My research is in algebraic and tropical geometry. I use tools from combinatorics and non-archimedean geometry to explore the geometry of curves, moduli spaces, and enumerative geometry.
This is a survey paper that provides a self-contained introduction to the theory of chip-firing games, leading up to the recent theory of tropical Prym varieties. It contains plenty of examples, figures, exercises, and open problems. It is an extension of the notes developed for the 2022 Cambridge summer school on combinatorial methods in algebraic geometry.
17. Tropical tangents for complete intersection curves (with Nathan Ilten), Mathematics of Computation (2023).
We introduce a tropical version of the Gauss map, which sends a point of a curve to its tangent line (regarded as a point in Grassmannian). As a result, under mild assumptions, we describe the tropical dual and tangential variety of a complete intersection curve.
16. Kirchhoff's theorem for Prym varieties (with Dmitry Zakharov, and appendix by Sebastian Casalaina-Martin), Forum of Mathematics, Sigma (2022).
We prove an analogue of Kirchhoff's matrix tree theorem for computing the order of the Prym group of a free double cover of graphs, and the volume of the tropical Prym variety in the metric setting. We interpret the latter result in terms of a semi-canonical decomposition of the tropical Prym variety, via a careful study of the tropical Abel--Prym map. In particular, we show that the map is harmonic, determine its degree at every cell of the decomposition, and prove that its global degree is 2g-1. As a counterpart, the appendix by Sebastian Casalaina-Martin shows that the degree of the algebraic Abel--Prym map is 2g-1 as well.
15. Prym–Brill–Noether loci of special curves (with Steven Creech, Caelan Ritter, and Derek Wu), International Mathematics Research Notices (2020).
We use tropical techniques to obtain new bounds on algebraic Brill–Noether loci in Prym varieties. In addition, we prove tropological (topological properties of tropical varieties) results such as equi-dimensionality and path connectedness of such loci.
We classify invariant G-covers of tropical curves, namely unramified covers of a fixed tropical base curve Γ, with an action of a finite abelian group G that acts transitively on the fibres. We introduce the notion of a dilated cohomology group for a tropical curve Γ, which generalizes simplicial cohomology groups of Γ with coefficients in G, by allowing nontrivial stabilizers at vertices and edges. We show that G-covers of Γ with a given collection of stabilizers are in natural bijection with the elements of the corresponding first dilated cohomology group of Γ.
13. Skeletons of Prym varieties and Brill–Noether theory (with Martin Ulirsch), Algebra & Number Theory (2020).
The paper is concerned with Prym varieties - a class of Abelian varieties that appears in the presence of double covers. We show that the Prym construction commutes with tropicalization, thus confirming a conjecture by Dave Jensen and myself. As an application, we prove new bounds on the dimensions of Brill–Noether loci in Prym varieties.
12. Lifting tropical self intersections (with Matt Satriano), Journal of Combinatorial Theory Series A (2019).
Our goal in the paper is to investigate the discrepancy between tropicalization of intersection and intersection of tropicalization. We focus on the case of algebraic curves with the same tropicalization, and show that the set of tropical divisors that arise in this manner is a pure dimensional balanced polyhedral complex and compute its dimension. When the genus is at most 1, we show that all the cells of the linear system that have this dimension are realizable. Some of the tools that we introduce as part of the proof are interesting in their own right, and may be useful for explicitly constructing large families of realizable tropical divisors.
11. Projective duals to algebraic and tropical hypersurfaces (with Nathan Ilten), Proceedings of the London Mathematical Society (2019).
We describe the dual varieties of tropical planes curves and tropical surfaces in 3-space, and provide a partial description in higher dimension. As a result, we find the transformation of Newton polygons of curves under projective duality, and recover classical formulas for the degree of dual plane curves.
We classify tropical bitangents to plane curves that lift to algebraic bitangents, along with their lifting multiplicity. Using this framework, we show that each of the 7 bitangents of a smooth tropical plane quartic lifts in sets of 4 to algebraic bitangents. This implies a weak Plücker theorem: a generic tropically smooth quartic admits 28 bitangent lines. We also examine how to extend our methods to count real bitangents.
We show that with appropriate natural multiplicities, every tropical plane quartic (either smooth or not) has 7 equivalence classes of bitangent lines. Moreover, the multiplicity of bitangent lines varies continuously in families of tropical plane curves.
8. Tropicalization of theta characteristics, double covers, and Prym varieties (with Dave Jensen), Selecta Mathematica, (2018).
We study the behaviour of theta characteristics on an algebraic curve under the specialization map to a tropical curve. We show that each effective theta characteristic on the tropical curve is the specialization of 2g−1 even theta characteristics and 2g−1 odd theta characteristics. We then study the relationship between unramified double covers of a tropical curve and its theta characteristics, and use this to define the tropical Prym variety.
7. Tritangent planes to space sextics: the algebraic and tropical stories (with Corey Harris), Combinatorial Algebraic Geometry, Fields Institute Communications (2017).
We study planes tangent to general canonical sextic curves at three points. We determine the number of real tritangents when such a curve is real. We then revisit a curve constructed by Emch with the greatest known number of real tritangents, and conversely construct a curve with very few real tritangents. Using recent results on the relation between algebraic and tropical theta characteristics, we show that the tropicalization of a canonical sextic curve has 15 tritangent planes.
We prove a version of Clifford's theorem for metrized complexes. Namely, a metrized complex that carries a divisor of degree 2r and rank r (for 0<r<g−1) also carries a divisor of degree 2 and rank 1. We provide a structure theorem for hyperelliptic metrized complexes, and use it to classify divisors of degree bounded by the genus. We discuss a tropical version of Martens' theorem for metric graphs.
5. Enumerative geometry of elliptic curves on toric surfaces (with Dhruv Ranganathan), Israel Journal of Mathematics (2017).
We establish the equality of classical and tropical curve counts for elliptic curves on toric surfaces with fixed j-invariant, refining results of Mikhalkin and Nishinou–Siebert. As an application, we determine a new formula for such counts on Hirzebruch surfaces. This formula relates the count of elliptic curves with the number of rational curves on the surface satisfying a small number of tangency conditions with the toric boundary. Furthermore, the combinatorial tropical multiplicities of Kerber and Markwig for counts on ℙ2 are given algebro-geometric interpretations. As a consequence, a new proof of Pandharipande's formula for counts of elliptic curves in ℙ2 with fixed j-invariant is obtained.
We show that the algebraic rank of divisors on certain graphs is related to the realizability problem of matroids. As a consequence, we produce a series of examples in which the algebraic rank depends on the ground field. We use the theory of metrized complexes to show that equality between the algebraic and combinatorial rank is not a sufficient condition for smoothability of divisors, thus giving a negative answer to a question posed by Caporaso, Melo, and the author.
3. Bitangents of tropical plane quartic curves (with Matt Baker, Ralph Morrison, Nathan Pflueger and Qingchun Ren), Mathematische Zeitschrift (2016).
We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic geometry. For example, we show that every such curve admits either infinitely many or exactly 7 bitangent lines. We also prove that a smooth tropical plane quartic curve cannot be hyperelliptic.
2. Algebraic and combinatorial rank of divisors on finite graphs (with Lucia Caporaso and Margarida Melo), Journal de Mathématiques Pures et Appliquées (2015).
We study the algebraic rank of a divisor on a graph, an invariant defined using divisors on algebraic curves dual to the graph. We prove it satisfies the Riemann–Roch formula, a specialization property, and the Clifford inequality. We prove that it is at most equal to the (usual) combinatorial rank, and that equality holds in many cases, though not in general.
We construct a space classifying divisor classes of a fixed degree on all tropical curves of a fixed combinatorial type and show that the function taking a divisor class to its rank is upper semicontinuous. We extend the definition of the Brill–Noether rank of a metric graph to tropical curves and use the upper semicontinuity of the rank function on divisors to show that the Brill–Noether rank varies upper semicontinuously in families of tropical curves. Furthermore, we present a specialization lemma relating the Brill–Noether rank of a tropical curve with the dimension of the Brill–Noether locus of an algebraic curve.
Collaborators: Matt Baker, Lucia Caporaso, Corey Harris, Nathan Ilten, Dave Jensen, Margarida Melo, Ralph Morrison, Nathan Pflueger, Dhruv Ranganathan, Qingchun Ren, Matt Satriano, Martin Ulirsch, Dmitry Zakharov.