Conical resolutions and cohomology of the moduli spaces of nodal hypersurfaces

TL;DR

This paper computes the rational cohomology of spaces of nodal hypersurfaces, including cubic and quartic plane curves and cubic surfaces, using conical resolutions and GIT methods.

Contribution

It introduces a generalized conical resolution method for computing cohomology of nodal hypersurfaces and applies it to specific cases like plane quartics.

Findings

Cohomology of spaces of nodal cubic and quartic plane curves calculated.

Cohomology of spaces of nodal cubic surfaces determined.

Explicit GIT quotient description used for cubic surfaces.

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.