Non-hyperelliptic curves of genus three are smooth quartics. Is the moduli space of such curves affine?
I think this follows from a more general result on smooth complete intersections, but I'm looking for a simple proof.
One idea would be to use a "good" compactification, i.e., such that the boundary divisor is ample. This can be done by using a suitable Grassmannian containing the Hilbert scheme.
I'd like to avoid something like this and give a more elementary argument. Is that possible?