Title pre-talk: review of Castelnuovo’s theorem and blow-ups

Abstract pre-talk: I will recall the main ideas of the proof of Castelnuovo’s theorem, and how one can use blow-ups to resolve the indeterminacy locus of a rational map of surfaces.

Title: Smooth weighted blowdowns

Abstract: An analogue of blow-ups are weighted blow-ups. Those are transformations, in nature similar to a blow-up, but which are a bit more flexible. For example, weighted blow-ups give better algorithms for resolving singularities of algebraic varieties, and often appear in moduli spaces of interest. The price that one has to pay for the extra flexibility is that the result of a weighted blow-up might no longer be a variety, but rather an algebraic stack. Therefore one natural question is: when is an algebraic stack a weighted blow-up of a simpler space? My coauthors and I give some criteria for when this question has a positive answer. This is a joint work with Arena, Di Lorenzo, Mathur, Obinna and Pernice.