Title

Slicing surfaces

Abstract

We study a map of terrain by examining height lines, essentially slicing a surface representing the terrain horizontally into curves. These curves are typically smooth, but there are usually a few 'bad slices', even for a smooth surface. In geometry, number theory, and physics these bad slices turn out to be very interesting, as they capture a lot of information about the surface. They may be called degenerations, singular fibres, bad reduction, phase transitions, mixed structures or qualitative changes, depending on the context, and are often more informative than the good slices. I will discuss a classification of these bad slices in the context of algebraic geometry, and number theory. This is an intricate question, and I will outline some of the arithmetic, geometry and combinatorics involved in this classification.