3:45: Tame stacks and reduction of coversDan AbramovichThis is joint work with Martin Olsson and Angelo Vistoli, in which we describe a class of algebraic stacks, called tame stacks, which for many purposes is a good replacement for Deligne-Mumford stacks in finite and mixed characteristics. This is based on a classification of finite flat linearly reductive group schemes, which should be of independent interest if indeed it is not yet known (a fact that seems correct yet hard to believe).Tame stacks admit nice constructions, in particular the theory of twisted stable maps works well with target a tame stack. The special case of target BG with G a linearly reductive finite group-scheme gives an approach for (rather singular) reduction of spaces of Galois admissible covers. |
5:00: The past, present, and future of limit linear seriesBrian OssermanMany classical questions on curves can be either rephrased in terms of or approached via linear series, which are objects closely related to morphisms to projective spaces. Many of the most fundamental results on linear series of the past 30 years have been proved via degeneration techniques, and there is plenty of evidence that such techniques will continue to play an important role.We will give a survey of the most powerful and general such degeneration technique, the theory of limit linear series originally developed by Eisenbud and Harris in the 1980's. After describing the Eisenbud-Harris point of view, we will sketch a newer construction which functorializes and compactifies the Eisenbud-Harris construction, discuss ongoing work (with Deepak Khosla) to construct a universal limit linear series moduli space, and suggest further directions for generalization which are likely to prove important in the future. |
PDF notes, courtesy of Bjorn Poonen