Geometry, Topology, and Physics Seminar, Winter 2016

Part of the NSF/UCSB ‘Research Training Group’ in Topology and Geometry

Organizers: Dave Morrison and Zhenghan Wang.
Meets 4:00 - 5:30 p.m. Fridays in South Hall 6635.

January 29

### Elliptic Calabi-Yau torsors

Abstract: Every elliptic fibration $\pi: X \to B$ with a rational section determines an elliptic curve $E$ defined over the function field $K=K(B)$ of the base; if two elliptic fibrations determine the same elliptic curve, then they are birationally equivalent. As a group-scheme over $K$, $E$ may admit "torsors," i.e., projective curves $C$ over $K$ with a transitive action $E \times C \to C$ having trivial stabilizers. The question we will address is: if $X$ is an elliptically fibered Calabi-Yau variety,and $C/K$ is a torsor for the associated elliptic curve $E/K$, when does $C/K$ have a birational model which is itself a Calabi-Yau variety? The question is an important one to answer for application to F-theory.

I will discuss various partial results concerning this question, some old and some new. I will also briefly describe the application to F-theory.

February 12

### Elliptic Calabi-Yau torsors, II

Abstract: Every elliptic fibration $\pi: X \to B$ with a rational section determines an elliptic curve $E$ defined over the function field $K=K(B)$ of the base; if two elliptic fibrations determine the same elliptic curve, then they are birationally equivalent. As a group-scheme over $K$, $E$ may admit "torsors," i.e., projective curves $C$ over $K$ with a transitive action $E \times C \to C$ having trivial stabilizers. The question we will address is: if $X$ is an elliptically fibered Calabi-Yau variety,and $C/K$ is a torsor for the associated elliptic curve $E/K$, when does $C/K$ have a birational model which is itself a Calabi-Yau variety? The question is an important one to answer for application to F-theory.

I will discuss various partial results concerning this question, some old and some new. I will also briefly describe the application to F-theory.

February 19

### Brauer groups and elliptic torsors

Abstract: We continue our discussion of torsors for elliptic fibrations, relating them to the Brauer group of the total space of the fibration (following Dolgachev and Gross). Explicit examples will be given.

March 11

### More about elliptic torsors

Abstract: We continue our discussion of torsors for elliptic fibrations.