Students' Algebraic Geometry Seminar : Brill-Noether theory

The most fundamental invariant of a curve in projective space is its degree. This is the number of intersection points of the curve with a general hyperplane and can be thought of roughly as “how many twists the curve has”. Given an abstract curve $C$, it is natural to ask for which $d$ and $r$ does $C$ admit a degree $d$ map into $\mathbb{P}^r$. The Brill-Noether theorem gives a complete answer to this question for a general curve as well as a partial answer for any curve. We will discuss the key ideas of the proof, namely the reduction of the problem to studying a map of vector bundles on the Picard scheme of $C$.