A Sextic with 35 Cusps
Abstract
Recently, W. Barth and S. Rams discussed sextics with up to 30 A2-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 D5. This narrows the possibilities for the maximum number μA2(6) of A2-singularities on a sextic to 35 μA2(6) 37. To construct this surface, we use a general algorithm in characteristic zero for finding hypersurfaces with many singularities within a family.
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