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Is there a smooth modular compactification of the moduli space of smooth curves of genus $ g > 1 $ over $ \mathbb{C} $?

I am willing to allow for enrichments such as level structures. The compactification should be a projective variety rather than a stack. Any references are highly appreciated.

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  • $\begingroup$ As comment: If $\bar \partial_J$ is transverse to zero section then $\mathcal M_{g,k}(M,g,\beta)$ will be smooth orbifold $\endgroup$
    – user21574
    Commented May 27, 2017 at 8:50

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Sure: see Eduard Looijenga, "Smooth Deligne-Mumford compactifications by means of Prym level structures", which completely answers your question.

There is also later work by de Jong-Pikaart, Boggi-Pikaart and Abramovich-Corti-Vistoli where more general non-abelian level structures are considered, and over more general base schemes.

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$\mathbb{P}^3$ compactifies the moduli space of genus 2 curves with level 3 structure and the choice of an odd theta characteristic.

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  • $\begingroup$ Thanks. Where can I find the details? $\endgroup$
    – Partha Solapurkar
    Commented Apr 20, 2013 at 1:48
  • $\begingroup$ for exemple here arxiv.org/abs/math/0601251 there should be also a compactification for the event theta by van der geer (math ann in the 80s) but I can't rememeber if it is smooth $\endgroup$
    – IMeasy
    Commented Apr 22, 2013 at 10:09

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