A Brill-Noether theorem for curves of a fixed gonality

Algebra Seminar
Monday, March 6, 2017 - 15:00
1 hour (actually 50 minutes)
Skiles 005
The Brill-Noether varieties of a curve C parametrize embeddings of C of prescribed degree into a projective space of prescribed dimension. When C is general in moduli, these varieties are well understood: they are smooth, irreducible, and have the “expected” dimension. As one ventures deeper into the moduli space of curves, past the locus of general curves, these varieties exhibit intricate, even pathological, behaviour: they can be highly singular and their dimensions are unknown. A first measure of the failure of a curve to be general is its gonality. I will present a generalization of the Brill—Noether theorem, which determines the dimensions of the Brill—Noether varieties on a general curve of fixed gonality, i.e. “general inside a chosen special locus". The proof blends a study of Berkovich skeletons of maps from curves to toric varieties with tropical linear series theory. The deformation theory of logarithmic stable maps acts as the bridge between these ideas. This is joint work with Dave Jensen.