Fast reduction in the de Rham cohomology groups of projective hypersurfaces

Fast reduction in the de Rham cohomology groups of projective hypersurfaces

Mon, 08/02/2010
16:00 		Sebastian Pancratz (Mathematical Institute, Oxford) Junior Number Theory Seminar Add to calendar SR1
Fast reduction in the de Rham cohomology groups of projective hypersurfaces
Let $ X $ be a smooth hypersurface in projective space over a field $ K $ of characteristic zero and let $ U $ denote the open complement. Then the elements of the algebraic de Rham cohomology group $ H_{dR}^n(U/K) $ can be represented by $ n $-forms of the form $ Q \Omega / P^k $ for homogeneous polynomials $ Q $ and integer pole orders $ k $, where $ \Omega $ is some fixed $ n $-form. The problem of finding a unique representative is computationally intensive and typically based on the pre-computation of a Groebner basis. I will present a more direct approach based on elementary linear algebra. As presented, the method will apply to diagonal hypersurfaces, but it will clear that it also applies to families of projective hypersurfaces containing a diagonal fibre. Moreover, with minor modifications the method is applicable to larger classes of smooth projective hypersurfaces.