Hypersurfaces with many Aj-singularities: Explicit constructions

Some Singular Curves and Surfaces Arising from Invariants of Complex Reflection Groups

“…It is a classical problem to determine the maximal number of singularities of a given type that a curve or a surface might have. Several kinds of upper bounds have been given [Sak], [Bru], [Miy], [Var], [Wah]..., and these bounds have been approached for small degrees [Iv], [Bar], [Esc1], [Esc2], [End], [EnPeSt], [Lab], [Sar3], [Sta]... or general degrees [Chm].…”

On a family of complex algebraic surfaces of degree 3n

“…We find 3n(n − 1) more singularities than in the Chmutov hypersurfaces P 3n (u 1 , u 2 ) − P 3n (u 3 , u 4 ) = 0. Hypersurfaces with A j -singularities in P n (C) can be obtained along the lines of [5]. In particular, there is a family of Belyi polynomials associated with a series of planar trees obtained by a substitution process, which, when used in combination with Q d + 2, allows us to show the existence of surfaces with a high number of cusps (see Appendix B and Fig.…”

“…In particular, there is a family of Belyi polynomials associated with a series of planar trees obtained by a substitution process, which, when used in combination with Q d + 2, allows us to show the existence of surfaces with a high number of cusps (see Appendix B and Fig. 1 in [5]).…”