Conical resolutions and cohomology of the moduli spaces of nodal hypersurfaces

Abstract

A projective hypersurface is nodal if it does not have singularities worse than simple nodes. We calculate the rational cohomology of the spaces of equations of nodal cubic and quartic plane curves and also nodal cubic surfaces in the projective 3-space. We deduce the cohomology of the corresponding spaces of equations and moduli spaces. In the case of plane cubics we use the method of conical resolutions. We handle plane quartics by applying a more general version of this method which we construct in this paper. For cubic surfaces we use an explicit description of the compact GIT quotient.