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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?

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    $\begingroup$ In response to some comments, my question is about complex curves only. No non-zero characteristic discussion please. $\endgroup$
    – Mohammad Farajzadeh-Tehrani
    Commented Sep 29, 2011 at 4:25

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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.

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    $\begingroup$ 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 $\mathbb Z$. $\endgroup$
    – Qing Liu
    Commented Sep 28, 2011 at 21:22
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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.

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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.

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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.

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