The genus 1 case is one of the first well-understood cases of moduli spaces, at least over the complex numbers, because isomorphism classes of elliptic curves are classified by the J-invariantwhere . Topologically, is just the affine line, but it can be compactified to a stack with underlying topological space by adding a stable curve at infinity. This is an elliptic curve with a single cusp. The construction of the general case over was originally completed by Deligne and Rapoport. Note that most authors consider the case of genus one curves with one marked point as the origin of the group since otherwise the stabilizer group in a hypothetical moduli space would have stabilizer group at the point given by the curve, since elliptic curves have an Abelian group structure. This adds unneeded technical complexity to this hypothetical moduli space. On the other hand, is a smooth Deligne–Mumford stack. Genus 2 Affine parameter space In genus 2 it is a classical result that all such curves are hyperelliptic,pg 298 so the moduli space can be determined completely from the branch locus of the curve using the Riemann–Hurwitz formula. Since an arbitrary genus 2 curve is given by a polynomial of the form for some uniquely defined , the parameter space for such curves is given by where corresponds to the locus . Weighted projective space Using a weighted projective space and the Riemann–Hurwitz formula, a hyperelliptic curve can be described as a polynomial of the form where are parameters for sections of . Then, the locus of sections which contain no triple root contains every curve represented by a point . Genus 3 This is the first moduli space of curves which has both a hyperelliptic locus and a non-hyperelliptic locus. The non-hyperelliptic curves are all given by plane curves of degree 4 (using the genus degree formula), which are parameterized by the smooth locus in the Hilbert scheme of hypersurfaces . Then, the moduli space is stratified by the substacks . Birational geometry Unirationality conjecture In all of the previous cases, the moduli spaces can be found to be unirational, meaning there exists a dominant rational morphismand it was long expected this would be true in all genera. In fact, Severi had proved this to be true for genera up to . Although, it turns out that for genus all such moduli spaces are of general type, meaning they