# Research

# Pre-prints

## Connectedness of the moduli space of all reduced curves

We prove that the moduli space of all reduced n-pointed algebraic curves of fixed arithmetic genus is connected as an application of moduli of equinormalized curves and Ishii's theory of territories.

## A stratification of moduli of arbitrarily singular curves

with Christopher Guevara and David Smyth

Submitted

We introduce a new moduli stack "equinormalized curves," closely related to the moduli space of all reduced, connected algebraic curves. We construct a stratification of the moduli space of equinormalized curves indexed by generalized dual graphs and prove

that each stratum is a fiber bundle over a finite quotient of a product of M_g,n's. The fibers are moduli schemes parametrizing subalgebras of a fixed algebra, and are in principle explicitly computable as locally closed subschemes of products of Grassmannians. We thus obtain a remarkably explicit geometric description of moduli of reduced curves with arbitrary singularities.

# Papers

## Hyperelliptic Gorenstein curves and logarithmic differentials

with Luca Battistella

Accepted in Forum of Mathematics, Sigma

We produce a flexible tool for contracting subcurves of logarithmic hyperelliptic curves, which is local around the subcurve and commutes with arbitrary base-change. As an application, we prove that hyperelliptic multiscale differentials determine a sequence of Gorenstein contractions of the underlying nodal curve, whose dualising bundle they descend to generate. This is the first piece of evidence for a more general conjecture about limits of differentials.

## On Compactifications of Mg,n with Colliding Markings

One way of forming modular compactifications of the space of n-pointed smooth algebraic curves is by allowing marked points to collide, as in the spaces of weighted pointed stable curves constructed by Hassett. This paper studies all such ways of constructing compactifications by allowing markings to collide.

We find that for any modular compactification, collisions of markings are controlled by a simplicial complex which we call the collision complex. Conversely, we identify modular compactifications with essentially arbitrary collision complexes, including complexes not associated to any space of weighted pointed stable curves. These moduli spaces classify the modular compactifications by nodal curves with smooth markings as well as the modular compactifications in genus one with Gorenstein curves and smooth markings.

## The Scheme of Monogenic Generators II: Local Monogenicity and Twists

with Sarah Arpin, Leo Herr, and Hanson Smith

Published May 2023 in Research in Number Theory

By the results of the first paper in this series, a choice of a generator for an A-algebra B is a point of a scheme. In this paper, we study and relate several notions of local monogenicity that emerge from this perspective. We first consider the conditions under which the extension B/A admits monogenerators locally in the Zariski and finer topologies, recovering a theorem of Pleasants as a special case. We next consider the case in which B/A is etale, where the local structure of etale maps allows us to construct a universal monogenicity space and relate it to an unordered configuration space. Finally, we consider when B/A admits local monogenerators that differ only by the action of some group (usually the multiplicative group or affine transformation group), giving rise to a notion of twisted monogenerators. In particular, we show a number ring A has class number one if and only if each twisted monogenerator is in fact a global monogenerator.

## The Scheme of Monogenic Generators I: Representability

with Sarah Arpin, Leo Herr, and Hanson Smith

Published January 2023 in Research in Number Theory

Given an extension of algebras B/A, when is B generated by a single element of B over A? In this paper, we show there is a scheme parameterizing the choice of a generator in B, a ``moduli space'' of generators. This scheme relates naturally to Hilbert schemes and configuration spaces. We give explicit equations and ample examples.

## A Classification of Modular Compactifications of the Space of Pointed Elliptic Curves by Gorenstein Curves

We classify the Deligne-Mumford stacks compactifying the moduli space of smooth n-pointed curves of genus one using Gorenstein curves with distinct smooth markings. This classification uncovers new moduli spaces, which we may think of coming from an enrichment of the notion of level used to define Smyth's m-stable spaces. Finally, we construct a cube complex of Artin stacks interpolating between the new moduli spaces, a multidimensional analogue of the wall-and-chamber structure seen in the log minimal model program for \overline{M}_g.

## Planar Representations of Group Actions on Surfaces

with Samuel Lippert and Aaron Wootton

Published in 2022 in Contemporary Mathematics volume 776

The space of skeletal signatures was introduced as a simple but generally much coarser two-dimensional representation of the space of all signatures with which a group can act on a compact oriented surface. In the following, we provide a complete characterization of the groups for which the skeletal signature space provides a complete and accurate picture of the structure of the full space of signatures. We show that such groups fall into three distinct families, and for two of these families, we show that for sufficiently large genus, the skeletal signature space depends only on group order, and we explicitly describe these spaces. For the third family, we show that the skeletal signature strongly spends depends upon group structure and provide some partial analysis of this space.

## Contractions of Subcurves of Log Curves

This is a technical paper based on my thesis. The main result is a construction of a contraction of subcurves of families of logarithmic curves compatible with base change from tropical data. Earlier contraction constructions tend to rely on first finding a smoothing family, and it can be obscure how the smoothing family affects the resulting contracted curves. The convenience of this paper's construction enabled the classification of Gorenstein compactifications of M_1,n.

## Rank Drops of Recurrence Matrices

Published August 2015 in Electronic Journal of Linear Algebra

We say a recurrence matrix is a matrix whose terms are sequential members of a linear homogeneous recurrence sequence of order k and whose dimensions are both greater than or equal to k. In this paper, the ranks of recurrence matrices are determined. In particular, it is shown that the rank of such a matrix differs from the previously found upper bound of k in only two situations: When (a_j) satisfies a recurrence relation of order less than k, and when the nth powers of distinct eigenvalues of (a_j) coincide.

## Asymptotic Equivalence of Group Actions on Surfaces and Riemann-Hurwitz Solutions

with Aaron Wootton

Published in 2014 in Archiv der Mathematik

The topological data of a group action on a compact Riemann surface can be encoded using a tuple (h; m 1, ..., m s ) called its signature. There are two number theoretic conditions on a tuple necessary for it to be a signature: the Riemann–Hurwitz formula is satisfied and each m i equals the order of a non-trivial group element. We show on the genus spectrum of a group that asymptotically, satisfaction of these conditions is in fact sufficient. We also describe the order of growth for the number of tuples which satisfy these conditions but are not signatures in the case of cyclic groups.