Pencils of symmetric surfaces in P_3

TL;DR

This paper explores three new symmetric surface pencils in complex projective three-space, analyzing their equations, symmetries, singularities, and confirming a conjecture about the maximum number of nodes.

Contribution

It introduces three new pencils of symmetric surfaces with specific degrees, computes their equations, and establishes a new lower bound for the maximum number of nodes.

Findings

Degree 12 surface has 600 nodes, confirming a conjecture.

Pencils are invariant under specific symmetry groups including Heisenberg and reflection groups.

Described the base locus and singularities of each pencil.

Abstract

In this note we investigate three new pencils of symmetric surfaces in complex projective three-space. 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.