are not unirational. They accomplished this by studying the Kodaira dimension of the coarse moduli spaces and found for . In fact, for , and hence is of general type. Geometric implication This is significant geometrically because it implies any linear system on a ruled variety cannot contain the universal curve . Stratification of boundary The moduli space has a natural stratification on the boundary whose points represent singular genus curves. It decomposes into strata , where for . where the action permutes the two marked points. whenever is even. The curves lying above these loci correspond to A pair of curves connected at a double point. The normalization of a genus curve at a single double point singularity. A pair of curves of the same genus connected at a double point up to permutation. Stratification for genus 2 For the genus case, there is a stratification given by . Further analysis of these strata can be used to give the generators of the Chow ring proposition 9.1. Moduli of marked curves One can also enrich the problem by considering the moduli stack of genus g nodal curves with n marked points, pairwise distinct and distinct from the nodes. Such marked curves are said to be stable if the subgroup of curve automorphisms which fix the marked points is finite. The resulting moduli stacks of smooth (or stable) genus g curves with n marked points are denoted (or ), and have dimension . A case of particular interest is the moduli stack of genus 1 curves with one marked point. This is the stack of elliptic curves. Level 1 modular forms are sections of line bundles on this stack, and level N modular forms are sections of line bundles on the stack of elliptic curves with level N structure (roughly a marking of the points of order N). Boundary geometry An important property of the compactified moduli spaces is that their boundary can be described in terms of moduli spaces for genera . Given a marked, stable, nodal curve one can associate its dual graph, a graph with vertices labelled by nonnegative integers and allowed to have loops, multiple edges and also numbered half-edges. Here the vertices of the graph correspond to irreducible components of the nodal curve, the labelling of a vertex is the arithmetic genus of the corresponding component, edges correspond to nodes of the curve and the