## Commutative Algebra and Algebraic Geometry: The Fellowship of the Ring: The Maximal Rank Theorem

Seminar | May 1 | 3:45-5 p.m. | 939 Evans Hall

Joe Harris, Harvard University

Department of Mathematics

The Brill-Noether theorem establishes a fundamental link between the classical notion of a curve in projective space, given as the zero locus of polynomials, and the (relatively) modern notion of an abstract curve. Specifically, it tell us when and how a general abstract curve can be embedded in $\mathbf{P^r}$.

But that’s just the opening line of the story: having embedded our abstract curve in projective space, we can ask about the geometry and algebra of the image. In particular, we ask what sort of polynomial equations define the image — what their degrees are, and how many of them there are. The Maximal Rank Conjecture, recently proved by Eric Larson, gives the answer to this question. In this talk, I’ll describe the moving parts of Larson's proof (several of which are interesting theorems in their own right), and how they fit together to resolve the conjecture.

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