Equivariant and invariant theory of nets of conics

TL;DR

This paper develops the invariant theory of nets of conics by calculating equivariant classes of orbit closures, enabling solutions to enumerative problems and determining Thom polynomials of contact singularities.

Contribution

It introduces the calculation of equivariant fundamental classes for orbit closures of nets of conics and applies this to invariant theory and enumerative geometry.

Findings

Calculated equivariant fundamental classes of orbit closures.

Developed the invariant theory of nets of conics.

Determined Thom polynomials for contact singularities.

Abstract

Two parameter families of plane conics are called nets of conics. There is a natural group action on the vector space of nets of conics, namely the product of the group reparametrizing the underlying plane, and the group reparametrizing the parameter space of the family. We calculate equivariant fundamental classes of orbit closures. Based on this calculation we develop the invariant theory of nets of conics. As an application we determine Thom polynomials of contact singularities of type (3,3). We also show how enumerative problems---in particular the intersection multiplicities of the determinant map from nets of conics to plane cubics---can be solved studying equivariant classes of orbit closures.