Birational study of (mildly) noncommutative surfaces

Tuesday, 24-Oct-2023 

Abstract

Given the lack of morphisms in algebraic geometry, algebraic geometers often study varieties using rational maps, which are morphisms defined only on some dense open subset. Birational maps are the isomorphisms in the category of rational maps, and studying the equivalence classes they generate is the subject of  birational geometry. The birational study of projective curves is relatively straightforward since  each curve has a unique smooth model birational to it, and in particular, any birational map of smooth projective  curves extends uniquely to an isomorphism. For surfaces, the story becomes much more interesting as this no longer holds. In this talk, we will review the classical birational theory of algebraic surfaces which explains what does happen, and see how many of the results have noncommutative analogues in the setting of orders. The latter is joint work with Ingalls.