Dhruv Ranganathan

Tropical compactification and the Gromov–Witten theory of \mathbb {P}^1

We use tropical and non-archimedean geometry to study the moduli space of genus 0, stable maps to

Brill–Noether theory of maximally symmetric graphs

\mathbb {P}^1

Moduli of rational curves in toric varieties and non-Archimedean geometry

P1 relative to two points. This space is exhibited as a tropical compactification in a toric variety. Moreover, the fan of this toric variety may be interpreted as a moduli space for tropical relative stable maps with the same discrete data. As a consequence, we confirm an expectation of Bertram and the first two authors, that the tropical Hurwitz cycles are tropicalizations of classical Hurwitz cycles. As a second application, we obtain a full descendant correspondence for genus 0 relative invariants of

Superabundant Curves and the Artin Fan

\mathbb {P}^1

Toric Symmetry Of CP3

P1.

Brill-Noether theory for curves of a fixed gonality

We develop a theory of multi-stage degenerations of toric varieties over finite rank valuation rings, extending the Mumford--Gubler theory in rank one. Such degenerations are constructed from fan-like structures over totally ordered abelian groups of finite rank. Our main theorem describes the geometry of successive special fibers in the degeneration in terms of the polyhedral geometry of a system of recession complexes associated to the fan.