arXiv subjects

Graham Keiper

Publications and source records attributed to Graham Keiper.

7 recordsLinked to original sources

arXiv2026

Interpolation matrices and jumping lines of logarithmic bundles

We study jumping lines loci of logarithmic bundles associated with finite sets of points in the projective plane. Using the interpolation matrix introduced in [DMTG25], we describe these loci as the zero sets of explicit determinants depending on parameters $(d,m)$ determined by the number of points. We show that for points in general position the determinant defines an irreducible curve of the expected degree, while for special configurations it acquires fixed components related to the combinatorics of the arrangement. The approach provides a new geometric interpretation of the classical jumping lines of Dolgachev--Kapranov and Barth, and connects them to the framework of unexpected curves and hypersurfaces.

math.AG↗

arXiv2025

Splittings of Ideals of Points in $\mathbb{P}^{1}\times\mathbb{P}^{1}$

Let $I_\mathbb{X}$ be the bihomogeneous ideal of a finite set of points $\mathbb{X} \subseteq \mathbb{P}^1 \times \mathbb{P}^1$. The purpose of this note is to consider ``splittings'' of the ideal $I_\mathbb{X}$, that is, finding ideals $J$ and $K$ such that $I_\mathbb{X} = J+K$, where $J$ and $K$ have prescribed algebraic or geometric properties. We show that for any set of points $\mathbb{X}$, we cannot partition the generators of $I_\mathbb{X}$ into two ideals of points. The best case scenario is where at most one of $J$ or $K$ is an ideal of points. To remedy this we introduce the notion of unions of lines and ACM (Arithmetically Cohen-Macaulay) points which allows us to say more about splittings. For a set $\mathbb{W}$ of unions of lines and ACM sets of points, we can write $I_\mathbb{W} = J + K$ where both $J$ and $K$ are ideals of unions of lines and ACM points as well. When $\mathbb{W}$ is a union of lines and ACM points, we discuss some consequences for the graded Betti numbers of $I_{\mathbb{W}}$ in terms of these splittings.

math.AC↗

arXiv2025

Symbolic Powers of Toric Ideals

This paper investigates the symbolic powers of toric ideals. We first describe them in terms of the kernel of certain linear maps derived from the lattice structure of the toric ideal. Furthermore, we apply our results to show that symbolic powers of a toric ideal can also be expressed as saturations of regular powers with the monomial given by the product of all the variables. Finally, we conclude with a computationally significant result for computing symbolic powers of toric ideals.

math.AC↗

arXiv2020

Density of $f$-ideals and $f$-ideals in mixed small degrees

A squarefree monomial ideal is called an $f$-ideal if its Stanley-Reisner and facet simplicial complexes have the same $f$-vector. We show that $f$-ideals generated in a fixed degree have asymptotic density zero when the number of variables goes to infinity. We also provide novel algorithms to construct $f$-ideals generated in small degrees.

math.AC↗

arXiv2020

Regularity and h-polynomials of toric ideals of graphs

For all integers $4 \leq r \leq d$, we show that there exists a finite simple graph $G= G_{r,d}$ with toric ideal $I_G \subset R$ such that $R/I_G$ has (Castelnuovo-Mumford) regularity $r$ and $h$-polynomial of degree $d$. To achieve this goal, we identify a family of graphs such that the graded Betti numbers of the associated toric ideal agree with its initial ideal, and furthermore, this initial ideal has linear quotients. As a corollary, we can recover a result of Hibi, Higashitani, Kimura, and O'Keefe that compares the depth and dimension of toric ideals of graphs.

math.AC↗

arXiv2019

Splittings of Toric Ideals

Let $I \subseteq R = \mathbb{K}[x_1,\ldots,x_n]$ be a toric ideal, i.e., a binomial prime ideal. We investigate when the ideal $I$ can be "split" into the sum of two smaller toric ideals. For a general toric ideal $I$, we give a sufficient condition for this splitting in terms of the integer matrix that defines $I$. When $I = I_G$ is the toric ideal of a finite simple graph $G$, we give additional splittings of $I_G$ related to subgraphs of $G$. When there exists a splitting $I = I_1+I_2$ of the toric ideal, we show that in some cases we can describe the (multi-)graded Betti numbers of $I$ in terms of the (multi-)graded Betti numbers of $I_1$ and $I_2$.

math.AC↗

arXiv2018

Betti numbers of toric ideals of graphs: A case study

We compute the graded Betti numbers for the toric ideal of a family of graphs constructed by adjoining a cycle to a complete bipartite graph. The key observation is that this family admits an initial ideal which has linear quotients. As a corollary, we compute the Hilbert series and $h$-vector for all the toric ideals of graphs in this family.

math.AC↗