1,720,996 research outputs found
Ideals of powers and powers of ideals: intersecting algebra, geometry, and combinatorics
No full textThis book discusses regular powers and symbolic powers of ideals from three perspectives– algebra, combinatorics and geometry – and examines the interactions between them. It invites readers to explore the evolution of the set of associated primes of higher and higher powers of an ideal and explains the evolution of ideals associated with combinatorial objects like graphs or hypergraphs in terms of the original combinatorial objects. It also addresses similar questions concerning our understanding of the Castelnuovo-Mumford regularity of powers of combinatorially defined ideals in terms of the associated combinatorial data. From a more geometric point of view, the book considers how the relations between symbolic and regular powers can be interpreted in geometrical terms. Other topics covered include aspects of Waring type problems, symbolic powers of an ideal and their invariants (e.g., the Waldschmidt constant, the resurgence), and the persistence of associated primes
The Waldschmidt constant for squarefree monomial ideals
Get PDFGiven a squarefree monomial ideal I⊆R=k[x1,...,xn], we show that αˆ(I), the Waldschmidt constant of I, can be expressed as the optimal solution to a linear program constructed from the primary decomposition of I. By applying results from fractional graph theory, we can then express αˆ(I) in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of I. Moreover, expressing αˆ(I) as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on αˆ(I), thus verifying a conjecture of Cooper–Embree–Hà–Hoefel for monomial ideals in the squarefree case. As an application, we compute the Waldschmidt constant and the resurgence for some families of squarefree monomial ideals. For example, we determine both constants for unions of general linear subspaces of Pn with few components compared to n, and we compute the Waldschmidt constant for the Stanley–Reisner ideal of a uniform matroid
Comparing invariants of toric ideals of bipartite graphs
Get PDFGiven a finite simple graph G, one can associate to G an ideal I_G, called the toric ideal of G. There are a number of algebraic invariants of ideals which are frequently studied in commutative algebra. In general, understanding these invariants is very difficult for arbitrary ideals. However, when the ideals are related to combinatorial objects, in this case, graphs, a deeper investigation can be conducted. If, in addition, the graph G is bipartite, even more can be said about these invariants. In this thesis, we explore a comparison of invariants of toric ideals of bipartite graphs. Our main result describes all possible values for the tuple (reg(K[E]/I_G), deg(h_{K[E]/I_G}), pdim(K[E]/I_G), depth(K[E]/I_G), dim(K[E]/I_G)) when G is a bipartite graph on n ≥ 1 vertices.ThesisMaster of Science (MSc
Toric Ideals of Finite Simple Graphs
Get PDFThis thesis deals with toric ideals associated with finite simple graphs. In particular we
establish some results pertaining to the nature of the generators and syzygies of toric
ideals associated with finite simple graphs.
The first result dealt with in this thesis expands upon work by Favacchio, Hofscheier,
Keiper, and Van Tuyl which states that for G, a graph obtained by
"gluing" a graph H1 to a graph H2 along an induced subgraph, we can obtain the toric ideal associated to G from the toric ideals associated to H1 and H2 by taking their sum as ideals in the larger ring and saturating by a particular monomial f. Our contribution is to
sharpen the result and show that instead of a saturation by f, we need only examine the colon ideal with f^2.
The second result treated by this thesis pertains to graded Betti numbers of toric
ideals of complete bipartite graphs. We show that by counting specific subgraphs one
can explicitly compute a minimal set of generators for the corresponding toric ideals as well as minimal generating sets for the first two syzygy modules. Additionally we provide formulas for
some of the graded Betti numbers.
The final topic treated pertains to a relationship between the fundamental group
the finite simple graph G and the associated toric ideal to G. It was shown by
Villareal as well as Hibi and Ohsugi that the generators of a toric ideal associated to a finite simple graph correspond to the closed even walks of the graph G, thus linking algebraic properties to combinatorial ones. Therefore it is a natural question whether there is a relationship between the toric ideal associated to the graph G and the fundamental group of the graph G. We show, under the assumption that G is a bipartite graph with some additional assumptions, one can conceive of the set of binomials in the toric ideal with coprime terms, B(IG), as a group with an appropriately chosen operation ⋆ and establish a group isomorphism (B(IG), ⋆) ∼= π1(G)/H where H is a normal subgroup. We exploit this relationship further to obtain information about the generators of IG as well as bounds on the Betti numbers. We are also able to characterise all regular sequences and hence compute the depth of the toric ideal of G. We also use the framework to prove that IG = (⟨G⟩ : (e1 · · · em)^∞) where G is a set of binomials which correspond to a generating set of π1(G).ThesisDoctor of Philosophy (PhD
Monomial ideals with a prescribed Waldschmidt constant
Get PDFLet I be a monomial ideal in R=K[x_1,x_2,..,x_n], a polynomial ring over a field K. The Waldschmidt constant of I is an asymptotic invariant of I. The Waldschmidt constant manifests in many ways in commutative algebra and algebraic geometry, and is related to open problems such as the ideal containment problem and Nagata's conjecture. For a monomial ideal, the computation of its Waldschmidt constant reduces to solving a linear optimization problem. This thesis shows how to construct a monomial ideal with Waldschmidt constant equal to any rational number greater than or equal to 1. The family of monomial ideals investigated are intersections of powers of prime monomial ideals (in Chapter 3) and square-free principal Borel ideals (in Chapter 4).ThesisDoctor of Philosophy (PhD)Let I be a monomial ideal in R=K[x_1,x_2,..,x_n], a polynomial ring over a field K. The Waldschmidt constant of I is an asymptotic invariant of I. The Waldschmidt constant manifests in many ways in commutative algebra and algebraic geometry, and is related to open problems such as the ideal containment problem and Nagata's conjecture. For a monomial ideal, the computation of its Waldschmidt constant reduces to solving a linear optimization problem. This thesis shows how to construct a monomial ideal with Waldschmidt constant equal to any rational number greater than or equal to 1. The family of monomial ideals investigated are intersections of powers of prime monomial ideals (in Chapter 3) and square-free principal Borel ideals (in Chapter 4)
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Virtual Resolutions of Points in Sufficiently General Position in P^1 × P^1
Get PDFMinimal free resolutions are an important notion in algebraic geometry and commutative algebra. The minimal free resolution of a subvariety in projective spaces provides geometric properties of the subvariety. However, if the ambient space is the product of projective spaces, the minimal free resolution can be too long. On the other hand, virtual resolutions of a subvariety of products of projective spaces can be shorter and they still provide information about the subvariety. In this thesis, we investigate sets of points in
P^1 × P^1 with generic Hilbert function and in particular, points in a sufficiently general position. We find an explicit virtual resolution of ideals of a sufficiently general set of points in P^1 × P^1. Our proof depends upon computing some values of the mutigraded Castelnuovo-Mumford regularity and using a result of Berkesch, Erman and Smith. We also generalize one of the Berkesch, Erman and Smith’s result in a special case.ThesisMaster of Science (MSc
Hilbert functions of schemes of double and reduced points
No full textIt remains an open problem to classify the Hilbert functions of double
points in the projective plane.
Given a valid Hilbert function H of a zero-dimensional scheme inthe projective plane, we show how to construct a set of fat points Z of double and reduced points such that the Hilbert function of Z is the same as H. In other words, we show that any valid Hilbert function H of a zero-dimensional scheme is the Hilbert function of some set of double and reduced points. In addition, we give necessary and sufficient conditions for the Hilbert function of a scheme of a double points, or double points plus one additional reduced point, to be the Hilbert function of points with support on a star configuration of lines
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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