Consider the affine curve $C_1 = V(y^2 – (x^4+1)) \subset \Bbb A^2_k$.
In the answers to this question, they claim that there is a (unique?) nonsingular projective curve $C_2$ corresponding to $C_1$ (using "using weighted projective space, or by gluing affine models", or "blow-ups").
Could someone explain to me : 1) what does it mean that a nonsingular projective curve $C_2$ "corresponds" to $C_1$ ? and 2) what is an explicit equation for $C_2$ (they might be several $C_2$…) ?
The naive idea of a projective curve associated to $C_1$ is the projective closure under the inclusion map $\Bbb A^2 \to \Bbb P^2, (x,y) \mapsto [x:y:1]$, which gives $S_2 = V(y^2z^2 – (x^4 + z^4)) \subset \Bbb P^2_k$. But this is a singular curve.
My question 1) is to understand what we define as being a non-singular projective curve associated to $C_1$. And then my question 2) is to know what this definition gives very explicitly in our specific case.
For question 1), a possible definition would be "there exists an open immersion $j : \Bbb A^2 \to \Bbb P^2$ such that the closure of $j(C_1)$ is a nonsingular curve $C_2$". Or maybe "the (unique) projective smooth curve $C_2$ with function field equal to $Frac(k[x,y]/(y^2-x^4-1))$" as here? Would that answer correctly my question 1)?
Thank you!
Best Answer
In general, I don't know of a way to write down the normalization given the equation of the original curve, but for a hyperelliptic curve you can be very explicit. Suppose for simplicity you are given an equation $$y^2 = f(x)$$ corresponding to a hyperelliptic curve over an algebraically closed field of characteristic 0. Then one can show that the curve in $\Bbb A^2$ given by the equation $$w^2 = v^{2g + 2}f(1/v)$$ ($g$ here being the genus of the curve, which you can write down in terms of the degree of $f$) glues together with the original curve via \begin{align*} (x,y)&\mapsto (v,w) = \left(\frac{1}{x},\frac{y}{x^{g+1}}\right)\\ (v,w)&\mapsto (x,y) = \left(\frac{1}{v},\frac{w}{v^{g+1}}\right) \end{align*} and that the glued curve is smooth. It is a good exercise to work all this out for yourself - try to show that a hyperelliptic curve over an algebraically closed field of characteristic $0$ is always given by an equation of this form! :)
As a last remark, you could not hope that the answer to question 1 would be "there exists an open immersion $j:\Bbb A^2\to\Bbb P^2$ such that the closure of $j(C_1)$ is a nonsingular curve $C_2,$" because not all curves can be embedded in $\Bbb P^2$! If you replace "$\Bbb P^2$" by "$\Bbb P^n$ for some $n$" (even only using $\Bbb P^3$ will suffice, in fact), then you would have a less canonical but equivalent description of the nonsingular projective curve corresponding to $C_1.$