\section{Normalized central configurations and curves of them}\label{sec:setup}

\subsection{The system of Albouy and Kaloshin}
Let $m=(m_1,\dots,m_5)$. Following \cite[p.~540]{AK12}, a \emph{normalized central configuration} is a solution $(x,y,\delta)\in\C^5\times\C^5\times\C^{10}$ of
\begin{equation}\label{eq:AK4}
x_k=\sum_{l\ne k}m_l\,\delta_{kl}^{3}(x_k-x_l),\quad y_k=\sum_{l\ne k}m_l\,\delta_{kl}^{3}(y_k-y_l),\quad \delta_{kl}^{2}\big((x_k-x_l)^2+(y_k-y_l)^2\big)=1,\quad y_1=y_2,
\end{equation}
for $k\ne l$ in $\{1,\dots,5\}$, with $\delta_{kl}=\delta_{lk}$. It is \emph{real} if $x,y\in\R^5$, and \emph{positive} if moreover all $\delta_{kl}$ are positive. Summing the first equation multiplied by $m_k$ shows that solutions have their centre of mass at the origin. The multiplier $\lambda$ of \eqref{eq:cc} is normalized to $1$.

\begin{lemma}\label{lem:realpos}
Let $m\in(\R_{>0})^5$. Every central configuration with masses $m$ is similar, by an orientation-preserving similarity, to exactly two positive normalized central configurations, which differ by the rotation by $\pi$. In particular, if \eqref{eq:AK4} has finitely many solutions, there are finitely many central configurations up to similarity.
\end{lemma}

\begin{proof}
Taking the scalar product of \eqref{eq:cc} with $m_i(q_i-c)$ and summing gives $\lambda I=U$ with $I=\sum_im_i|q_i-c|^2>0$ and $U=\sum_{i<j}m_im_j/r_{ij}>0$. So $\lambda>0$. We translate $c$ to the origin, scale by $\lambda^{1/3}$, and rotate so that $q_1-q_2$ is horizontal. Exactly two rotations do this. Conversely, the translation and the scale of a positive normalized central configuration are fixed by \eqref{eq:AK4}.
\end{proof}

We put $z=x+iy$, $w=x-iy$ and $r_{kl}=\delta_{kl}^{-1}$, so that $r_{kl}^2=z_{kl}w_{kl}$ with $z_{kl}=z_k-z_l$. For real configurations $w=\bar z$. For complex ones $z$ and $w$ are independent.

\subsection{Centred coordinates and the slice}\label{ss:slice}
For $h\in\C^5$ and $p=(p_{kl})_{k<l}$ with $p_{lk}=p_{kl}$, put $F_k(h)=h_k-\sum_{l\ne k}m_l\,p_{kl}^3(h_k-h_l)$. The unknowns are $u=(u_1,\dots,u_4)$, $v=(v_1,\dots,v_4)$ and $p$, with $u_5=v_5=0$, $u_{kl}=u_k-u_l$, $v_{kl}=v_k-v_l$. The eighteen \emph{rows} are
\begin{equation}\label{eq:rows}
D_{kl}=p_{kl}^2u_{kl}v_{kl}-1,\qquad Z_k=F_k(u)-F_5(u),\qquad W_k=F_k(v)-F_5(v)\qquad(k\le 4).
\end{equation}
Let $X_m\subset\C^{18}$ be their zero set. On $X_m$ the functions $u_k$, $v_k$, $u_{kl}$, $v_{kl}$ and $p_{kl}$ vanish nowhere. The map $(x,y,\delta)\mapsto(u,v,p)$ with $u_k=z_k-z_5$, $v_k=w_k-w_5$, $p=\delta$ identifies the solutions of the first three equations of \eqref{eq:AK4} with $X_m$, and the solutions of \eqref{eq:AK4} with $X_m\cap\{u_{12}=v_{12}\}$.

\begin{lemma}\label{lem:slice}
\begin{enumerate}[label=\textup{(\alph*)}]
\item The group $\C^*$ acts freely on $X_m$ by $s\cdot(u,v,p)=(su,s^{-1}v,p)$, the complex rotation $(z,w)\mapsto(sz,s^{-1}w)$. Let $Y_m=X_m\cap\{u_1=1\}$. Then $\C^*\times Y_m\to X_m$, $(s,y)\mapsto s\cdot y$, is an isomorphism, every orbit meets $\{u_{12}=v_{12}\}$ in exactly two points, and \eqref{eq:AK4} has exactly $2|Y_m|$ solutions.
\item For $t>0$ the map $(u,v,p)\mapsto(t^{1/3}u,t^{1/3}v,t^{-1/3}p)$ is an isomorphism $X_m\to X_{tm}$ that preserves $\{u_{12}=v_{12}\}$, and its composition with the rotation by $s=t^{-1/3}$, $(u,v,p)\mapsto(u,t^{2/3}v,t^{-1/3}p)$, is an isomorphism $Y_m\to Y_{tm}$. Relabelling the bodies together with their masses, followed by a translation and a rotation, is an isomorphism $Y_m\to Y_{m'}$ for the permuted masses $m'$. In particular finiteness of $Y_m$ is invariant under positive scaling and permutation of the masses.
\item A configuration satisfies \eqref{eq:cc} for the masses $m$ with multiplier $\lambda$ if and only if it satisfies \eqref{eq:cc} for $tm$ with multiplier $t\lambda$, and relabelling the bodies together with their masses maps the central configurations for $m$ to those for the permuted masses. Both bijections preserve similarity and orientation. Hence the number of central configurations up to similarity, and up to orientation-preserving similarity, is invariant under positive scaling and permutation of the masses.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) The rows $D_{kl}$ are invariant and $Z_k$, $W_k$ are multiplied by $s$, $s^{-1}$. The action is free because $u_1\ne0$ on $X_m$. The inverse of the map is $x\mapsto(u_1,u_1^{-1}\cdot x)$. The point $s\cdot x$ lies in $\{u_{12}=v_{12}\}$ if and only if $s^2=v_{12}/u_{12}$. (b) The rows for $tm$ at the image point are multiples of the rows for $m$. For a relabelling, we translate the new body $5$ to the origin and rotate to restore $u_1=1$. (c) Multiplying the masses by $t$ multiplies the right side of \eqref{eq:cc} by $t^2$ and the left side by $t$ for fixed $\lambda$. Relabelling permutes the equations.
\end{proof}

For $\{i,j,k,l,n\}=\{1,\dots,5\}$ the \emph{4-product} of \cite[p.~580]{AK12} is $\cP_{ij}=r_{ij}^2r_{kl}^2r_{ln}^2r_{nk}^2$.

\subsection{The potential form of the rows}\label{ss:potential}
Let
\[
U=\sum_{k<l}m_km_l\,p_{kl},\qquad I=\sum_{k}m_ku_kv_k-\frac1M\Big(\sum_km_ku_k\Big)\Big(\sum_km_kv_k\Big),\qquad V=U+\tfrac12I,
\]
where $M=\sum_km_k$. We regard the $p_{kl}$, on the hypersurface defined by the rows $D_{kl}$, as functions of $(u,v)$.

\begin{lemma}\label{lem:potential}
On $\{D_{kl}=0\}$, with $\nabla_u$ and $\nabla_v$ the gradients in $(u_1,\dots,u_4)$ and $(v_1,\dots,v_4)$,
\[
(Z_1,\dots,Z_4)=2A\,\nabla_vV,\qquad (W_1,\dots,W_4)=2A\,\nabla_uV,\qquad A=\operatorname{diag}(m_1^{-1},\dots,m_4^{-1})+m_5^{-1}\one\one^{\mathsf T}.
\]
The matrix $A$ is invertible, with $A^{-1}=\operatorname{diag}(m_1,\dots,m_4)-mm^{\mathsf T}/M$ (restricted to the first four coordinates).
\end{lemma}

\begin{proof}
We write $\tilde F_k=\sum_{l\ne k}m_l(u_l-u_k)p_{kl}^3$, so that $Z_k=u_k+\tilde F_k-\tilde F_5$. From $p_{kl}^2u_{kl}v_{kl}=1$ we get $\partial p_{kl}/\partial v_k=-\tfrac12p_{kl}^3u_{kl}$. So $\partial U/\partial v_k=\tfrac12m_k\tilde F_k$. Also $\partial I/\partial v_k=m_k(u_k-u_G)$ with $u_G=\sum_lm_lu_l/M$. Hence $(2/m_k)\,\partial V/\partial v_k=u_k-u_G+\tilde F_k$. By antisymmetry $\sum_{l=1}^5m_l\tilde F_l=0$. Since $u_5=0$, also $\sum_{l\le4}m_l(u_l-u_G)=m_5u_G$. Hence $(2/m_5)\sum_{l\le4}\partial V/\partial v_l=u_G-\tilde F_5$ and $(2A\nabla_vV)_k=u_k+\tilde F_k-\tilde F_5=Z_k$. Exchanging $u$ and $v$ gives the rows $W_k$.
\end{proof}

A point of $X_m$ is thus a critical point of $V$ restricted to $\{D_{kl}=0\}$. At fixed masses all points of a family in $X_m$ are critical points of the same $V$. The next lemma is \cite[Lemma~2]{AK12}, in the coordinates of the slice.

\begin{lemma}\label{lem:U}
Let $M\ne0$.
\begin{enumerate}[label=(\alph*)]
\item On $X_m$, $U=\sum_km_kz_kw_k=I$, with $z,w$ recentred at the centre of mass.
\item $U$ is constant on every irreducible component of $X_m$ and of $Y_m$, in particular on every irreducible curve $C\subseteq Y_m$.
\item For $m\in(\R_{>0})^5$, $U=I>0$ at the point of $Y_m$ given by a real central configuration.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) In the coordinates $z,w$ of \eqref{eq:AK4} the centre of mass is at the origin and $I=\sum_km_kz_kw_k$. We multiply the equation for $z_k$ by $m_kw_k$ and sum over $k$. By antisymmetry and $p_{kl}^2z_{kl}w_{kl}=1$, $\sum_km_kz_kw_k=\sum_{k<l}m_km_lp_{kl}^3z_{kl}w_{kl}=\sum_{k<l}m_km_lp_{kl}=U$. (b) By Lemma~\ref{lem:potential} and the invertibility of $A$, the gradient of $V$ on $\{D_{kl}=0\}$ vanishes at every point of $X_m$. The differential of $V$ therefore vanishes on the tangent space of each irreducible component at its smooth points, which form a connected dense open subset. Hence $V$ is constant on the component. By (a), $V=\tfrac32U$ on $X_m$. (c) is the identity $\lambda I=U$ of the proof of Lemma~\ref{lem:realpos}, with $\lambda=1$.
\end{proof}

\begin{corollary}\label{cor:U}
Let $m\in(\R_{>0})^5$. If $Y_m$ contains no irreducible curve on which some $r_{kl}^2$ is non-constant and $U$ is not identically zero, then the planar five-body problem with masses $m$ has finitely many central configurations up to similarity.
\end{corollary}

\begin{proof}
Infinitely many central configurations give infinitely many points of $Y_m$ with $U>0$ (Lemmas~\ref{lem:realpos}, \ref{lem:slice} and \ref{lem:U}(c)), hence an irreducible curve $C\subseteq Y_m$ on which $U$ is a positive constant. The fibres of $(u,v,p)\mapsto(r_{kl}^2)_{k<l}$ on $Y_m$ have at most $2^{13}$ points (fix the distances: $u_1=1$ determines $v_1$, each $u_k$ is a root of a quadratic, and the signs of the $p_{kl}$ are free). Hence some $r_{kl}^2$ is not constant on $C$.
\end{proof}

We call $m\in(\R_{>0})^5$ \emph{settled} if $Y_m$ contains no irreducible curve on which some $r_{kl}^2$ is non-constant and $U\not\equiv0$. We call such a curve \emph{nontrivial}. A settled mass vector has finitely many central configurations up to similarity. The isomorphisms of Lemma~\ref{lem:slice}(b) multiply the $r_{kl}^2$ and $U$ by nonzero constants or permute them. Thus settledness is invariant under positive scaling and permutation of the masses.

\subsection{Places and balancing}\label{ss:balancing}
Let $C\subseteq Y_m$ be a nontrivial curve. The functions $u_k$, $v_k$, $u_{kl}$, $v_{kl}$, $p_{kl}$, $r_{kl}^2$ and $\cP_{ij}$ are units of the coordinate ring $\C[C]$: for instance $1/p_{kl}=p_{kl}u_{kl}v_{kl}$. A non-constant unit has a zero and a pole on the smooth projective model $\bar C$ of $C$. All of them lie at the finitely many \emph{places at infinity}, the points of $\bar C$ not over $C$. For a place $P$ at infinity with valuation $\val_P$ put
\[
\rho(P)=\big(-\val_P(p_{kl})\big)_{k<l}\in\Z^{10}.
\]
Since the divisor of each $p_{kl}$ has degree zero,
\begin{equation}\label{eq:bal}
\sum_{P\text{ at }\infty}\rho(P)=0,
\end{equation}
and $\rho(P)\ne0$ for some $P$, because some $p_{kl}$ is not constant.

On $X_m$ we put $z_k=u_k-F_5(u)$ and $w_k=v_k-F_5(v)$ for $k\le5$. Since the rows $Z_k$, $W_k$ say $F_k(u)=F_5(u)$ and $F_k(v)=F_5(v)$, the pair $(z,w)$ solves the first two equations of \eqref{eq:AK4} (in the variables $z=x+iy$, $w=x-iy$), with $z_k-z_l=u_{kl}$ and $w_k-w_l=v_{kl}$. Following \cite[p.~545]{AK12} we put $Z_{kl}=p_{kl}^3z_{kl}$, $W_{kl}=p_{kl}^3w_{kl}$ (\cite{AK12} write $z_{kl}$ for $z_l-z_k$, but signs play no role below),
\[
\cZ=(z_1,\dots,z_5,Z_{12},\dots,Z_{45}),\qquad \cW=(w_1,\dots,w_5,W_{12},\dots,W_{45}),
\]
and $\cN=\|\cZ\|\,\|\cW\|$ with the maximum norm. The rotation $s\cdot$ of Lemma~\ref{lem:slice} multiplies $\cZ$ by $s$ and $\cW$ by $s^{-1}$. Thus $\cN$ is a function on $Y_m$. Since $Z_{kl}W_{kl}=p_{kl}^4$ and $|z_{kl}w_{kl}|\le4\cN$,
\begin{equation}\label{eq:Nbounds}
|r_{kl}|^{-4}\le\cN,\qquad |r_{kl}|^{2}\le4\cN .
\end{equation}
A place $P$ at infinity is \emph{singular} if $\cN$ is unbounded near $P$, that is, if $\val_P(\cZ)+\val_P(\cW)<0$, where $\val_P$ of a vector is the minimum over its components. At a place $P$ that is not singular, \eqref{eq:Nbounds} bounds every $r_{kl}^2$ above and away from zero. So $\rho(P)=0$. Hence the sum \eqref{eq:bal} runs over the singular places and at least one place is singular.

\subsection{Diagrams}\label{ss:diagrams}
Along a singular place, \cite[p.~545]{AK12} draws a circle around the body $k$ if $z_k$ is of maximal order among the components of $\cZ$, and a stroke between $k$ and $l$ if $Z_{kl}$ is of maximal order. This gives the $z$-diagram. In the same way $\cW$ gives the $w$-diagram. An edge is a $z$-edge, a $w$-edge or a $zw$-edge according to which strokes are present. The conclusion of \cite[\S7]{AK12} is that every singular sequence approaches one of the sixteen diagrams of their Fig.~11, up to renumbering of the bodies and exchange of the colours $z$ and $w$. Table~\ref{tab:diagrams} in Appendix~\ref{app:types} lists them with the types used below. We say that a singular place has \emph{diagram} $d$ and \emph{labelling} $\lambda$ if it approaches diagram $d$ with the renumbering $\lambda$.

\subsection{Inputs from \cite{AK12}}\label{ss:akinputs}
We use the following results of \cite{AK12}, as stated there:
\begin{enumerate}[label=(\roman*)]
\item the conclusion of \S7: every singular sequence approaches one of the sixteen diagrams of Fig.~11, which are listed in Table~\ref{tab:diagrams};
\item Estimates~1 and~2 of \S3, which say, in the notation there, that for each pair, $\varepsilon\preceq r_{kl}\preceq\varepsilon^{-2}$ and $\varepsilon^2\preceq z_{kl}\preceq\varepsilon^{-2}$, with $r_{kl}\approx\varepsilon$ exactly at the $zw$-edges and $z_{kl}\approx\varepsilon^2$ exactly at the maximal $w$-edges; and on a $z$-stroke, $\varepsilon\preceq r_{kl}\preceq1$, $\varepsilon\preceq z_{kl}\preceq\varepsilon^{-2}$ and $\varepsilon^2\preceq w_{kl}\preceq\varepsilon$, where $r_{kl}\approx\varepsilon$, $z_{kl}\approx\varepsilon$ and $w_{kl}\approx\varepsilon$ each characterize a $zw$-edge, and $r_{kl}\approx1$, $z_{kl}\approx\varepsilon^{-2}$ and $w_{kl}\approx\varepsilon^2$ each characterize a maximal $z$-edge; Rules~1b--1e and~2e of \S3, Proposition~1 of \S5.4 and Proposition~4 of \S7.1;
\item the mass conditions of \S8: conditions (23), (29), (32), (36) and (40), the facts of \S8.7 quoted in \S\ref{ss:d7} (among them the reduction to the system of \S5.4), and \S\S8.9.1--8.9.2;
\item the proof of Proposition~9, which we use in one place where we also give a second argument.
\end{enumerate}
The hypotheses $m_k\ne0$, $M\ne0$, $M_Q\ne0$ for each four-body subset $Q$, $m_a+m_b\ne0$ and $m_k^3+m_l^3\ne0$, under which these results are stated, hold for positive masses. In diagram~$3$ of Fig.~11 the colours of the four sides alternate around the square, $z$ on $14$ and $23$ and $w$ on $12$ and $34$. Note that the alternation is also forced by Rule~2g of \cite[\S3]{AK12} and by the order calculus of \S\ref{ss:orders}, which admits only the two alternating colourings of the sides. It occurs at places of the modified Roberts continuum \cite[pp.~543--544]{AK12}.