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Essential dimension of moduli of curves and other algebraic stacks

Patrick BrosnanZinovy ReichsteinAngelo 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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