Algebraic and Tropical Moduli Spaces and Brill-Noether Theory

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Algebraic geometers study spaces that are (locally) defined by polynomial equations. One powerful method of studying these spaces, which has seen exciting recent developments, is the method of degenerations. Applying this method allows to reduce complicated geometric objects to configurations of simple ones replacing some geometric aspects by combinatorial ones. Such degenerations can be useful for study of geometric problems: the very rough philosophy is that one studies the combinatorics (i.e., the discrete data) of the pieces in order to deduce things about the more complicated original space. The research supported by this award will center on using modern degeneration techniques, especially those from the field of tropical geometry, to study classical spaces from algebraic geometry. The main goal of tropical geometry is transforming questions about algebraic varieties into questions about polyhedral complexes. A process called tropicalization attaches a polyhedral complex to an algebraic variety. The polyhedral complex, a combinatorial object, encodes some of the geometry of the original algebraic variety. The first main project involves using new techniques from tropical geometry and combinatorial topology to compute top-weight rational cohomology of the moduli space of curves, finding explicit new cohomology classes therein. A complementary aspect of this project is to develop stack-theoretic foundations for tropical moduli spaces. The second project studies the geometry of Brill-Noether varieties of curves, i.e. moduli spaces of linear series on curves. The approach builds on recent advances in the moduli theory of limit linear series, and will yield a refined understanding of Brill-Noether varieties. This project will also uncover further connections between Brill-Noether theory and the combinatorics of graphs and Young tableaux. These connections are then amplified in the third main direction of research, which consists of several combinatorial investigations that will shed additional light on the close connection between graphs and algebraic curves.