Arrangements of Real Lines and Surfaces with A and D Singularities

Abstract

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, A_{3n + 1}, and D₄ are constructed.

2000 AMS Subject Classification::

• 14J17
• 14J70

Keywords:

• algebraic surfaces
• arrangements
• singularities