MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
In response to some comments, my question is about complex curves only. No non-zero characteristic discussion please. – Mohammad F. Tehrani Sep 29 '11 at 4:25
up vote 19 down vote accepted

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

J. Igusa, Arithmetic Variety of Moduli for Genus Two, Annals of Mathematics, Vol. 72, No. 3 (1960), pp. 612-649.

Brendan Hassett's paper Classical and minimal models of the moduli space of curves of genus two is also a nice paper studying explicit compactifications for $M_2$ and their birational geometry properties.

share|cite|improve this answer
Dear J.C., the description of $M_2$ in terms quotient space of $(\mathbb P^1)^6$ is valid only in characteristic different from $2$. But Igusa worked in any characteristic and even over $\mathbb Z$. – Qing Liu Sep 28 '11 at 21:22

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.

share|cite|improve this answer

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.

Over a field of any characteristic, in my paper § 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.

share|cite|improve this answer

For a birational viewpoint:

W. Rulla, The birational geometry of $\overline{M}_{3}$ and $\overline{M}_{2,1}$, Ph.D. thesis, University of Texas at Austin, 2001.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.