\subsection{From a square root to a double cover}
\label{sec:DoubleCover}

In addition to the associated hypersurface defined in the previous subsection, one can also associate another variety to a square root of a polynomial.
We will see that these two varieties are not isomorphic in general but always birationally equivalent to each other.
Hence, in view of our rationalizability study, the two approaches are equivalent.
The approach described in this subsection is particularly convenient for a more geometrical analysis;
the approach described in Subsection~\ref{sec:ProjSurf} is more suitable for a generalization and, a priori, requires less advanced geometrical tools.
Both approaches have advantages and disadvantages in different contexts, and we will use both throughout this paper.

In this subsection,
let $k$ be any field and $f\in R=k[X_1,...,X_n]$ be a non-constant squarefree polynomial of degree $d$.
Define $r:=\ceil{d/2}$.
Consider the square root  $\sqrt{f}$.
Let $\A_k^{n+1}$ be the affine space over $k$ with variables $X_1,...,X_n,W$.
Let $\PP_k=\PP_k(1,...,1,r)$ be the weighted projective space over $k$ with coordinates $s,y_1,...,y_n,u$ of weights $1,1,...,1,r,$ respectively.
The relations between the affine and projective coordinates are $X_i=y_i/s$ for $i=1,...,n$ and $W=u/s$.

\begin{definition}\label{def:AssDoubleCover}
We define {\em the double cover associated to} $\sqrt{f}$ to be the hypersurface in $\PP_k$ given by 
$$
\overline{S}\colon u^2-s^{2r}f(y_1/s,...,y_n/s)=0.
$$
\end{definition}
\begin{remark}
The associated affine hypersurface $V\subset \A_k^{n+1}$ 
(cf. Definition~\ref{def:AffAssHyp}) 
has a natural structure of double cover of $\A_k^n$.
If $S$ denotes $V$ viewed as a double cover, then
$\overline{S}$ is the projective closure of $S$ in $\PP_k$.
\end{remark}

\begin{proposition}\label{prop:Birational}
Let $\overline{V}$ and $\overline{S}$ be the hypersurface and the double cover associated to $\sqrt{f}$.
Then $\overline{V}$ and $\overline{S}$ are birationally equivalent.
\end{proposition}
\begin{proof}
Define the rational map $\Phi\colon \PP_k \dashrightarrow \PP_k^{n+1}$ via 
$$
\Phi\colon (s,y_1,...,y_n,u)\mapsto (s,y_1,...,y_n,u/s^{r-1}).
$$
Then, $\Phi$ is well-defined over a Zariski open subset of $\PP_k=\PP_k (1,1,...,1,r)$, sends $\overline{S}$ to $\overline{V}$, and admits an inverse,
namely the rational map $\Psi\colon\mathbb{P}_k^{n+1}\dashrightarrow\mathbb{P}_k$ defined by
$$
\Psi\colon (z,x_1,...,x_n,w)\mapsto (z,x_1,...,x_n,z^{r-1}w).
$$
Hence, $\Phi$ is a birational map from $\overline{S}$ to $\overline{V}$.
\end{proof}


\begin{example}
Take $k$ to be the field of complex numbers $\CC$. 
In Example~\ref{e:EllipticCubic}, we have seen that the associated hypersurface of the square root $\sqrt{X(X-1)(X-\lambda)}$ is the elliptic curve 
$\overline{V}\subset\PP_k^2$ defined by
$$
\overline{V}\colon zw^2-x(x-z)(x-\lambda z)=0.
$$

The double cover associated to the square root is the curve $\overline{S}$ in the weighted projective space $\PP_k(1,1,2)$ with coordinates $s,y,u$ of weights $1,1,2$, respectively,
defined by the equation
$$
\overline{S}\colon u^2-sy(y-s)(y-\lambda s)=0.
$$
Using the map $\overline{S}\to\PP_k^1$ defined by  $(s:y:u)\mapsto (s:y)$, 
one sees that $\overline{S}$ is a double cover of $\PP_k^1$ ramified at four points: $(0:1:0), (1:0:0), (1:1:0), (1:\lambda:0)$. Hence, $\overline{S}$ is an elliptic curve (by Hurwitz's theorem, cf.~\cite[Corollary IV.2.4]{Har77}).
Finally, Proposition~\ref{prop:Birational} shows that $\overline{S}$ and $\overline{V}$ are indeed birationally equivalent, although a priori they might look very different.
\end{example}
{"config": "arxiv", "file": "2006.07121/04DoubleCovers.tex"}