  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.
