Pencils of symmetric surfaces in P₃

2001/06/11 by Alessandra Sarti, A. Sarti, Sarti, Alessandra · 4 citations
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Computational Geometry and Mesh Generation #FOS: Mathematics #Mathematical Approximation and Integration #Mathematics and Applications #math.AG

paper · pdf · doi:10.48550/arxiv.math/0106080

24 pages, 2 figures

arxiv created 2001/06/11 · openalex publication_date 2001/06/11 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 F₄ resp. H₄. 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.

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