|
3
|
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. |
||
|
|
|
5
|
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? |
||
|
|
