Geometry Seminar

The Hurwitz space H means the space of all covers of CP1 with given degree D and simple branching over N points. It is a non-Galois unramified cover of the moduli space M of unordered N-tuples in CP1. We explain the monodromy action of the base on the fiber, in the case D=4. Equivalently, we find the Galois group of the Galois cover associated to H-->M.

Recent work of Sturmfels and Xu establishes an intriguing connection between the Hilbert functions of important projective varieties from phylogenetic algebraic geometry and a genus 0, sl_2(\C) case of the celebrated Verlinde formula from mathematical physics. We will discuss how this relationship can be constructed and generalized with the representation theory of affine Kac-Moody algebras and the associated theory of conformal blocks.

A (minimal) complex algebraic manifold is of general type if its canonical class (the determinant line bundle of the cotangent bundle) is positive. This is the typical behaviour, often interpreted loosely as being the hyperbolic case. Such manifolds have canonical embeddings into (weighted) projective space, simplifying questions of isomorphism and classification. Given equations of a manifold in some embedding, it is possible to compute the canonical embedding using various computer algebra packages. The images of these embeddings are defined by Gorenstein systems of equations. The easiest cases of this are when the image is cut out by one or two independent equations. Such complete intersection 3-folds of general type have been the subject of lists (by Reid and Fletcher in the 1980s) that were recently shown to be complete (by Chen-Chen-Chen). Turning the question around, I consider more complicated systems of Gorenstein equations, and construct analogous classifications of orbifold 3-folds in higher codimension.