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Adam Van Tuyl

Publications and source records attributed to Adam Van Tuyl.

At least 73 records · Page 4Linked to original sources
arXiv2012

Cohen-Macaulay Circulant Graphs

Let G be the circulant graph C_n(S) with S a subset of {1,2,...,\lfloor n/2 \rfloor}, and let I(G) denote its the edge ideal in the ring R = k[x_1,...,x_n]. We consider the problem of determining when G is Cohen-Macaulay, i.e, R/I(G) is a Cohen-Macaulay ring. Because a Cohen-Macaulay graph G must be well-covered, we focus on known families of well-covered circulant graphs of the form C_n(1,2,...,d). We also characterize which cubic circulant graphs are Cohen-Macaulay. We end with the observation that even though the well-covered property is preserved under lexicographical products of graphs, this is not true of the Cohen-Macaulay property.

math.AC↗
arXiv2012

Fat lines in P^3: powers versus symbolic powers

We study the symbolic and regular powers of ideals I for a family of special configurations of lines in P^3. For this family, we show that I^(m) = I^m for all integers m if and only if I^(3) = I^3. We use these configurations to answer a question of Huneke that asks whether I^(m) = I^m for all m if equality holds when m equals the big height of the ideal I.

math.AC↗
arXiv2012

Balanced vertex decomposable simplicial complexes and their h-vectors

Given any finite simplicial complex Δ, we show how to construct a new simplicial complex Δ_χ that is balanced and vertex decomposable. Moreover, we show that the h-vector of the simplicial complex Δ_χ is precisely the f-vector, denoted f(Δ), of the original complex Δ. We deduce this result by relating f(Δ) with the graded Betti numbers of the Alexander dual of Δ_χ. Our construction generalizes the "whiskering" construction of Villarreal, and Cook and Nagel. As a corollary of our work, we add a new equivalent statement to a theorem of Björner, Frankl, and Stanley that classifies the f-vectors of simplicial complexes. We also prove a special case of a conjecture of Cook and Nagel, and Constantinescu and Varbaro on the h-vectors of flag complexes.

math.AC↗
arXiv2012

Star configurations on generic hypersurfaces

Let $F$ be a homogeneous polynomial in $S = \mathbb{C}[x_0,...,x_n]$. Our goal is to understand a particular polynomial decomposition of $F$; geometrically, we wish to determine when the hypersurface defined by $F$ in $\mathbb{P}^n$ contains a star configuration. To solve this problem, we use techniques from commutative algebra and algebraic geometry to reduce our question to computing the rank of a matrix.

math.AG↗
arXiv2012

Bounding invariants of fat points using a coding theory construction

Let $Z \subseteq \proj{n}$ be a fat points scheme, and let $d(Z)$ be the minimum distance of the linear code constructed from $Z$. We show that $d(Z)$ imposes constraints (i.e., upper bounds) on some specific shifts in the graded minimal free resolution of $I_Z$, the defining ideal of $Z$. We investigate this relation in the case that the support of $Z$ is a complete intersection; when $Z$ is reduced and a complete intersection we give lower bounds for $d(Z)$ that improve upon known bounds.

math.AC↗
arXiv2012

Asymptotic resurgences for ideals of positive dimensional subschemes of projective space

Recent work of Ein-Lazarsfeld-Smith and Hochster-Huneke raised the problem of determining which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci-Harbourne defined a quantity called the resurgence to address this problem for homogeneous ideals in polynomial rings, with a focus on zero dimensional subschemes of projective space; the methods and results obtained there have much less to say about higher dimensional subschemes. Here we take the first steps toward extending this work to higher dimensional subschemes. We introduce new asymptotic versions of the resurgence and obtain upper and lower bounds on them for ideals of smooth subschemes, generalizing what is done by Bocci-Harbourne. We apply these bounds to ideals of unions of general lines in ${\bf P}^N$. We also pose a Nagata type conjecture for symbolic powers of ideals of lines in ${\bf P}^3$.

math.AG↗
arXiv2012

Symbolic powers versus regular powers of ideals of general points in P^1 x P^1

Recent work of Ein-Lazarsfeld-Smith and Hochster-Huneke raised the problem of which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci-Harbourne developed methods to address this problem, which involve asymptotic numerical characters of symbolic powers of the ideals. Most of the work done up to now has been done for ideals defining 0-dimensional subschemes of projective space. Here we focus on certain subschemes given by a union of lines in ${\bf P}^3$ which can also be viewed as points in ${\bf P}^1\times {\bf P}^1$. We also obtain results on the closely related problem, studied by Hochster and by Li-Swanson, of determining situations for which each symbolic power of an ideal is an ordinary power.

math.AC↗
arXiv2011

Separators of Arithmetically Cohen-Macaulay fat points in P^1 x P^1

Let Z be a set of fat points in P^1 x P^1 that is also arithmetically Cohen-Macaulay (ACM). We describe how to compute the degree of a separator of a fat point of multiplicity m for each point in the support of Z using only a numerical description of Z. Our formula extends the case of reduced points which was previously known.

math.AC↗
arXiv2010

Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals

There is a natural one-to-one correspondence between squarefree monomial ideals and finite simple hypergraphs via the cover ideal construction. Let H be a finite simple hypergraph, and let J = J(H) be its cover ideal in a polynomial ring R. We give an explicit description of all associated primes of R/J^s, for any power J^s of J, in terms of the coloring properties of hypergraphs arising from H. We also give an algebraic method for determining the chromatic number of H, proving that it is equivalent to a monomial ideal membership problem involving powers of J. Our work yields two new purely algebraic characterizations of perfect graphs, independent of the Strong Perfect Graph Theorem; the first characterization is in terms of the sets Ass(R/J^s), while the second characterization is in terms of the saturated chain condition for associated primes.

math.AC↗
arXiv2010

Potentially Nilpotent Patterns and the Nilpotent-Jacobian Method

A nonzero pattern is a matrix with entries in {0,*}. A pattern is potentially nilpotent if there is some nilpotent real matrix with nonzero entries in precisely the entries indicated by the pattern. We develop ways to construct some potentially nilpotent patterns, including some balanced tree patterns. We explore the index of some of the nilpotent matrices constructed,and observe that some of the balanced trees are spectrally arbitrary using the Nilpotent-Jacobian method. Inspired by an argument in [R. Pereira, Nilpotent matrices and spectrally arbitrary sign patterns. Electron. J. Linear Algebra, 16 (2007), 232--236], we also uncover a feature of the Nilpotent-Jacobian method. In particular, we show that if N is the nilpotent matrix employed in this method to show that a pattern is a spectrally arbitary pattern, then N must have full index.

math.RA↗
arXiv2010

Separators of fat points in P^n

In this paper we extend the definition of a separator of a point P in P^n to a fat point P of multiplicity m. The key idea in our definition is to compare the fat point schemes Z = m_1P_1 + ... + m_iP_i + .... + m_sP_s in P^n and Z' = m_1P_1 + ... + (m_i-1)P_i + .... + m_sP_s. We associate to P_i a tuple of positive integers of length v = deg Z - deg Z'. We call this tuple the degree of the minimal separators of P_i of multiplicity m_i, and we denote it by deg_Z(P_i) = (d_1,...,d_v). We show that if one knows deg_Z(P_i) and the Hilbert function of Z, one will also know the Hilbert function of Z'. We also show that the entries of deg_Z(P_i) are related to the shifts in the last syzygy module of I_Z. Both results generalize well known results about reduced sets of points and their separators.

math.AC↗
arXiv2010

Separators of fat points in P^n x P^m

We introduce definitions for the separator of a fat point and the degree of a fat point for a fat point scheme Z in P^n x P^m, and we study some of their properties.

math.AC↗
arXiv2010

A conjecture on critical graphs and connections to the persistence of associated primes

We introduce a conjecture about constructing critically (s+1)-chromatic graphs from critically s-chromatic graphs. We then show how this conjecture implies that any unmixed height two square-free monomial ideal I, i.e., the cover ideal of a finite simple graph, has the persistence property, that is, Ass(R/I^s) \subseteq Ass(R/I^{s+1}) for all s >= 1. To support our conjecture, we prove that the statement is true if we also assume that χ_f(G), the fractional chromatic number of the graph G, satisfies χ(G) -1 < χ_f(G) <= χ(G). We give an algebraic proof of this result.

math.AC↗
arXiv2010

Star configuration points and generic plane curves

Consider l lines in P^2 such that no three lines meet in a point. Let X(l) denote all points of intersections of these l lines. We describe all pairs (d,l) such that generic degree d curve in P^2 contains a X(l).

math.AG↗
arXiv2010

Associated primes of monomial ideals and odd holes in graphs

Let $G$ be a finite simple graph with edge ideal $I(G)$. Let $J(G)$ denote the Alexander dual of $I(G)$. We show that a description of all induced cycles of odd length in $G$ is encoded in the associated primes of $J(G)^2$. This result forms the basis for a method to detect odd induced cycles of a graph via ideal operations, e.g., intersections, products and colon operations. Moreover, we get a simple algebraic criterion for determining whether a graph is perfect. We also show how to determine the existence of odd holes in a graph from the value of the arithmetic degree of $J(G)^2$.

math.AC↗
arXiv2009

Algebraic properties of the path ideal of a tree

The path ideal (of length t >=2) of a graph G is the monomial ideal, denoted I_t(G), whose generators correspond to the directed paths of length t in G. We study some of the algebraic properties of I_t(G) when G is a tree. We first show that I_t(G) is the facet ideal of a simplicial tree. As a consequence, the quotient ring R/I_t(G) is always sequentially Cohen-Macaulay, and the Betti numbers of R/I_t(G) do not depend upon the characteristic of the field. We study the case of the line graph in greater detail at the end of the paper.

math.AC↗