
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.
Other Quarters: [
Spring, 2017;
Wnter, 2017;
Fall, 2016;
Spring, 2016;
Winter, 2016;
Fall, 2015;
Spring, 2015;
Winter, 2014;
Fall, 2013;
Fall, 2012;
Fall, 2011;
Winter, 2011;
Spring, 2010;
Winter, 2010;
Fall, 2009;
Spring, 2009;
Winter, 2009;
Fall, 2008;
Spring, 2008;
Winter, 2008;
Fall, 2007;
Spring, 2007;
Winter, 2007;
Fall, 2006
]
January 29 
David R. Morrison (UCSB)
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 groupscheme 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 CalabiYau 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 CalabiYau variety? The question is an
important one to answer for application to Ftheory.
I will discuss various partial results concerning this question, some old
and some new. I will also briefly describe the application to Ftheory.
Audio [ mp3, m4a ]; Lecture notes version 1; Lecture notes version 2.

February 12 
David R. Morrison (UCSB)
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 groupscheme 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 CalabiYau 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 CalabiYau variety? The question is an
important one to answer for application to Ftheory.
I will discuss various partial results concerning this question, some old
and some new. I will also briefly describe the application to Ftheory.
Audio [ mp3, m4a ]; Lecture notes version 1; Lecture notes version 2.

February 19 
David R. Morrison (UCSB)
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.
Audio [ mp3, m4a ]; Lecture notes version 1; Lecture notes version 2.

March 11 
David R. Morrison (UCSB)
Abstract:
We continue our discussion of torsors for elliptic fibrations.
Audio [ mp3, m4a ]; Lecture notes version 1; Lecture notes version 2.


