Pencils of symmetric surfaces in P_3
Abstract
In this note we investigate three new pencils of symmetric surfaces in complex projective threespace. These have degree 6, 8 resp. 12 and are invariant under the action of subgroups of SO(4) containing the Heisenberg group. The pencils of degree 6 and 12 are invariant under the action of bigger groups, precisely under the action of the reflection groups F_4 resp. H_4. We compute equations for the generators of the pencils and describe the base locus of each pencil. We find also the singular surfaces and their number of singularities, which are, in fact, ordinary double points. In degree 12, we get a surface with 600 nodes. This confirms a conjecture of V. Goryounov and presents a new lower bound for the maximal number of nodes of such a surface.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2001
 arXiv:
 arXiv:math/0106080
 Bibcode:
 2001math......6080S
 Keywords:

 Algebraic Geometry
 EPrint:
 24 pages, 2 figures