While in the 19th century an algebraic curve was synonymous with a one-dimensional subset of projective space specified by polynomial equations, the modern study of curves makes use of the definition of an abstract curve independent of a projective embedding. Brill--Noether theory is the bridge between these two perspectives. The fundamental question: given an abstract curve C, what is the geometry of the space of maps of C to projective space with certain invariants?
As a crowning achievement of the modern study of linear series in the 1980s, this geometry is well-understood when the curve C is sufficiently generic. However, in nature, curves are often encountered via a realization specified by polynomial equations of relatively small degree, which might force the curve to be too special for the classic Brill--Noether theorem to apply. In this talk, I will discuss joint work with Eric Larson and Hannah Larson which provides the first complete analogue of all of the main theorems of Brill--Noether theory when the curve is equipped with a low degree map to the line. Speaker(s): Isabel Vogt (Brown University)