• Department of Math & Stats
• Room 519, Jeffery Hall
• 613 533 2392
• Andrew H. Hoefel
• Post-Doctoral Researcher
• Queen's University

# Research

My general interests are in combinatorial aspects of commutative algebra. I've done work on Stanley-Reisner complexes of certain homological types, monomial ideals with extremal Hilbert functions, and powers of edge ideals with linear resolutions.

I'm currently working on various conjectures that compare symbolic powers to regular powers of fat point ideals. I also have been looking at moduli spaces of hyperplane arrangements and Hilbert functions of Artinian Gorenstein ideals which are invariant under the action of $\sym_n$. See my research statement for a detailed look at what I'm working on.

# Projects

The following is an overview of various projects I'm working on.

## Moduli Spaces of Hyperplane Arrangements

Greg Smith and I have been examining the equations of toric varieties associated to hyperplane arrangements through the Gelfand-MacPherson correspondence. Take a class of ordered hyperplane arrangements $[\mathcal A]_{\sim}$ (of $n$ hyperplanes in $\mathbb P^{r-1}$) where $\mathcal A \sim \mathcal A'$ if $\mathcal A$ and $\mathcal A'$ are equal after a change coordinates (i.e., after multiplying the matrix of $\mathcal A$ by $P \in PGL(r)$). The equation of a hyperplane is determined only up to a scalar multiple. Thus, if $h_1, \ldots, h_n$ are equations for $\mathcal A$, then so are $\lambda_1 h_1, \ldots, \lambda_n h_n$. If we put these equations in a matrix, $\left[ \begin{matrix} \lambda_1 h_1 \\ \lambda_2 h_2 \\ \vdots \\ \lambda_n h_n \end{matrix} \right]_{n\times r}$ then the column spaces of these matrices sweep out a torus in the Grassmannian $G(r,n)$ as the $\lambda_i$ vary.

This relationship between classes of hyperplanes $[\mathcal A]$ and tori in $G(r,n)$ is the Gelfand-MacPherson correspondence: $(\mathbb P^{r-1})^n / PGL(r) \cong G(r,n)/(\field^*)^n.$ What we'd like is a way to describe the closure of the torus in $G(r,n)$ (the toric variety) associated to class of arrangements $[A]$ in terms of the combinatorics of $\mathcal A$. There are a number of ways we might answer this question. I'm keeping some rough notes to help keep track of what we find.

## Hilbert Functions

The Hilbert function of a graded module $M$ is $H_M(d) = \dim M_d$. Macaulay's theorem describes all Hilbert functions of quotients of the polynomial ring, while the Kruskal-Katona Theorem describes Hilbert functions of quotients of the exterior algebra (equiv., $f$-vectors of simplicial complexes). There are a number of other rings where analogous results hold and it would nice to understand what they have in common.

Another problem I've been thinking about is how to determine the maximum degree of a generator in the lexification $L$ of a homogeneous ideal $I$. For a given ideal, there are a number of interesting degrees where the behaviour of the Hilbert function changes, or becomes determined in some way. Here's a list:

• $D_I$, the maximum degree of a generator of $I$;
• $D_L$, the maximum degree of a generator of the lexification of $I$;
• $D_G$, the minimum degree greater than or equal to $D_I$ where $I_d$ is Gotzmann for $d \geq D_G$ (this equals $D_L$);
• $D_\beta$, the maximum degree of a syzygy of $I$;
• $D_\beta - (n-1)$, the smallest degree where no cancellation of Betti numbers is needed to give $H_I(d) = HP_I(d)$;
• $D_F$, the number of binomials occuring in a certain factorization of $HP_I$;
• $D_S$, the regularity of the saturation of $I$;
• $D_P$, the smallest degree where $H_I(d) = HP_I(d)$ for $d \geq D_P$;

The problem of understanding how these invariants relate is certainly interesting. I suspect, however, that many of these relationships are known since we have so many tools at our disposal. As I read more and plug away at the details, I'll record the answers I find in a section on Gotzmann Regularity.