Brill-Noether theory of curves on ${\mathbb{P}}^1\times {\mathbb{P}}^1$: tropical and classical approach

Abstract

The gonality sequence $(d_r{)}_{r\ge 1}$ of a smooth algebraic curve comprises the minimal degrees $d_r$ of linear systems of rank $r$. We explain two approaches to compute the gonality sequence of smooth curves in ${\mathbb{P}}^1\times {\mathbb{P}}^1$: a tropical and a classical approach. The tropical approach uses the recently developed Brill--Noether theory on tropical curves and Baker's specialization of linear systems from curves to metric graphs. The classical one extends the work of Hartshorne on plane curves to curves on ${\mathbb{P}}^1\times {\mathbb{P}}^1$.

References in corpus (1)

• The sandpile model on K_{m,n} and the rank of its configurations