Since 2021 I have been conducting research in algebraic geometry. Before then I was primarily interested in category theory.

Publications

A preprint of my upcoming paper, Compact Moduli Spaces of Marked Cubic Curves, can be found at the bottom of this page or by clicking on this link.

For the curious,

algebraic geometers study shapes which can be defined as the set of solutions to polynomial equations. For instance, the zero set of a single polynomial in two variables, \(f(x,y)\), is the set of all points \((x,y)\) such that \(f(x,y)=0\). This defines a curve in the plane, such as the circle of radius \(r\) which is the set of solutions to \(x^2 + y^2 - r^2 = 0\). When the polynomial \(f(x,y)\) changes, so does the curve defined by \(f(x,y) =0\). By fixing a degree \(d\) and treating the coefficients \(c_{i,j}\) of

\[f(x,y) = c_{0,0} + c_{1,0} x + c_{0,1} y + c_{2,0} x^2 + c_{1,1} xy + c_{0,2} y^2 + c_{3,0} x^3 + \dots + c_{0,d}y^d\]

as variables we obtain a space of parameters in which every point corresponds to a plane curve. Questions of interest include: How does the geometry of the parameter space reflect the geometry of the curves it parametrizes? What are the ways in which the generic curves with well behaved geometry can gradually degenerate? And how are spaces of plane curves related to other (moduli) spaces of geometric objects?

Technically,

in my thesis I construct GIT compactifications of the moduli space of smooth, projective, plane curves of degree \(d\) with \(n\) distinct marked points. These spaces are obtained as geometric quotients of a parameter space of marked plane curves \(\mathcal{C}_{n,d} \subset \mathbb{P}(\Gamma (\mathcal{O}(d), \mathbb{P}^2 )) \times (\mathbb{P}^2)^n\) by the group \(SL(3)\) acting as linear automorphisms of the plane. To obtain a geometric quotient \(\overline{M}_{g(d),\vec{w}}^{git} := \mathcal{C}_{n,d}^{ss} //_{\vec{w}} SL(3)\) one must first remove from \(\mathcal{C}_{n,d}\) a closed locus of unstable marked curves. It is of interest that there are several choices of stability conditions one can make to determine which locus to remove from \(\mathcal{C}_{n,d}\), corresponding to a choice of an \(SL(3)-\) linearized line bundle on \(\mathcal{C}_{n,d}\). The foundational research of Thaddeus and Dolgachev and Hu on Variations of Geometric Invariant Theory quotients (VGIT) describes how such a choice of line bundle corresponds to the choice of a vector \(\vec{w}\) in a finite-dimensional, rational, polyhedral cone. This cone is divided by hyperplanes into a finite set of convex, polyhedral chambers such that \(\vec{w_1}\) and \(\vec{w_2}\) lie in the same chamber if and only if \(\overline{M}_{g(d), \vec{w_1}}^{git} \cong \overline{M}_{g(d),\vec{w_2}}^{git}\) . Furthermore, if \(\vec{w_2}\) and \(\vec{w_3}\) lie in adjacent chambers then there are explicit birational morphisms \(\overline{M}_{g(d),\vec{w_2}}^{git} \dashleftarrow \dashrightarrow \overline{M}_{g(d),\vec{w_3}}^{git}\) closely related to the birational morphisms which appear in the Minimal Model Program of algebraic geometry. I give explicit description of the cone of line bundles \(\vec{w}\) that give rise to a suitable moduli space \(\overline{M}_{g(d), \vec{w}}^{git}\). Special attention is paid to the case \(d=3\) of marked elliptic curves. In this case I give a complete description of the wall and chamber decomposition and the wall-crossing behavior. I also discuss how this space is related to other moduli spaces such as \(\overline{M}_{1,n}\) and the moduli space of cubic surfaces with a marked Eckhardt point.