Why Not Add Cuspidal Curves in Moduli Space of Stable Curves? – Algebraic Geometry

Question

Let $\mathcal{M}_{g,n}$ be the moduli space (stack) of stable smooth curves of genus $g$ with $n$ marked points over $\mathbb{C}. $ It's known that by adding stable nodal curves to $\mathcal{M}_{g,n}$, the resulting space $\overline{\mathcal{M}}_{g,n}$ is compact. But why is it so? For example, consider the following family of elliptic curves in $\mathcal{M}_{1,1}$, $$y^2=x^3+t,$$ where $t \in \mathbb{C^*}$. Then this family of elliptic curves degenerates into the cuspidal cubic curve $$y^2 = x^3.$$ So why is $\overline{\mathcal{M}}_{1,1}$ compact?

Answer 1

If you add cuspidal curves, then $\overline{\mathcal{M}}_{1,1}$ will no longer be separated, which is the scheme/stack analogue of Hausdorff. Specifically, consider the families $$y_1^2 = x_1^3 + t^6 \ \mbox{and}\ y_2^2 = x_2^3 + 1$$ (so the second family is a constant family with no $t$-dependence). For all nonzero $t$, they are isomorphic by the change of variables $y_1 = t^3 y_2$, $x_1 = t^2 x_2$. So they should give the same map from $\mathbb{C}^{\ast}$ to moduli space (namely, a constant map). If the cuspidal curve corresponded to a point of moduli space, then this map would have two limits.

The situation is similar with regard to the family $y_3^2 = x_3^3 + t$ that you consider. On the level of coarse moduli spaces, this family also corresponds to a constant map $\mathbb{C}^{\ast} \to \overline{\mathcal{M}}_{1,1}$. The subtlety is that the families $y_2^2 = x_2^3 + 1$ and $y_3^2 = x_3^3 + t$ are not isomorphic over $\mathrm{Spec}\ \mathbb{C}[t^{\pm 1}]$, but only over the cover where we adjoin a $6$-th root of $t$. The definition of coarse moduli space is meant exactly to accommodate families which are not isomorphic but become isomorphic after a finite cover.

In general, when choosing a definition for a moduli space, if you allow too many objects, you will fail to be separated and, if you allow too few objects, you will fail to be proper (analogue of compact). So the answer to "why don't we include" is usually "that would break separatedness" and the answer to "why must we include" is "in order to be proper".