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Essential dimension of moduli of curves and other algebraic stacks

Patrick Brosnan, Zinovy Reichstein, Angelo Vistoli (2011)

Journal of the European Mathematical Society

In this paper we consider questions of the following type. Let k be a base field and K / k be a field extension. Given a geometric object X over a field K (e.g. a smooth curve of genus g ), what is the least transcendence degree of a field of definition of X over the base field k ? In other words, how many independent parameters are needed to define X ? To study these questions we introduce a notion of essential dimension for an algebraic stack. Using the resulting theory, we give a complete answer to...

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