Irish Algebraic Geometry Seminar          

March 11-12, 2011                                                

School of Mathematical Sciences,         

University College Cork

 

           

 

This is our second Algebraic Geometry meeting. The first was organized by Madeeha Khalid in November 2010.

 

Venue: Western Gateway Building (a.k.a. IT building, UCC)

on Western Road, Cork.

 

Accommodation: Garmish house on Western Road.

 

Schedule:

 

                                     Friday, March 11

Time

Speaker

Title

Room

1:00 pm

Damiano Testa

(University of Oxford)

Conics on the Fermat quintic threefold

 

WGB G08

3:00 pm

Diane Maclagan

(University of Warwick)

Tropical bounds on effective cycles

WGB G13

4:30 pm

Vijay Singh

(University College Dublin)

Explicit Honda-Tate theorem for supersingular Abelian Varieties

WGB G13

                                    Saturday, March 12

Time

Speaker

Title

Room

9:00 am

Madeeha Khalid

(St. Patrick’s College, Drumcondra)

Azumaya algebras on K3 surfaces

 

WGB 106


 


Conference dinner: Friday 6:30pm.

Abstracts:

 

Damiano Testa: Conics on the Fermat quintic threefold

 

Many interesting features of algebraic varieties are encoded in the spaces of rational curves that they contain.  For instance, a smooth cubic surface in complex projective three-dimensional space contains exactly 27 lines; exploiting the configuration of these lines it is possible to find a (rational) parameterization of the points of the cubic by the points in the complex projective plane.

 

After a general overview, we focus on the Fermat quintic threefold X, namely the hypersurface in four-dimensional projective space with equation x^5+y^5+z^5+u^5+v^5=0.  The space of lines on X is well-known.  I will explain how to use a mix of algebraic geometry, number theory and computer-assisted calculations to study the space of conics on X.

 

This talk is based on joint work with R. Heath-Brown.


Vijay Singh: Explicit Honda-Tate theorem for supersingular Abelian Varieties

I will give the list of characteristic polynomials of supersingular abelian varieties of dimensions up to 7, and the simple procedure to find them which can in principle be extended to all dimensions.

 

Madeeha Khalid: Azumaya algebras on K3 surfaces

 

There has been considerable interest in various notions of non commutative geometry in the last decade or so, particularly twisted sheaves on Calabi Yau manifolds, in the context of mathematical physics.

An Azumaya algebra on a variety is essentially a vector bundle of matrices on the variety.  It corresponds to a twisted sheaf on the variety and hence is a natural objects to study.

In this talk we will mostly explain a classical example of an Azumaya algebra on a K3 surfaces, and present some of our results about its invariants.

We will conclude with some recent results on moduli spaces of Azumaya algebras on K3 surfaces.