A Sextic with 35 Cusps

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 $\mu_{A_2}(6)$ of $A_2$-singularities on a sextic to $35 \le \mu_{A_2}(6) \le 37$. To construct this surface, we use a general algorithm in characteristic zero for finding hypersurfaces with many singularities within a family.