Title: Essential dimension and algebraic stacks
Abstract: I will report on joint work with Patrick Brosnan and
Zinovy Reichstein. We extend the notion of "essential dimension",
which has been studied so far for algebraic group, to algebraic
stacks. The problem is the following: given a geometric object X
over a field K (e.g., an algebraic variety), what is the least
transcendence degree of a field of definition of X over the prime
field? In other words, how many independent parameters do we need
to define X? We have complete results for smooth, or stable, curves
in characteristic 0. Furthermore the stack-theoretic machinery that
we develop can also be applied to the case of case of algebraic
groups, showing for example that the essential dimension of the
group Spin_n grows exponentially with n.