Stratifications of moduli spaces of curves in positive characteristic

Associate Professor Jonas Bergström will receive funding from the Knut and Alice
Wallenberg Foundation to recruit an international researcher for a postdoctoral position at the Department of Mathematics, Stockholm University.

This project is about properties of curves. These are among the oldest objects in mathematics, having been studied since ancient Greece and they are still one of the most important objects of study in the field of algebraic geometry.

A central question in algebraic geometry is to classify curves. That is, trying to answer what types of curves there are. To answer this question, mathematicians have often introduced different invariants. These can be thought of as fingerprints that can allow one to distinguish curves even when they appear similar. As geometric objects we picture curves over the real (or complex numbers), but in this project we will consider them in the weirder world of positive characteristic, where a multiple of the number 1 will equal 0. The reason for turning to positive characteristic is that in this case there are deep and important invariants of curves that are not present over the real or complex numbers. To make these invariants in positive characteristic visible, we need to consider the Jacobian of the curve. The Jacobian is an abelian variety, which is a higher dimensional space with the very special property that its points can be added (and form a group).

If we consider all abelian varieties then the properties of our invariants are well understood. But if restrict ourselves to only Jacobians of curves then these properties are much more mysterious, and little is known. The aim of this project is to shed new light on these questions.

Photo: Sara Despres