, to show that has finite stabilizers, hence it is a Deligne–Mumford stack. Moreover, they find a stratification of as , where is the subscheme of smooth stable curves and is an irreducible component of . They analyze the components of (as a GIT quotient). If there existed multiple components of , none of them would be complete. Also, any component of must contain non-singular curves. Consequently, the singular locus is connected, hence it is contained in a single component of . Furthermore, because every component intersects , all components must be contained in a single component, hence the coarse space is irreducible. From the general theory of algebraic stacks, this implies the stack quotient is irreducible. Properness Properness, or compactness for orbifolds, follows from a theorem on stable reduction on curves. This can be found using a theorem of Grothendieck regarding the stable reduction of Abelian varieties, and showing its equivalence to the stable reduction of curves.section 5.2 Coarse moduli spaces One can also consider the coarse moduli spaces representing isomorphism classes of smooth or stable curves. These coarse moduli spaces were actually studied before the notion of moduli stack was introduced. In fact, the idea of a moduli stack was introduced by Deligne and Mumford in an attempt to prove the projectivity of the coarse moduli spaces. In recent years, it has become apparent that the stack of curves is actually the more fundamental object. The coarse moduli spaces have the same dimension as the stacks when ; however, in genus zero the coarse moduli space has dimension zero, and in genus one, it has dimension one. Examples of low genus moduli spaces Genus 0 Determining the geometry of the moduli space of genus curves can be established using deformation Theory. The number of moduli for a genus curve, e.g. , is given by the cohomology groupWith Serre duality this cohomology group is isomorphic tofor the dualizing sheaf . But, using Riemann–Roch shows the degree of the canonical bundle is , so the degree of is , hence there are no global sections, meaningshowing there are no deformations of genus curves. This proves is just a single point, and the only genus curves is given by . The only technical difficulty is the automorphism group of is the algebraic group , which rigidifies once three points on are fixed, so most authors take to mean . Genus 1