Understanding the singularities of algebraic curves and surfaces is important for understanding the geometry of these curves and surfaces.  A difficult problem in CAGD is the handling of self-intersections, and the theory of singularities of algebraic varieties is potentially a tool for handling this problem. In the GAIA project special emphasis has been put on detecting and locating singularities appearing on parameterized and implicitly given curves and surfaces of low degree.

• The presence of singularities affects the geometry of complex and real projective hypersurfaces and of their complements. We illustrate the general principles and the main results by many explicit examples involving curves and surfaces. 
• We classify and analyze the singularities of a surface patch given by a parameterization in order to proceed to an early detection. We distinguish algebraically defined surface patches and procedural surfaces given by evaluation of a program. Also we distinguish between singularities which can be detected by a local analysis of the parameterization and those which require a global analysis, more difficult to achieve.
• The detection of singularities is a critical ingredient of many geometrical problems, in particular in intersection operations. Once these critical points are located, one can for instance safely use numerical methods to follow curve branches. Detecting a singularity in a domain may also help in combining several types of methods, in our case symbolic and numerical methods.