Symmetric Surfaces with Many Singularities

Abstract

Let G ⊂ SO(4) denote a finite subgroup containing the Heisenberg group. In this paper we classify all such groups, we find the dimension of the spaces of G-invariant polynomials and we give equations for the generators whenever the space has dimension two. Then we complete the study of the corresponding G-invariant pencils of surfaces in ℙ₃ which we started in Sarti [Sarti, A. (2000). Pencils of symmetric surfaces in ℙ³(C). J. Algebra 246:429–452]. It turns out that we have five more pencils, two of them containing surfaces with nodes.

Key Words:

• Invariant polynomials
• Group actions on varieties
• Singularities

Mathematics Subject Classification:

• Primary 14JL7, 14J25, 14L3
• Secondary 16W22

Acknowledgment

I thank Prof. Wolf Barth at the University of Erlangen for many helpful comments and discussions.

Notes

^#Communicated by R. Piene.