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I'm not an expert in this topic, so please excuse my negligence. I'd also appreciate references to the literature. Throughout, I will work over the complex numbers, although the analogous questions could be asked over other fields, and they may even be more interesting in that situation. Let $M_g$ denote the moduli space of smooth projective curves of genus $g$. Is $M_g$ rational (i.e., birational to a projective space) for a general integer $g\geq 1$?

This is true when $g=1$, as $M_1$ is then just the moduli space of elliptic curves. I'm not sure how to show this for genus $2$, but note that as genus $2$ curves are hyperelliptic, the moduli space $M_2$ can be identified with $(\mathbf{P}^1)^6/(\mathrm{SL}_2\cdot\Sigma_6) = \mathbf{P}^6/\mathrm{SL}_2$, so it's at least unirational. (I think that this is even true when we work over a field of characteristic not $2$.) This suggests that the first step in proving rationality would be showing that $M_g$ is in general birationally equivalent to quotient of $\mathbf{P}^N$ by a reductive group, which implies that it is unirational. Rationality seems harder. The article https://arxiv.org/abs/0804.1509 shows that $M_3$ is also rational, and my thoughts about unirationality seem to be confirmed in the introduction. What about genus greater than $3$?

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    $\begingroup$ This is a well-studied subject, doing some research on the internet would have been more efficient than posting here. Have a look at the Bourbaki talk by C. Voisin about 10 years ago. $\endgroup$ – abx Jun 20 '18 at 5:53
  • $\begingroup$ $M_g$ is rational for $g\le 6$; for $g=2$ this is due to Igusa, for $g=3,5$ to Katsylo and for $g=4,6$ to Shepherd-Barron. $\endgroup$ – inkspot Jun 26 '18 at 23:25
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Your question is a plausible guess; indeed, Severi conjectured that $\mathcal{M}_g$ is unirational for all $g$. He proved this conjecture for $g\leq 10$. Sernesi was first to prove unirationality of $\mathcal{M}_g$ in genus 12, then Chang and Ran proved it in genera 11 and 13, Verra in genus 14. Bruno and Verra have shown that $\mathcal{M}_{15}$ is rationally connected.

The picture is quite different for higher genera. For instance, Harris and Mumford proved that $\bar{\mathcal{M}_g}$ is a variety of general type for $g\geq 24$; Farkas did this for $g=22$. Farkas proved that the Kodaira dimension of $\bar{\mathcal{M}_{23}}$ is at least 2 and conjectures that this is, in fact, an equality.

A good exposition on this topic can be found in Farkas' "Birational aspects of the geometry of $\bar{\mathcal{M}_g}$".

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