Deligne-Mumford moduli space of curves

From Wikipedia, the free encyclopedia

In mathematics, the Deligne-Mumford moduli space of curves is a refined construction of a moduli space of algebraic curves, that is work from 1969 by Pierre Deligne and David Mumford. It combined two novel techniques in algebraic geometry, Mumford's geometric invariant theory and Michael Artin's algebraic stacks, to construct a moduli object (not a scheme), including enough stable curves, and for which the irreducibility of the space of curves could be proved. This last was a classical theorem, but one whose status had been the subject of dispute.

The construction gives what is now called a Deligne-Mumford stack, something less general than a typical Artin stack. The space is home to the Mumford measure.

1 Notation
2 Properties
3 References
4 External links

[edit] Notation

It is denoted by $\mathcal{M}_g$ , and the compactification by $\overline \mathcal{M}_g$ , where g is the genus.

[edit] Properties

[edit] Dimension

For $g > 1$ , $\mathrm{dim} \mathcal{M}_g = 3g-3$ . This number was known classically as the number of parameters on which a compact Riemann surface depends.