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?

4$\begingroup$ I believe this has to do with the fact that $\mathscr M_{g,n}$ is stacky and so the valuative criterion doesn't hold on the nose. You might need to take an etale extension of your curve before you can extend the map from the generic point. In this particular case, this is related to the fact that the cuspidal singularity is unstable: After an unramified base change of $\mathbb C[[t]]$, the family will become either a node or good reduction. $\endgroup$– AsvinSep 25, 2019 at 18:03

1$\begingroup$ Also, a cusp would have a $\mathbb{C}^*$worth of automorphisms, while the usual moduli spaces of stable curves are designed to be as rigid as possible and in fact are DeligneMumford. $\endgroup$– QfwfqSep 25, 2019 at 20:57

$\begingroup$ @Qfwfq Do you mean the cuspidal curve $y^2=x^3$ has an automorphism group $\mathbb{C^*}$, even with a marked point at the cusp? $\endgroup$– Yuhang ChenSep 26, 2019 at 0:10

$\begingroup$ @Asvin I don't follow your last sentence. What do you mean by saying "the cuspidal singularity is unstable"? $\endgroup$– Yuhang ChenSep 26, 2019 at 0:15

$\begingroup$ @yYuhangChen i was basically referring to the last part of David's answer where he talks about how after adding a 6th root of t, you get an isotrivial (hence good redn) curve. $\endgroup$– AsvinSep 26, 2019 at 1:09
1 Answer
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".

3$\begingroup$ Could you explain better "The definition of coarse moduli space is meant exactly to accommodate families which are not isomorphic but become isomorphic after a finite cover"? I would've said that what is meant to accomodate this phenomenon is exactly the use of the étale topology on the base category (whether we use algebraic spaces as c.m.s.'s or we go full stacky); and that the definition of c.m.s. is meant to avoid stackiness (automorphisms) while still retaining the same geometric points and some geometry. Is my intuition incorrect about this? $\endgroup$– QfwfqSep 25, 2019 at 21:46

4$\begingroup$ Thanks for the enlightening answer (especially the last paragraph)! Only after seeing your example, then I realize that the family of curves I considered have a same $j$invariant, which is zero. So this "zero map" is the constant map $\mathbb{C}^* \to \overline{\mathcal{M}}_{1,1}$ you mentioned, on the level of coarse moduli spaces. $\endgroup$ Sep 26, 2019 at 2:24

$\begingroup$ @Qfwfq If $S$ is reduced and $X \to S$ and $Y \to S$ are two families which become isomorphic after being pulled back along a finite surjective cover $T \to S$, then the two families correspond to the same map from $S$ to the coarse moduli space. $\endgroup$ Sep 27, 2019 at 3:01

$\begingroup$ I was under the impression that something like a converse held as well  that if $X \to S$ and $Y \to S$ were two families corresponding to the same map from $S$ to the coarse moduli space, that there was a finite surjection $\pi: T \to S$ such that $\pi^{\ast} X \to T$ and $\pi^{\ast} Y \to T$ are isomorphic. But having failed to find either a proof or reference, I am coming to suspect I'm wrong. $\endgroup$ Sep 27, 2019 at 3:01