How to Calculate Homology Groups of Spaces of Nonsingular Algebraic Projective Hypersurfaces

A general method of computing cohomology groups of the space of nonsingular algebraic hypersurfaces of degree $d$ in $\mathbf{CP}^n$ is described. Using this method, rational cohomology groups of such spaces with $n=2$, $d\le 4$ and $n=3=d$ and also of the space of nondegenerate quadratic vector fields in $\mathbf C^3$ are calculated.