103 research outputs found
Star configuration points and plane curves
No full textLet
ℓ
1
,
…
,
ℓ
l
\ell _1,\ldots ,\ell _l
be
l
l
lines in
P
2
\mathbb {P}^2
such that no three lines meet in a point. Let
X
(
l
)
\mathbb {X}(l)
be the set of points
{
ℓ
i
∩
ℓ
j
|
1
≤
i
>
j
≤
l
}
⊆
P
2
\{\ell _i \cap \ell _j ~|~ 1 \leq i > j \leq l\} \subseteq \mathbb {P}^2
. We call
X
(
l
)
\mathbb {X}(l)
a star configuration. We describe all pairs
(
d
,
l
)
(d,l)
such that the generic degree
d
d
curve in
P
2
\mathbb {P}^2
contains an
X
(
l
)
\mathbb {X}(l)
. Our proof strategy uses both a theoretical and an explicit algorithmic approach. We also describe how one may extend our algorithmic approach to similar problems.</p
Plane curves containing a star configuration
No full textGiven a collection of l general lines L_1,...,L_l in P^2 the star configuration X is the set of points constructed from all pairwise intersections of these lines. For each non-negative integer d, we compute the dimension of the family of curves of degree d that contain a star configuratio
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
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
Toric Ideals of Finite Simple Graphs
No full textThis 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
A note on the van der Waerden complex
No full textEhrenborg, Govindaiah, Park, and Readdy recently introduced the van der Waerden complex, a pure simplicial complex whose facets correspond to arithmetic progressions. Using techniques from combinatorial commutative algebra, we classify when these pure simplicial complexes are vertex decomposable or not Cohen-Macaulay. As a corollary, we classify the van der Waerden complexes that are shellable
Partial Betti splittings with applications to binomial edge ideals
No full textWe introduce the notion of a partial Betti splitting of a homogeneous ideal, generalizing the notion of a Betti splitting first given by Francisco, Hà, and Van Tuyl. Given a homogeneous ideal and two ideals and such that , a partial Betti splitting of relates some of the graded Betti of with those of , and . As an application, we focus on the partial Betti splittings of binomial edge ideals. Using this new technique, we generalize results of Saeedi Madani and Kiani related to binomial edge ideals with cut edges, we describe a partial Betti splitting for all binomial edge ideals, and we compute the total second Betti number of binomial edge ideals of trees.26 page
The Waldschmidt constant for squarefree monomial ideals
No full textGiven 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
No full textGiven 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
Monomial ideals with a prescribed Waldschmidt constant
No full textLet 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)
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