# smooth curves of genus 3 over an algebraic closed field

Is there a way to "easily" compute and describe the Moduli space of smooth curves of genus 3 without stacks and stable curves?

In Hartshorne's Algebraic Geometry there is a nice excercise (Chapter IV Curves, Excercise 2.2) doing this for g=2. I'm searching for something similar.

Thanks a lot.

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Given a genus three curve, consider the canonical map. It is either two to one onto a plane conic, or it is an embedding into the plane as a quartic. By counting parameters, you can see that most curves fall into the second (nonhyperelliptic) case. Thus a Zariski open subset of the moduli space $\mathcal{M}_3$ can be described as the space of nonsingular plane quartics mod $PGL_3$. Is this what you were after?