Moduli spaces of Riemann surfaces

The Deligne-Mumford moduli space is the space M g,n, of isomorphism classes of stable nodal Riemann surfaces of arithmetic genus g with n marked points. We explicitly construct an unfolding of a stable marked nodal Riemann surface using a pair of pants decomposition and varying the gluing parameters. We show that our unfolding satisfies a universal property and therefore gives a chart on Mg,n. This construction gives a geometric interpretation to the unique complex structure on the moduli space. We also explore the relationship between pairs of pants and hexagons in the upper half plane. In particular, we study the behavior of a pair of pants as a boundary component degenerates to a cusp. Included is a proof of the Riemann Roch theorem for surfaces with boundary.