissertação - Kariny de Andrade Dirino - 2017.pdf: 1986993 bytes, checksum: fd843aaf3aee361fd3b4dd512828d8f0 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-08-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Considering a layered directed graphs we may associate it to an algebra, denoted as , whose generators are the edges of the graph and the relations are defined through: every ways with the same initial vertex and the same final vertex determine different fractorizations for the same polynomial with coefficients in a non-commutative ring. We present a study about these algebras and their main properties, presenting some classes of examples and having as central focus the Hasse graph of the partially ordered set of k -faces of Petersen graph, . We discuss the results on basis for algebras of type we calculate their Hilbert series and the automorphisms group of these algebras, we determine the subgraphs induced by the set of vertices fixed by each and we calculate the graded trace generating functions, in order to introduce problems related to koszulity. / Dado um grafo orientado em níveis podemos associar a ele uma álgebra, denotada por cujos geradores são as arestas do grafo e as relações são definidas mediante: todos os caminhos com o mesmo vértice inicial e mesmo vértice final determinam fatorações distintas para o mesmo polinômio com coeficientes em um anel não comutativo. Exibimos um estudo sobre essas álgebras e suas principais propriedades, apresentando algumas classes de exemplos e tendo como foco central o grafo de Hasse do conjunto parcialmente ordenado das k-faces do grafo de Petersen, . Abordamos resultados sobre bases para álgebras do tipo , calculamos as suas séries de Hilbert e o grupo dos automorfismos dessas álgebras, determinamos os subgrafos induzidos pelo conjunto dos vértices fixados por cada e calculamos as funções geradoras do traço graduado, a fim de introduzirmos problemas relacionados à koszulidade.

We compute the Hilbert-Kunz functions of two-dimensional rings of type ADE by using representations of their indecomposable, maximal Cohen-Macaulay modules in terms of matrix factorizations, and as first syzygy modules of homogeneous ideals.