Division theorems for the rational cohomology of some discriminant   complements
  The main purpose of this paper is to show that the mixed Hodge polynomial of the ``space of equations'' for smooth complete intersections of given multidegree in $\mathbb{C} P^n$ is divisible by the mixed Hodge polynomial of the group $\mathrm{GL}_{n+1}(\mathbb{C})$, the quotient being the mixed Hodge polynomial of the corresponding quotient space. As a by-product of the method used in the proof, we obtain expressions divisible by the order the automorphism group of any smooth projective hypersurface of given dimension and degree 
Alexei G. Gorinov