Moduli spaces $\mathcal{M}_{2,1}$ and $\mathcal{M}_{3,1}$

The cell structure of the spaces $\mathcal{M}_{2,1}$ and $\mathcal{M}_{3,1}$ is considered. These are the spaces of complex curves of genus 2 and 3 with one marked point. For the space $\mathcal{M}_{2,1}$, nine cells of the highest dimension 8 are described and their adjacency is studied. For the space $\mathcal{M}_{3,1}$, a list of all $1726$ cells of the highest dimension 14 (with orientation) is obtained. The list of adjacent couples of cells is also obtained. These lists can be found on the web.