I watched wikipedia page Moduli space, and the definition of $n$-marked moduli space as follow
One can also enrich the problem by considering the moduli stack of genus $g$ nodal curves with $n$ marked points. Such marked curves are said to be stable if the subgroup of curve automorphisms which fix the marked points is finite. The resulting moduli stacks of smooth (or stable) genus $g$ curves with $n$-marked points are denoted $\mathcal{M}_{g,n}$ (or $\overline {\mathcal{M}_{g,n}})$, and have dimension $3g−3+n$.
I have two questions in the case $n\ge 2$.
- Are the $n$ points orderd?
- Can be there the same point in the $n$ points?