Essential dimension and algebraic stacks.

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.