On Castryck and Cools's gonality conjecture.
Jan Draisma
This talk describes a beautiful conjecture by Belgian mathematicians
Castryck and Cools, which concerns the minimal degree of a non-constant
rational function (=the "gonality") on a plane curve with prescribed
Newton polygon Delta. The so-called "lattice width" of Delta is an easy
upper bound on the gonality, and Castryck and Cools conjecture that this
upper bound is attained for general curves with Newton polygon Delta
(modulo two well-understood exceptions). Using Matt Baker's
specialisation lemma and chip-firing, they deduce their conjecture from
a second, purely combinatorial conjecture, which they prove in many
interesting cases. There will be homework for the audience!