Equivariant and invariant theory of nets of conics with an application to Thom polynomials

M. Domokos, L.M. Fehér, and R. Rimányi

Journal of Singularities
volume 7 (2013), 1-20

Received: 18 June 2012. Received in revised form: 1 January 2013.

DOI: 10.5427/jsing.2013.7a

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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.

Author(s) information:

M. Domokos L. M. Fehér R. Rimányi
Rényi Institute of Mathematics Department of Analysis Department of Mathematics
Hungarian Academy of Sciences Eotvos University University of North Carolina
Budapest, Hungary Budapest, Hungary Chapel Hill, NC, USA
email: domokos.matyas@renyi.mta.hu email: lfeher@renyi.mta.hu email: rimanyi@email.unc.edu