Moduli of algebraic curves

Summary

The moduli space $M_g$ is a geometric space whose points are the isomorphism classes of smooth projective algebraic curves of genus $g$. Over $\C$ these are the compact Riemann surfaces of genus $g$, and $M_g$ is the quotient of Teichmüller space by the mapping class group. For $g \ge 2$, $M_g$ is an irreducible quasi-projective variety of dimension $3g - 3$, and it has a compactification $\overline{M}_g$ by stable curves due to Deligne and Mumford. It is a central object of algebraic geometry, linking geometry, topology and number theory.