Eric Ahlqvist

This talk is about algebraic stacks with finite abelian stabilizers.
We will start with some simple but important examples of the type of
constructions that we are interested in and then give an overview of
two projects in a more general context:

The first is about ``tame stacky covers’’—a class of stacks that
includes flat root stacks and flat stacky modifications. These
constructions naturally appear in Gromov–Witten theory, birationalgeometry, the compactification of tame Deligne–Mumford stacks, and the
theory of parabolic vector bundles, among other areas.  Inspired by
Pardini’s work in the 90s on abelian covers of algebraic varieties, we
classify tame stacky covers in terms of certain intrinsically defined
``building data’’. It turns out that a stacky cover can always be
realized as the stack classifying trivializing 1-cochains of a certain
2-cocycle with values in a symmetric monoidal category.

The second project is ongoing work with David Rydh about ``wild stacky
covers’’ in positive characteristic. These play an important role in,
for example, the compactification of wild Deligne-Mumford stacks and
moduli of stable vector bundles on punctured curves in positive
characteristic. I will sketch how we construct and classify these wild
stacky covers.