UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

October 06, 2009

939 Evans Hall


3:45PM: Relative Rounding in Toric Geometry

Arthur Ogus

Attached to a toric variety X is a topological space X_\lag on which polar coordinates are well-defined. There is a natural map X_\lag \to X, which is a kind of real blowing up and can greatly simplify singularities: X_\lag is a manifold with boundary. This technique applies also to equivariant mappings between toric varieties, and makes sense globally in the context of log geometry. Our main result says that if f: X --> Y is an exact and smooth morphism of log schemes over C, then the associated map f_\lag :X_\lag --> Y_\lag is a topological submersion whose fibers are orientable manifolds with boundary. Since the result is local, it reduces to the case of affine toric varieties, and my talk will concentrate on this case, so knowledge of log geometry will not be required. The proof depends on a new look at the moment mapping (inspired by Birch's theorem in statistics) and a way to ``force'' functoriality. This is joint work with Chikara Nakayama.

5:00PM: Constructing Curves of Low Genus

Charles Crissman

It is well-known that the moduli space $M_g$ of genus-$g$ curves is unirational for $g\leq 14$. What this means in down-to-Earth terms is that these spaces ``have coordinates,'' in the sense that there is an affine space where a generic point corresponds to a generic curve of genus $g$, and only finitely many points correspond to the same curve. As a simple example, every curve of genus 2 has an equation of the form $y^2 = x(x-1)(x-a)(x-b)(x-c)$ where $0,1,a,b,c$ are distinct. The coordinates a,b,c serve as coordinates for $M_2$ in the sense above.\\ I will discuss how to give coordinates for curves of genus $\leq 10$ by finding nodal plane models or by finding the equations of their canonical embeddings. If time permits, I will venture into the murkier waters of genus 11-14. All necessary facts about $M_g$ will be recalled, so no expertise will be required.

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