Classification of algebraic surfaces of low degrees
Mentor:Ivona Grzegorczyk, Professor of Mathematics, California State University Channel Islands
In this project we study algebraic surfaces in 3-dimensional real space R3. These surfaces consist of points satisfying a finite number of polynomial equations of the form Ax2+By2+Cz2+Dxz+Exy+Fyz+Gx+Hy+Iz+J = 0. Note that the case of degree one linear polynomials defines planes only, hence classifying them is a trivial task. We will discuss the case of degree two, that was completely resolved by classification of quadratic surfaces into 17 classes. The case of degree three (cubic) surfaces, the classification problem in terms of algebraic and geometric properties remains unsolved. In this research project, we study these objects, including the lines and curves that lie on them, their symmetries and the number and types of singular points. By studying the effects of deformation on a surface, we attempt to identify properties that may be indicative of possible classes.
The first papers concerned with classification of cubic surfaces include Cayley-Salmon theorem (1849) that states that there are exactly 27 straight lines lying on any smooth, real-valued cubic surface. Next Schläfli (1858) has created a division of cubic surfaces into species with respect to the number of real straight lines and tritangent planes on them, but his classification is incomplete, as he only distinguishes surfaces by types of their singular points, hence putting every smooth cubic surface into the same class and species. In 1987, Knörrer and Miller presented a complete topological classification of surfaces with assigned singularities. Even in this case, their classification does not incorporate geometric invariants of each surface. We will present results for some special classes of these cubic surfaces.