Braid groups in algebraic geometry

Arnol'd and Brieskorn discovered that the braid groups and their generalizations the Artin groups occur naturally in the study of singularities of algebraic varieties. I will explain how group-theoretic properties of the Artin groups contributes to the understanding of several classical moduli spaces, including the moduli spaces of quartic curves in CP2, cubic surfaces in CP3, and cubic threefolds in CP4. (Joint work with J. Carlson and D. Toledo.)