Back to Departmental Colloquium: Spring 2000
Departmental Colloquium
Date: Friday, Feb 11, 2000
Time: 4:15PM
Location: JWB 335
Brendan Hassett
University of Chicago
Title |
The locus of rational cubic hypersurfaces |
Abstract |
Let X be a smooth cubic hypersurface of dimension d in complex projective space. By definition, X is rational if its field of algebraic functions is purely transcendental over the complex numbers. Classically, it was known that X is irrational when d=1 and rational when d=2. Clemens and Griffiths proved that X is irrational when d=3, but no cubic of dimension d>3 is known to be irrational. My talk will focus on the case of cubic fourfolds. First, I will review the known examples of rational cubic fourfolds, which form a countably infinite union of subvarieties in the moduli space of cubic fourfolds. Then I will discuss how these subvarieties are special' in the moduli space. For instance, the known examples of rational cubic fourfolds possess associated K3 surfaces,’ isomorphic to surfaces blown up in a birational map P^4 —> X. The definition of an associated K3 surface is intrinsic to (the Hodge structure of) the cubic fourfold. Unfortunately, experimental evidence suggests the presence of such a surface does not guarantee rationality. This led to recent joint work with Tschinkel. If X contains an algebraic surface with certain invariants then X is necessarily rational. We study the problem of representing homology classes on X by such algebraic surfaces. This leads to general conjectures describing the effective 1-cycles on symplectic varieties naturally arising from X. |
Algebraic Geometry