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There are many alternative compactifications of $M_{g,n}$ which live naturally under the classical Deligne-Mumford compactification $\overline{M}_{g,n}$. For instace the moduli spaces of weighted curves $\overline{M}_{g,A[n]}$ where $A=(a_{1},...,a_{n})$ is a vector of rational weights $0<a_{i}\leq 1$. These spaces were introduced in http://arxiv.org/abs/math/0205009.

Starting with weights $A = (1,...,1)$ by lowering them to $A=(a_{1},...,a_{n})$ we get a reduction morphism $$\rho:\overline{M}_{g,n}\rightarrow\overline{M}_{g,A[n]}.$$ The morphism $\rho$ is birational and in general it contracts some boundary divisors. Other moduli space, like moduli of Prym $\overline{R}_{g,n}^{r}$ and of Spin curves $\overline{S}_{g,n}^{r}$, admit natural forgetful morphisms over $\overline{M}_{g,n}$ which are finite and not birational.

Are there moduli spaces (perhaps moduli of curves with some addictional structure) admitting modular birational morphisms to $\overline{M}_{g,n}$ ?

In some sense a version of moduli of weighted curves birationally living above $\overline{M}_{g,n}$.

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  • $\begingroup$ What about the Artin stack of pre-stable curves? $\endgroup$ – Dan Petersen Nov 7 '13 at 8:40
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Of course since every space is a moduli space, one can get a positive answer to this question by simply blowing up $\overline{M}_{g,n}$.

But here is something more in the spirit of what you are asking. How about the moduli space of pairs $((C,x),p)$ consisting of a stable 1-marked, genus $g$ curve $(C,x)$ along with a point $p$ in the projective space $ \mathbb{P}H^0(C, \mathcal{O}( g\cdot x))$. If I've set this up correctly, the projective space is generically a point and so the forgetful map of this space to $\overline{M}_{g,1}$ is generically an isomorphism, but will have extra moduli when $\mathcal{O}_C(g\cdot x)$ has more sections than expected.

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