2004/09/20 by Oliver Labs, Labs, Oliver · 3 citations
Mathematics · #14J17 #14Q10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #math.AG #msc:14J17 #msc:14Q10
paper · pdf · doi:10.48550/arxiv.math/0409348
11 pages, 4 figures. For more images/movies, see http://www.AlgebraicSurface.net
arxiv created 2004/09/20 · openalex publication_date 2004/09/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We find a surface of degree 7 in real projective three-space P3(R) with 99 real nodes within a family of surfaces with dihedral symmetry: First, we consider this family over some small prime fields, which allows us to test all possible parameter sets using computer algebra. In this way we find some examples of 99-nodal surfaces over some of these finite fields. Then, the examination of the geometry of these surfaces allows us to determine the parameters of a 99-nodal septic in characteristic zero. This narrows the possibilities for μ(7), the maximum number of nodes on a septic, to: 99 <= μ(7) <= 104. When reducing our surface modulo 5, we even obtain a 100-nodal septic in P3(F5).