Matroid-Theoretic Approaches to Compactified Jacobians and Vector Bundles

Abstract

This thesis is composed of three distinct projects lying at the interface of combinatorics and algebraic geometry. Though not directly related to one another, these projects are united by the underlying themes of viewing graphs and matroids as discrete analogs of objects arising in algebraic geometry, and of viewing these combinatorial objects as tools for encoding discrete structure in algebraic-geometric contexts. The first project included in this thesis studies the combinatorics of certain subdivisions of Euclidean spaces into highly-structured polyhedra known as zonotopes. By taking periodic subdivisions of this type and passing to the quotient under the lattice of periodicity, one obtains a class of cell decompositions of real tori. We prove an enumerative result on the $f$-vectors of these decompositions. These subdivisions arise in the context of compactified Jacobians of nodal curves, where the cell structure encodes the inclusions of orbit closures under a torus action. We then use our $f$-vector result to give a formula for the classes of these compactifications in the Grothendieck ring of varieties. The second project is of a more tropical geometric nature, and represents a context in which the categorical and $K$-theoretic understanding of algebraic geometry carries over well into the tropical context. We establish in particular that many categories of matroids arising in modern matroid theory naturally possess the structure of proto-abelian categories, forming non-additive analogs of the abelian categories appearing in many areas of algebra. We extend this result to show that the category of vector bundles in tropical geometry possesses a similar categorical structure, and use this to give an account of tropical Harder-Narasimhan filtrations mirroring those appearing in the moduli theory of algebraic vector bundles. Finally, the third project describes a purely combinatorial result motivated by the Riemann-Roch theory of graphs. We study two group actions of the critical group of a graph on the set of its spanning trees. Prior to our work, it was known that these group actions coincide if this graph is planar, and we resolve a conjecture that the converse also holds. For each non-planar graph, these two actions are necessarily distinct. By viewing the acting group as a subgroup of the tropical Jacobian of the graph, this affords a connection to the first project mentioned above. The background for these projects is considerable and possesses little overlap. For the sake of readability, the thesis contains multiple sections and appendices dedicated to exposition of this background. The second and third projects mentioned above appear independently as papers, and the text of this thesis contains overlap with these documents.