Arrangements of Real Lines and Surfaces withAandDSingularities
Algebraic surfaces with many nondegenerate singularities can be constructed with the help of a class of bivariate polynomials with complex coefficients, associated with the affine Weyl group of the root system A2. Real variants of the polynomials are related to certain simple arrangements of real lines in the plane. In the study of its critical points, additional arrangements appear, which can be used to generate other singular surfaces. The existence of a high number of singularities in the associated surfaces is due to the fact that the polynomials based on the arrangements have many critical points with few critical values. Surfaces with singularities of types A2, A3n + 1, and D4 are constructed.