Hyperelliptic double covers and their genus

(The affine curve is a smooth Riemann surface chart-by-chart.) Let $F(x,y)=y^2-P(x)$, whose gradient $(2y,-P^{\mathrm{\prime }}(x))$ vanishes at a point of $X_0$ only if $y=0$, $P(x)=0$ and $P^{\mathrm{\prime }}(x)=0$, that is, only at a multiple root of $P$; since the $a_j$ are distinct, this never happens on $X_0$. Hence at every point of $X_0$ some partial derivative of $F$ is nonzero and [F4] exhibits a local holomorphic chart: where $y\mathrm{\ne }0$ the curve is a graph $y=\pm \sqrt{P(x)}$ over $x$ near a point with $P(x)\mathrm{\ne }0$, and where $y=0$ and $x=a_j$ the curve is a graph over $y$ near $(a_j,0)$, because $x-a_j=y^2\mathrm{/}{P}^{\mathrm{\prime }}(a_j)+O(y^3)$ can be solved holomorphically for $x$. Chart changes are restrictions of the holomorphic functions $x$ and $y$, hence holomorphic. Thus $X_0$ is a complex 1-manifold.

(Completion at infinity by explicit charts.) Write $t=1\mathrm{/}x$ and $v=y\mathrm{/}{x}^{g+1}$, so that on $X_0$ with $x\mathrm{\ne }0$ one has $v^2=P(1\mathrm{/}t)t^{2g+2}$.

(Compactness.) Choose $0<r<\epsilon$ small enough that the infinity charts of step 1.2 are defined for $\mathrm{\mid }t\mathrm{\mid }\le r$, and put $R=1\mathrm{/}r$. In either parity, for $\mathrm{\mid }x\mathrm{\mid }\le R$ one has $\mathrm{\mid }y{\mathrm{\mid }}^2=\mathrm{\mid }P(x)\mathrm{\mid }\le {\max }_{\mathrm{\mid }x\mathrm{\mid }\le R}\mathrm{\mid }P(x)\mathrm{\mid }$, the maximum being finite because $\mathrm{\mid }P\mathrm{\mid }$ is continuous on the compact disc and its image is compact, hence bounded by [F9]. Thus the affine piece with $\mathrm{\mid }x\mathrm{\mid }\le R$ is a closed bounded subset of ${\mathbb{C}}^2$, hence compact by [F9]. Every remaining affine point has $\mathrm{\mid }x\mathrm{\mid }>R$, equivalently $0<\mathrm{\mid }t\mathrm{\mid }<r$, and so lies in an infinity chart. In the even case, the images of the two closed chart discs $\mathrm{\mid }t\mathrm{\mid }\le r$ are compact and cover this end together with both added points. In the odd case, the image of the closed chart disc $\mathrm{\mid }s\mathrm{\mid }\le \sqrt{r}$ is compact and covers the end, since $t=s^2$, together with its added point. Therefore $X$ is the union of finitely many compact sets and is compact.

(Connectedness.) The projection $\pi$ restricts over $\mathbb{C}\setminus \{a_1,\dots ,a_m\}$ to a two-sheeted covering: over each $x$ there the two points $(x,\pm \sqrt{P(x)})$ are distinct. If this covering had two components, each would be one-sheeted over the connected base, giving a single-valued holomorphic branch $y$ on $\mathbb{C}\setminus \{a_1,\dots ,a_m\}$ with $y^2=P$. Now write $P=(x-a_j)u$ near $a_j$ with $u(a_j)\mathrm{\ne }0$ and ${\rho }^2=u$ by [F3]; on the punctured disc the two branches of $y$ are $\pm w\rho (x)$ where $w^2=x-a_j$. On the circle $x=a_j+ϵe^{i\theta }$, $0\le \theta \le 2\pi$, a continuous branch is $w={ϵ}^{1\mathrm{/}2}{e}^{i\theta \mathrm{/}2}$, and $w$ changes sign in the round trip while $\rho$ returns to itself; hence analytic continuation of the germ $y$ around this loop (which lies in $\mathbb{C}\setminus \{a_1,\dots ,a_m\}$) returns the germ $-y$, not $y$. A globally defined single-valued holomorphic function cannot have this behaviour: its continuation along any closed loop is itself. This contradiction shows the covering is connected; since the omitted fibres over the $a_j$ and the infinity points of step 1.2 are limit points of that connected part, $X$ is connected.

(Ramification at infinity.) Even case: in the chart $(t,v)$ of step 1.2 the target chart ${\varphi }_{\mathrm{\infty }}$ of [F5] reads $x\mapsto t$, so the chart expression of $\pi$ is $t\mapsto t$ and both points ${\mathrm{\infty }}_{\pm }$ have index $1$: infinity is not a branch value. Odd case: in the $s$-chart of step 1.2 the target chart reads $t=s^2$, so the single point $\mathrm{\infty }$ has index $2$ and infinity is a branch value.

($\pi$ is a degree-two proper holomorphic map.) In the charts of steps 1.1 and 1.2 the map $\pi$ has holomorphic expressions: $x$ in the affine charts and, in the infinity charts, respectively $t\mapsto t$ and $s\mapsto s^2$ towards the target chart ${\varphi }_{\mathrm{\infty }}$ of [F5]. Hence $\pi$ is holomorphic; it is nonconstant, and since $X$ is compact by step 2.1 it is proper because preimages of compact sets are closed in $X$, hence compact. Therefore the degree formula [F6] applies. Take a value $b\in \mathbb{C}\setminus \{a_1,\dots ,a_m\}$: its preimage is the two points $(b,\pm \sqrt{P(b)})$, at each of which the chart expression of $\pi$ is the identity in the affine coordinate, so both indices are $1$; hence $\mathrm{deg}\pi =2$.

(The finite roots are simple branch values.) Fix $j$ and write $P=(x-a_j)u$ with $u(a_j)\mathrm{\ne }0$, ${\rho }^2=u$ by [F3]. The only point of $X$ over $a_j$ is $(a_j,0)$, and near it the chart $w\mapsto (a_j+w^2,w\rho (a_j+w^2))$ presents the curve, with chart expression $w\mapsto w^2$ for $\pi$: thus ${\pi }^{-1}(a_j)$ is a single point of index $2$ by [F2]. Since a point of index $2$ is a critical point with branch value $a_j$, and the $a_j$ are distinct, the $m$ finite branch values are exactly the roots of $P$, each the image of one point of index $2$, and every other finite value has the two preimages of index $1$ from step 3.1.

(Riemann–Hurwitz gives $g(X)=g$.) The map $\pi$ is nonconstant holomorphic of degree $2$ between compact connected Riemann surfaces by steps 1.1, 1.2, 2.1 and 2.2, so [F7] applies with $Y=\widehat{\mathbb{C}}$: $2g(X)-2=2(2g(\widehat{\mathbb{C}})-2)+{\sum }_x(e_x(\pi )-1)$, the sum being finite. By the stereographic homeomorphism in [F8], $\widehat{\mathbb{C}}\cong S^2$, so the genus definition in [F8] gives $g(\widehat{\mathbb{C}})=0$. By steps 4.1 and 2.3 the ramification points are the $m$ roots, each contributing $2-1=1$, together with the point at infinity in the odd case, contributing $1$; hence the sum is $m$ for even $m$ and $m+1$ for odd $m$, equal to $2g+2$ in both cases. Therefore $2g(X)-2=2(-2)+(2g+2)=2g-2$, so $g(X)=g$.

(Conclusion and choice.) The charts, the branch points and the ramification indices are all computed explicitly from the polynomial and its finitely many roots, so this example uses no choice principle; the Axiom of Choice is inherited only through the genus interface [F8] used in [F7], as [F10] records.