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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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up vote 3 down vote accepted

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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Thanks. Where can I find the details? – Partha Solapurkar Apr 20 '13 at 1:48
for exemple here 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 – IMeasy Apr 22 '13 at 10:09

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