A Sextic with 35 Cusps

Oliver Labs

paper

A Sextic with 35 Cusps

arXiv:math/0502520

Abstract

Recently, W. Barth and S. Rams discussed sextics with up to 30 $A_{2}$ -singularities (also called cusps) and their connection to coding theory [math.AG/0403018]. In the present paper, we find a sextic with 35 cusps within a four-parameter family of surfaces of degree 6 in projective three-space with dihedral symmetry $D_{5}$ . This narrows the possibilities for the maximum number $μ_{A_{2}} (6)$ of $A_{2}$ -singularities on a sextic to $35 \leq μ_{A_{2}} (6) \leq 37$ . To construct this surface, we use a general algorithm in characteristic zero for finding hypersurfaces with many singularities within a family.

6 pages, 1 figure, for additional images/movies, see http://www.AlgebraicSurface.net