Brill-Noether theory via degeneration
Isabel Vogt (Brown)
Problem Sheets:
1 2

The motivating problem is: what is the geometry of the space of degree d maps from an abstract curve C of genus g to projective space of dimension r? The classic Brill-Noether theorem answers this question when the curve C is general in moduli. In this minicourse, we'll take the perspective that smooth curves are hard, and study this problem by degeneration to well-chosen reducible curves. This approach has the advantage of simplifying the geometric complexity, at the expense of introducing combinatorial complexity.

In the first lecture I'll introduce this topic and a nice degeneration to study the Brill-Noether problem. In the second lecture we'll show that the natural moduli space on the central fiber shows an interesting combinatorial structure. In the third lecture, I'll discuss how to recover global information about general curves from this degeneration, namely the theory of limit linear series and Eisenbud-Harris's regeneration theorem. Finally, in the last lecture, we'll move away from general curves to curves equipped with a (low degree) map to the projective line. In this case, the classic Eisenbud-Harris regeneration theorem does not apply, and I'll discuss how, with my collaborators Eric Larson and Hannah Larson, we solve this problem by mining the deep combinatorial structure on the central fiber related to the affine symmetric group.