4:00 pm Thursday, September 9, 2010

Damiano Testa (University of Oxford)

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.