Research - the questions that guide us 

I study the behavior of families of algebro-geometric objects and the structure of the spaces that parametrize them. Within this context, my work has three primary focuses: 

Next, I list my work in such areas. 

Moduli spaces of points and lines in affine and projective space

21) Polymatroids and moduli of points in flags with Javier González-Anaya, José Luis González (submitted) 

20) Higher-dimensional Losev-Manin spaces and their geometry with Javier González-Anaya, José Luis González, and Evangelos Routis (submitted)

19) An explicit wall crossing for the moduli space of hyperplane arrangements with Luca Schaffler. To appear in the Journal of the London Mathematical Society.

18) Eigenperiods and the moduli of points in the line with Haohua Deng. Nagoya Mathematical Journal (online 2025:1-24. doi:10.1017/nmj.2025.7)

17) Wonderful compactifications of the moduli space of points (joint with E. Routis). European Journal of Mathematics, 3(3), 520-564. 

16) Variation of stability for moduli spaces of unordered points in the plane (with B. Schmidt), Transactions of the American Mathematical Society, (2022)

15) The Fulton-MacPherson compactification is not a Mori dream space (with José Luis González and Evangelos Routis), Math Z, 2022.

14) Modular interpretation of a non-reductive Chow quotient (with N. Giansiracusa).  Proceedings of the Edinburgh Mathematical Society.

13) A slice of the moduli space of lines arrangements (joint with K. Ascher)  Algebra and Number theory.

Explicit Descriptions of Algebraic Varieties

12) Unimodal singularities and boundary divisors in the KSBA moduli of a class of Horikawa surfaces (with L. Schaffler, G. Pearlstein, Z. Zhang). Math Nachrichten, (2023)

11) Algebraic and analytic compactifications of moduli spaces (with M. Kerr). Notices of the American Mathematical Society 69.9 (2022).

10)  Geometric interpretation of toroidal compactifications of moduli of points in the line and cubic surfaces (with M. Kerr and L. Schaffler)  Advances in Mathematics, 381, 2021.

9) Applications of the moduli continuity method to log K-stable pairs (with J. Martinez-Garcia and C. Spotti) Journal of the London Mathematical Society 103.2 (2021): 729-759.

8) Moduli of cubic surfaces and their anticanonical divisors  (with J. Martinez-Garcia) Revista Matematica Complutense (RMC) 32.3 (2019): 853-873

7) Families of elliptic curves in Projective space and Bridgeland Stability  (with C. Lozano-Huerta, and B. Schmidt), Michigan Mathematical Journal.

6) On the GIT quotient of quintic surfaces. Transactions of the American Mathematical Society. 371 (2019), 4251-4275.

5)  Compactifications of the moduli space of plane quartics and two lines (with Z. Zhang and J. Martinez-Garcia) European Journal of Mathematics, 2018, Volume 4, Issue 3.

Software and computational tools

4) Computation of GIT quotients of semisimple groups with Jesus Martinez-Garcia, Han-Bom Moon, and David Swinarski (submitted)

3) Enumeration of max-pooling responses with generalized permutohedra  (with Laura Escobar, Javier González-Anaya, José L. González, Guido Montúfar, Alejandro H. Morales). (submitted)

2) Quivers and moduli of their thin sincere representations in Macaulay2   With Mary Barker, we developed a Macaulay2 package called ThinSincereQuivers,  which describes the geometry and moduli spaces of thin-sincere representations of acyclic quivers, i.e., the dimension vector is all equal to one (submitted).

1) VGIT for pairs, a computational approach (with J. Martinez-Garcia). The article appeared in the Proceedings of the AMS.  We also wrote a package in Python called Variations of GIT quotients. It calculates the computational information required to describe the GIT quotients, parametrizing a pair defined by a hypersurface and a hyperplane.