Hartshorne has an exercise given as follows (Chatper III 10.6):
Let $Y$ be the plane nodal cubic curve $y^2=x^2(x+1)$. Show that $Y$
has a finite etale covering $X$ of degree $2$, where $X$ is a union of two irreducible components, each one isomorphic to the normalization of $Y$.
This question/answer explores this exercise quite comprehensively. In particular, it says that the etale covering is given by the curve $\operatorname{Spec}k[s,t]/(t^2-(s^2-1)^2)$. While I understand why this candidate works, what's more puzzling to me is how one can approach this problem and arrive at this very curve.
What I understand is that we have a normalization of the curve $Y$: its the affine line with the morphism induced by $t\mapsto (t^2-1,t(t^2-1))$; there are plenty of answer here that address this: for example this one and these ones: Finding normalization of nodal cubic, Normalization of a variety, normalization of the nodal cubic, so I'm comfortable with this. In particular, it makes sense why this would be its normalization and the morphism describes a parametrization of the nodal cubic.
Now, with this knowledge equipped, how do one realize how to describe the curve $X$? We know that $X$ must be a union of two affine lines, where each is the normalization of $Y$ we saw earlier. Knowing just that, how do you the $X$ described as earlier is the correct idea? In fact, the irreducible components of $X$ are given by $\operatorname{Spec}k[s,t]/(t\pm(s^2-1))$; why are they isomorphic to the normalization of $Y$? Lastly, even if we knew the restriction morphism $(s,t)\mapsto (s^2-1,st)$ defined on $\mathbb{A}^2_k$ is the what describes $f$, how do we arrive at the description of $X$?
My question is quite fundamental: how does one approach questions of finding a (etale) covering of a given curve. For example, consider the cuspidal cubic curve $C=k[x,y]/(x^2-y^3)$. How does one go about finding an etale covering of $C$ of an appropriate degree? In fact, if it's possible, how does one go about finding etale coverings of the nodal cubic curve a given degree?
Any help will be extremely helpful! Thank you so much!
Best Answer
Here is a different way to realize the étale cover, inspired from the picture in Hartshorne (and as long as $\operatorname{char} k \neq 2$).
Set $C = \{y^2 = x^2(x+1)\} \subset \mathbb A^2$. The normalisation $\nu:\mathbb A^1 \to C, t \mapsto (t^2 - 1, t(t^2-1))$ factors over the closed immersion¹ $$\iota_1: \mathbb A^1 \to \mathbb A^3, t \mapsto (t^2 -1, t(t^2-1),t),$$ followed by the projection $p: \mathbb A^3 \to \mathbb A^2, (x,y,z) \mapsto (x,y)$. Clearly $p \circ \iota_1 = \nu$. You can also consider $$\iota_2: \mathbb A^1 \to \mathbb A^3, t \mapsto (t^2 - 1, t(t^2-1), -t),$$ which also satisfies $p \circ \iota_2 = \nu$.
As one can see, the curves intersect in the three points $$(0,0,1), (0,0,-1) \quad \text{and} \quad (-1,0,0).$$
Exercise. Those are the only intersection points.
So if we consider the union $X = \iota_1(\mathbb A^1) \cup \iota_2(\mathbb A^2)$, the map $X \to C$ is not étale at $P = (-1, 0,0)$, but étale everywhere else. Also, the maps $\iota_i(\mathbb A^1) \to C$ are étale at $P$, so we only have to separate the two points $\iota_1(0)$ and $\iota_2(0)$. To do this, consider the blow-up $\operatorname{Bl}_P(\mathbb A^3) \to \mathbb A^3$, and let $Y \subset \operatorname{Bl}_P(\mathbb A^3)$ be the strict transform of $X$. Then $Y \to C$ is étale.
I don't think a similar thing can be done for the cuspidal curve, because the cusp is locally irreducible, i.e. the completion $k[[x,y]] / (x^2 - y^3)$ is an integral domain. Since étale covers induce isomorphisms on completions², every connected étale cover over the cusp has to be irreducible.
¹ the $\iota_k$'s are closed immersions because they are basically graphs of the composition $\mathbb A^1 \xrightarrow{\nu} C \hookrightarrow \mathbb A^2$.
² I'm pretty sure this is true, but not totally sure if this holds in finite characteristic. Maybe someone more knowledgable could comment on that?