Back to Departmental Colloquium: Fall 2000
Departmental Colloquium
Date: Thursday, Nov 2, 2000
Time: 4:15PM
Location: JWB 335
Chris Peters
University of Grenoble
Title |
Discriminant varieties: how to compute their cohomology |
Abstract |
The zero set V(f) of any homogeneous polynomial f of degree d in (n+1) variables can be viewed as a degree d hypersurface in projective n -space. Fixing the degree d and n, all such hypersurfaces are parametrized by the coefficients of the corresponding polynomial up to a constant factor. It turns out that V(f) is singular precisely when the coefficients of f satisfy a polynomial equation \Delta=0 depending on d and n. this generalizes the well known relation b^2-4ac=0 for n=1, d=2 and therefore is called “discriminant relation”. The hypersurface V(\Delta) in the corresponding projective space is called “discriminant hypersurface”. It is known to be a highly singular object whose invariants (like homology groups etc.) are difficult to calculate. Some years ago Vassiliev outlined a method to approach this problem. Steenbrink has shown that this method gives much more information, namely that ALL of the mixed-Hodge theory is hidden in this approach. Together with Steenbrink I realized that this method allows to describe the mixed Hodge theory of the moduli space of SMOOTH hypersurfaces of fixed degree in a fixed projective space. In the talk I will address the more elementary aspects of this approach. |
Algebraic Geometry