Moduli space of genus 2 curves - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T05:37:57Z http://mathoverflow.net/feeds/question/76585 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76585/moduli-space-of-genus-2-curves Moduli space of genus 2 curves Mohammad F.Tehrani 2011-09-28T00:58:50Z 2011-09-28T21:19:27Z <p>Does any body know any reference in which the geometry of compactified moduli space of genus two curves ( Which is a three dimensional variety/stack/...) has been studied? </p> http://mathoverflow.net/questions/76585/moduli-space-of-genus-2-curves/76589#76589 Answer by J.C. Ottem for Moduli space of genus 2 curves J.C. Ottem 2011-09-28T01:34:09Z 2011-09-28T01:34:09Z <p>Genus 2 curves are hyperelliptic and so their coarse moduli space is just the Riemann-Hurwitz space $(\mathbb{P}^1)^6/(SL_2 \cdot S_6)$. So the description of $M_2$ is closedly linked with the invariants of binary sextic forms. The classic reference is the paper </p> <p>J. Igusa, <a href="http://www.jstor.org/stable/1970233" rel="nofollow">Arithmetic Variety of Moduli for Genus Two</a>, Annals of Mathematics, Vol. 72, No. 3 (1960), pp. 612-649.</p> <p>Brendan Hassett's paper <a href="http://math.rice.edu/~hassett/papers/genus2/logmodel3.pdf" rel="nofollow">Classical and minimal models of the moduli space of curves of genus two</a> is also a nice paper studying explicit compactifications for $M_2$ and their birational geometry properties.</p> http://mathoverflow.net/questions/76585/moduli-space-of-genus-2-curves/76601#76601 Answer by roy smith for Moduli space of genus 2 curves roy smith 2011-09-28T05:41:07Z 2011-09-28T05:41:07Z <p>I recommend part III, the case g=2, of Mumford's ":Towards an enumerative geometry of the moduli space of curves", in the Shafarevich 60th birthday volume.</p> http://mathoverflow.net/questions/76585/moduli-space-of-genus-2-curves/76696#76696 Answer by Qing Liu for Moduli space of genus 2 curves Qing Liu 2011-09-28T21:19:27Z 2011-09-28T21:19:27Z <p>Over the complex numbers, you might be interested in: Mostafa:Die Singularitäten der Modulmannigfaltigkeit $\overline M_g(n)$ der stabilen Kurven vom Geschlecht $g\geq 2$ mit $n$-Teilungspunktstruktur. (German) [The singularities of the moduli variety $\overline M_{g}(n)$ of stable curves of genus $g\geq 2$ with $n$-division point structure] J. Reine Angew. Math. 343 (1983), 81–98. </p> <p>Over a field of any characteristic, in my <a href="http://www.math.u-bordeaux.fr/~liu/articles/courbes_stables_genre2.pdf" rel="nofollow">paper</a> § 3, the scheme $\overline M_{2}$ over $\mathbb Z$ (and over any field $k$) is described as the normalization of the blowup of the weighted projective scheme $$\mathrm{Proj}\mathbb Z[J_2, J_4, J_6, J_8, J_{10}]/(J_4^2-J_2J_6+4J_8)$$ (the $J_i$'s are Igusa invariants and have weight $i$) along some explicit center. The singularities over $k$ are described as well. </p>