Papers and Preprints
Singular matroid realization spaces. We study smoothness of realization spaces of matroids for small rank and ground set. For $\C$-realizable matroids, when the rank is $3$, we prove that the realization spaces are all smooth when the ground set has $11$ or fewer elements, and there are singular realization spaces for $12$ and greater elements. For rank $4$ and $9$ or fewer elements, we prove that these realization spaces are smooth. As an application, we prove that $\Gr^{\circ}(3,n;\C)$---the locus of the Grassmannian where all Plücker coordinates are nonzero---is not schön for $n>=12$. This work has been submiited to the journal Annali della Scuola Normale Superiore di Pisa, Classe di Scienze. You can read the preprint here.
The Grassmannian of 3-planes in C^8 is schön. We prove that the open subvariety Gr_o(3,8) of the Grassmannian Gr(3,8) determined by the nonvanishing of Plücker coordinates is schön. That is, all its initial degenerations are are smooth. Furthermore, we find an initial degeneration that has two connected components, and show that the remaining initial degenrations, up to symmetry, are irreducible. As an application, we show that the Chow quotient of Gr(3,8) by the diagonal torus of PGL(8) is the log canonical compactification of 8 lines in P^2, resolving a conjecture of Hacking, Keele, and Tevelev. Along the way we develop techniques to study finite inverse limits of schemes. This is joint work with Daniel Corey. This work was accepted for publication by the journal Algebraic Combinatorics. You can read it here. The code for the computations can be found at this Github repository.
Generalized permutahedra and the positive flag Dressian. We show that the height functions that induce subdivisions of the regular permutahedron into cells that are generalized permutaheda can be found in the intersection of tropical hypersurfaces. We also prove under what conditions the cells of these subdivisions are Bruhat interval polytopes. This is joint work with Michael Joswig, Georg Loho, and Jorge Alberto Olarte. This work was accepted for publication by the journal International Mathematics Research Notices. You can read it here.
Boundary complexes of moduli spaces of curves in higher genus. Some of the results from my MA thesis are in a joint paper with Dr. Emily Clader and Kyla Quillin. This paper was accepted to be printed in the Proceedings of the American Mathematical Society. You can read the preprint here.
Boundary divisors on moduli spaces of curves with weighted marked points. My MA thesis, under the advisement of Dr. Emily Clader, was on moduli spaces of smooth curves of genus g with n marked points. As given, such a space is not compact. We can compactify the space by including nodal curves, or using a weight vector. Part of our work investigated the conditions under which codimension 1 subvarieties of the boundary called boundary divisors have nonempty intersection. This is equivalent to a question about when a closely related simplicial complex is a flag complex.