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? 



Moduli space of genus 2 curves
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](http://www.jstor.org/stable/1970233), 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](http://math.rice.edu/~hassett/papers/genus2/logmodel3.pdf) is also a nice paper studying explicit compactifications for $M\_2$ and their birational geometry properties.


