Hypersurfaces with many Aj-singularities: Explicit constructions

摘要

A construction of algebraic surfaces based on two types of simple arrangements of lines, containing the prototiles of substitution tilings, has been proposed recently. The surfaces are derived with the help of polynomials obtained from the lines generating the simple arrangements. One of the arrangements gives the generalizations of the Chebyshev polynomials known as folding polynomials. The other produces a family of polynomials which generates surfaces having more real nodes, and they can also be used, in combination with Belyi polynomials, to derive hypersurfaces in the complex projective space with many Aj-singularities. In some cases explicit expressions can be obtained from the classical Jacobi polynomials. The lower bounds for the maximum possible number of Aj-singularities in certain hypersurfaces of degree d are improved for several values of d and j.