1. (ii’)

$\exists m\in[r_{12}-1]\quad\nu_{1}<\frac{1}{m}\leq\nu_{2}$.

First assume condition (ii) is satisfied. Since $\nu_{1}\in\hat{T}$, due to
(66) there exists $m\in[r_{12}-1]$ such that
$\nu_{1}\in(\frac{1}{m+1},\frac{1}{m})$. Using (ii) with $i=m+1$ and $\ell=1$
shows that $\nu_{2}\geq\frac{1}{m}$, therefore (ii’) is satisfied.

Conversely, assume condition (ii’) is satisfied. We then have $m\in[r_{12}-1]$
such that

$\nu_{1}<\frac{1}{m}\leq\nu_{2}\,,$ (67)

and we want to check that condition (ii) is satisfied. For $i\in[m]$, we have
$\nu_{1}<\frac{1}{m}\leq\frac{1}{i}$ therefore $\nu_{1}\in T_{1}^{(i)}$, so
(ii) is automatically satisfied. For $i\geq m+1$ and $\ell\in[i-1]$, assume
that $\nu_{1}\in\hat{T}_{\ell}^{(i)}$, i.e.

$\frac{\ell}{i}<\nu_{1}<\frac{\ell}{i-1}\,.$ (68)

Combining (67) and (68) we see that $\frac{\ell}{i}<\frac{1}{m}$. This gives
$\ell m<i$, and as this inequality involves only integers, we have actually
$\ell m\leq i-1$, which gives $\frac{\ell}{i-1}\leq\frac{1}{m}$. The second
inequality in (67) yields $\frac{\ell}{i-1}\leq\frac{1}{m}\leq\nu_{2}$, which
is what we needed to check.

All in all, we have proved that, under the assumption $\nu_{1}\in\hat{T}$,
(ii) is equivalent to (ii’). Consequently, “(i) or (ii)” is equivalent to “(i)
or (ii’)”. Eventually, condition (i) where we remove the cases
$\nu_{1}=\frac{1}{m}$ for some $m\in[r_{12}-1]$ is exactly condition (C-ii),
and condition (ii’) where we include those cases is exactly condition (C-i). ∎

####  3.4.3. Applying the local criterion

We illustrate how the globalisation criterion of Theorem 3.14 applies in some
typical situations. Suppose that we have a non-resonant pair of points
$\\{p_{1},p_{2}\\}\subseteq x^{-1}(q)$. Then, globalisation holds provided
(C-i) or (C-ii) holds.

  * •

If $\omega_{0,1}$ has a pole at $p_{1}$ (i.e. $\bar{s}_{1}\leq 0$), (C-ii)
holds.

  * •

If $\omega_{0,1}$ is holomorphic but nonzero at $p_{1}$ (i.e.
$\bar{s}_{1}=s_{1}=1$) and $r_{2}>r_{1}$, then (C-i) holds with
$m=r_{1}<r_{12}=r_{2}$. Likewise if $\omega_{0,1}$ is holomorphic but nonzero
at $p_{2}$, with $m=r_{2}$.

  * •

If $\nu_{2}\geq 1$ and we want (C-i) or (C-ii), then we must have $\nu_{1}\leq
1$ (this realises (C-ii) if $r_{1}=r_{2}=1$ or (C-i) with $m=1$ if
$r_{12}>1$). In particular, if $p_{1}$ and $p_{2}$ are unramified
($r_{1}=r_{2}=1$), this means that $\omega_{0,1}$ cannot have a zero both at
$p_{1}$ and $p_{2}$.

  * •

Suppose that $\bar{s}_{1},\bar{s}_{2}\geq 2$, $r_{1},r_{2}\geq 2$, and
$0<\nu_{1}\leq\nu_{2}<1$. Then (C-ii) is never satisfied. We have
$\bar{s}_{1}=s_{1}$ and $\bar{s}_{2}=s_{2}$ (because $\nu_{k}\in(0,1)$ amounts
to $\bar{s}_{k}\in(0,r_{k})$, and thus $\bar{s}_{k}$ cannot be divisible by
$r_{k}$). By local admissibility, we can write $r_{k}=a_{k}s_{k}+b_{k}$, with
$0\leq a_{k}<r_{k}$ and $b_{k}\in\\{1,s_{k}-1\\}$. We claim that (C-i) is
satisfied if and only if $a_{1}>a_{2}$.

To justify the last claim, we rewrite (C-i) as the existence of
$m\in[r_{12}-1]$ such that $\frac{r_{2}}{s_{2}}\leq m\leq\frac{r_{1}}{s_{1}}$,
which is equivalent to $a_{2}+\frac{b_{2}}{s_{2}}\leq m\leq
a_{1}+\frac{b_{1}}{s_{1}}$. This is clearly satisfied if $a_{1}>a_{2}$, by
taking $m=a_{1}<r_{1}\leq r_{12}-1$. This is clearly not satisfied for
$a_{1}<a_{2}$. And, it were satisfied for $a_{1}=a_{2}$, we would have an
integer $m^{\prime}=m-a_{1}$ such that $\frac{b_{2}}{s_{2}}\leq
m^{\prime}\leq\frac{b_{1}}{s_{1}}$, which is impossible since
$\frac{b_{k}}{s_{k}}\in(0,1)$ for $k=1,2$.

Here is a noteworthy consequence of the two first points. Assume that
$P\subseteq P^{\prime}\subseteq x^{-1}(q)$ are given, that we already know
that globalisation over $P$ holds, and that $P^{\prime}\setminus P$ consists
only of points where $\omega_{0,1}$ does not have a zero. Then, if any pair
$\\{p_{1},p_{2}\\}$ with $p_{1}\in P^{\prime}\setminus P$ and $p_{2}\in
P^{\prime}$ is non-resonant, we can conclude that globalisation over
$P^{\prime}$ holds.

###  3.5. Globalisation above an open set

We go back to Proposition 3.4. Given a spectral curve, to globalise
topological recursion above an open set $V\subseteq
x(\Sigma)\subseteq\mathbb{P}^{1}$ (or above the whole $x(\Sigma)$), we need to
verify the conditions of Theorem 3.14 for all pairs of points in $x^{-1}(q)$,
and this for each $q\in V$ (or $q\in x(\Sigma)$). This looks like a daunting
task. Fortunately, for all points $q\in x(\Sigma)$ that are not branch points
of $x$, the conditions that guarantee vertical globalisation are rather simple
and easy to check. This is the content of the next proposition.

######  Theorem 3.15 (Conditions for unramified vertical globalisation).

Let $(\omega_{g,n})_{(g,n)\in\mathbb{Z}_{\geq 0}\times\mathbb{Z}_{>0}}$ be a
system of correlators on a locally admissible spectral curve
$\mathcal{S}=(\Sigma,x,\omega_{0,1},\omega_{0,2})$ that satisfies topological
recursion. Suppose that $q\in x(\Sigma)$ is not a branch point and let
$x^{-1}(q)=\\{p_{1},\ldots,p_{d}\\}$. Then topological recursion can be
globalised above $q$ if the following two conditions are satisfied:

  * •

At most one of the $p_{i}$ has $\bar{s}_{i}\geq 2$ (i.e. the $1$-form
$\omega_{0,1}$ cannot vanish at two distinct points in the same fibre of $x$).

  * •

For any $i,j$ such that $\bar{s}_{i}=\bar{s}_{j}$, we have
$\tau_{i}\neq\tau_{j}$ (i.e. if the $1$-form $\omega_{0,1}$ has the same order
at $p_{i}$ and at $p_{j}$, it must have distinct leading order coefficients
when read in the standard coordinate).

When these two conditions are met we say that $\omega_{0,1}$ _separates fibres
at $q$_.

###### Proof.

Since $q\not\in\mathsf{Br}$, we have $r_{i}=1$ for all $i\in[d]$. In
particular the standard coordinate is unambiguously defined. Therefore, by
Theorem 3.14, for all pairs $\\{p_{i},p_{j}\\}\in x^{-1}(q)$, with the
ordering $\bar{s}_{i}\leq\bar{s}_{j}$, we would like to verify
$\bar{s}_{i}\leq 1$ (condition (C-i) or (C-ii)). Since this has to hold for
all pairs of points in $x^{-1}(q)$, we conclude that at most one $p_{i}$ can
have $\bar{s}_{i}\geq 2$. Finally, condition (C3) of Lemma 3.12 imposes that
$\tau_{i}\neq\tau_{j}$ whenever $\bar{s}_{i}=\bar{s}_{j}$. ∎

Daunting it is no more! To show that topological recursion can be globalised
above an open set $V\subseteq x(\Sigma)$, or fully globalised, all we have to
check is that topological recursion can be globalised vertically above all
branch points (by verifying the conditions in Theorem 3.14, this is a finite
number of tests), and that $\omega_{0,1}$ separates fibres at other points, in
the sense of Theorem 3.15.

###  3.6. Digression: local vs. global topological recursion

In order to handle topological recursion on the normalisation of possibly
singular spectral curves, [BKS23] identified sufficient conditions as well as
necessary conditions for the global topological recursion to be well defined,
i.e. for the residue formula with a sum over subsets
$Z\subseteq\mathfrak{f}^{\prime}(z)$ of the whole fibre (instead of just
subsets $Z\subseteq\mathfrak{f}^{\prime}_{p}(z)$ of the local fibre at $p$ in
(12)) to produce symmetric multi-differentials. This is a weaker property than
globalisation: the latter requires the correlators produced by global
topological recursion to coincide with those produced by local topological
recursion (12).

It is worth pausing for a moment to compare our results with theirs. While the
sufficient conditions found in [BKS23] could potentially be weaker than the
sufficient conditions for globalisation given in Theorem 3.14 (since global
topological recursion could potentially produce symmetric differentials that
differ from those produced by local topological recursion), the necessary
conditions found in [BKS23] should be implied by the sufficient conditions of
Theorem 3.14 (since if topological recursion can be globalised, then certainly
global topological recursion should produce symmetric differentials). Yet,
there are three small differences between the setting of spectral curves
adopted in the present article and in [BKS23]. First, they allow
$\bar{s}_{p}=\infty$, i.e. $\omega_{0,1}$ can be identically zero on some
connected component, but we do not. Second, we allow $\bar{s}_{p}\neq s_{p}$,
while they impose $\bar{s}_{p}=s_{p}$ (i.e. $r_{p}$ does not divide
$\bar{s}_{p}$). Third, we allow $s_{p}\leq-1$, while they impose $s_{p}>0$.
For this reason, we quote results from [BKS23] only after specialisation to
the intersection of the two setting.

Let us start by checking that the necessary conditions for global topological
recursion to be well-defined are implied by the sufficient conditions for
globalisation above any branch point found in Theorem 3.14.

######  Theorem 3.16.

[BKS23, Theorem 2.13 and Proposition 2.16] Let $\mathcal{S}$ be a finite
spectral curve such that, for any $p\in\mathsf{Ram}$, we have
$s_{p}=\bar{s}_{p}\in\mathbb{Z}_{>0}$ and any pair of points in the fiber
$x^{-1}(x(p))$ is non-resonant. Then the global topological recursion produces
symmetric $n$-differentials $\omega_{0,n}$ for any $n\geq 3$, if and only if
for any branch point $q$ the three following conditions are simultaneously
satisfied

  * (C-a)

for any $p\in x^{-1}(q)$, we have $r_{p}=\pm 1\,\,{\rm mod}\,\,s_{p}$;

  * (C-b)

for all pairs $\\{p_{1},p_{2}\\}\subseteq x^{-1}(q)$ such that
$\min(s_{1},s_{2})\geq 3$ and for which there exists $\epsilon\in\\{\pm 1\\}$
with $r_{k}=\epsilon\,\,{\rm mod}\,\,s_{k}$ for $k=1,2$, we have
$\lfloor\frac{1}{\nu_{1}}\rfloor\neq\lfloor\frac{1}{\nu_{2}}\rfloor$;

  * (C-c)

for all triplets $\\{p_{1},p_{2},p_{3}\\}\subseteq x^{-1}(q)$ such that
$\lfloor\frac{1}{\nu_{1}}\rfloor=\lfloor\frac{1}{\nu_{2}}\rfloor=\lfloor\frac{1}{\nu_{3}}\rfloor$,
then $s_{k}=1$ for some $k\in\\{1,2,3\\}$.

######  Lemma 3.17.

Let $\mathcal{S}$ be a locally admissible finite spectral curve such that
$\bar{s}_{p}=s_{p}\in\mathbb{Z}_{>0}$ for any $p\in x^{-1}(\mathsf{Br})$. The
condition “(C-i) or (C-ii)” of Theorem 3.14 imply the conditions “(C-a) and
(C-b) and (C-c)” of Theorem 3.16.

###### Proof.

Since $s_{p}>0$, local admissibility imposes $r_{p}=\pm 1\,\,{\rm
mod}\,\,s_{p}$ , so (C-a) is part of the assumptions.

If a pair $\\{p_{1},p_{2}\\}\subseteq x^{-1}(q)$ has $\min(s_{1},s_{2})\geq 3$
and satisfy $r_{k}=\pm 1\,\,{\rm mod}\,\,s_{k}$ for $k=1,2$, then
$\frac{1}{\nu_{k}}=\frac{r_{k}}{s_{k}}$ is an irreducible fraction which
cannot be an integer. If (C-i) is satisfied this implies that there is an
integer in $(\frac{1}{\nu_{1}},\frac{1}{\nu_{2}})$ therefore
$\lfloor\frac{1}{\nu_{1}}\rfloor=\lfloor\frac{1}{\nu_{2}}\rfloor$ which is the
conclusion of (C-b). If (C-ii) holds, then the premise of (C-b) does not and
there is nothing to check.

Assume we have a triplet $\\{p_{1},p_{2},p_{3}\\}\subseteq x^{-1}(q)$ such
that
$\lfloor\frac{1}{\nu_{1}}\rfloor=\lfloor\frac{1}{\nu_{2}}\rfloor=\lfloor\frac{1}{\nu_{3}}\rfloor$.
If (C-i) is satisfied for the pair $\\{p_{1},p_{2}\\}$, then $\nu_{1}=\nu_{2}$
is an integer, in particular $s_{1}=s_{2}=1$. If (C-ii) is satisfied for the
pair $\\{p_{1},p_{2}\\}$, then $\nu_{1}>0$ imposes $r_{1}\geq r_{2}$ and
$\nu_{1}=\frac{1}{r_{1}}$, hence $s_{1}=1$. So, the conclusion of (C-c) holds.
∎

While sufficient conditions for globalisation should imply necessary
conditions for global topological recursion to be well defined, there need not
be a logical relation between sufficient conditions for globalisation and
sufficient conditions for global topological recursion to be well-defined. In
fact, the next lemma shows that the sufficient conditions for global
topological to be well defined from [BKS23] are weaker than the sufficient
conditions for globalisation from Theorem 3.14: there are exceptional cases
that do not satisfy the conditions of Theorem 3.14 for which global
topological recursion produces symmetric differentials.

######  Theorem 3.18.

[BKS23, Theorem 2.11] Let $\mathcal{S}$ be a locally admissible finite
spectral curve such that, for any $p\in x^{-1}(\mathsf{Br})$, we have
$s_{p}=\bar{s}_{p}\in\mathbb{Z}_{>0}$ and any pair of points in the fiber
$x^{-1}(x(p))$ is non-resonant. Above any branch point $q$, order the points
in $x^{-1}(q)$ as $p_{1},\ldots,p_{d}$ such that
$\nu_{1}\leq\nu_{2}\leq\cdots\leq\nu_{d}$. Assume that above any branch point
$q$ we have

$r_{1}=-1\,\,{\rm mod}\,\,s_{1}\qquad{\rm and}\quad\big{(}\forall
k\in\\{2,\ldots,d-1\\}\quad\,\,\,s_{k}=1\big{)}\qquad{\rm and}\qquad
r_{d}=1\,\,{\rm mod}\,\,s_{d}\,.$

Then global topological recursion (with sum over subsets
$Z\subseteq\mathfrak{f}^{\prime}(z)$ of the full fibre) is well defined.

######  Lemma 3.19.

Let $\mathcal{S}$ be a locally admissible finite spectral curve such that, for
any $p\in x^{-1}(\mathsf{Br})$, we have $s_{p}=\bar{s}_{p}\in\mathbb{Z}_{>0}$
and any pair of points in the fiber $x^{-1}(x(p))$ is non-resonant. Let
$q\in\mathsf{Br}$ and order the points in $x^{-1}(q)$ as above.

If $|x^{-1}(q)|>2$, then the sufficient conditions from Theorem 3.18 imply the
sufficient conditions “(C-i) or (C-ii)” for any pair of points in $x^{-1}(q)$
from Theorem 3.14.

The same holds for $|x^{-1}(q)|=2$, except when $(r_{1},s_{1},r_{2},s_{2})$
are such that $\min(s_{1},s_{2})\geq 2$ and
$\lfloor\frac{1}{\nu_{1}}\rfloor=\lfloor\frac{1}{\nu_{2}}\rfloor$, in which
case the pair $x^{-1}(q)$ does not satisfy “(C-i) or (C-ii)”.

###### Proof.

Assume the premises of Theorem 3.18. First consider $|x^{-1}(q)|>2$. For any
pair of points $\\{p_{i},p_{j}\\}\in x^{-1}(q)$ there are two possibilities:

  * •

either one of them is equal to $1$. Say it is $s_{i}$. Then
$\nu_{i}=\frac{1}{r_{i}}$, which implies that the pair satisfies“(C-i) or
(C-ii)”;

  * •

or the pair is $\\{p_{1},p_{d}\\}$. Then we have
$\nu_{1}\leq\nu_{2}=\frac{1}{r_{2}}\leq\nu_{d}$, hence (C-i) is satisfied with
$m=r_{2}$.

We now turn to $|x^{-1}(q)|=2$. If $s_{1}=1$ , then for $r_{1}\geq r_{2}$ we
have $\nu_{1}=\frac{1}{r_{12}}$ and (C-ii) is satisfied, and for $r_{1}<r_{2}$
we have $\nu_{1}=\frac{1}{r_{1}}\leq\nu_{2}$ so (C-i) is satisfied with
$m=r_{1}$. If $\min(s_{1},s_{2})\geq 2$, we may write
$r_{1}=r_{1}^{\prime}s_{1}+s_{1}-1$ and $r_{2}=r_{2}^{\prime}s_{2}+1$ for
$r_{1}^{\prime},r_{2}^{\prime}\in\mathbb{Z}_{\geq 0}$. Then
$r_{i}^{\prime}=\lfloor\frac{r_{i}}{s_{i}}\rfloor$ and

$r_{1}^{\prime}+\frac{s_{1}-1}{s_{1}}=\frac{1}{\nu_{1}}\geq\frac{1}{\nu_{2}}=r_{2}^{\prime}+\frac{1}{s_{2}}\,.$

If $r_{1}^{\prime}\neq r_{2}^{\prime}$, then (C-i) is satisfied with
$m=r_{1}^{\prime}$. However, if $r_{1}^{\prime}=r_{2}^{\prime}$, then (C-i) is
not satisfied, and $\frac{s_{1}}{r_{1}}>\frac{1}{r_{2}}\geq\frac{1}{r_{12}}$
prevents (C-ii) to be satisfied. ∎

In the exceptional cases of Lemma 3.19, global topological recursion produces
well-defined symmetric multi-differentials, but we do not know if these are
the same as the multi-differentials produced by local topological recursion.
We do not study this further here but it would be interesting to understand
what happens for such spectral curves. The exception is realised for instance
with $(r_{1},s_{2})=(3,2)$ and $(r_{2},s_{2})=(4,3)$.

##  4\. Spectral curves, algebraic curves and normalisations

So far we talked about spectral curves abstractly, according to Definition
2.1. In practice, many spectral curves come from algebraic curves. In the
literature on topological recursion, spectral curves are often obtained by
first considering an affine curve $P(x,y)=0$ and then “parametrising it” by
thinking of $x$ and $y$ as two meromorphic functions on a compact Riemann
surface. There are subtleties with this statement, in particular if the affine
curve (or its projectivisation) is singular, which is usually the case. In
this section we construct spectral curves from algebraic curves carefully, and
determine which kind of algebraic curves give rise to spectral curves: (1)
which are locally admissible; (2) on which topological recursion can be fully
globalised. Along the way we review classical aspects of the resolution of
singularities of algebraic curves which are relevant to the construction.

###  4.1. Non-compact spectral curves from affine curves

####  4.1.1. Setting

Consider an affine curve

$\mathfrak{C}=\\{(x,y)\in\mathbb{C}^{2}\,\,|\,\,P(x,y)=0\\}\,,$ (69)

where $P(x,y)$ is a polynomial of positive degrees $d_{x}$ in $x$ and $d_{y}$
in $y$. We assume that $P$ is reduced, which means that it can be decomposed
into pairwise distinct irreducible factors $P=P_{1}\cdots P_{N}$ for some
positive integer $N$. We assume that $P$ has no factors in $\mathbb{C}[x]$,
and that $\\{y=0\\}\not\subseteq\mathfrak{C}$. Let us write
$\mathfrak{C}_{i}=\\{p\in\mathbb{C}^{2}\,\,|\,\,P_{i}(p)=0\\}$ for the
irreducible affine curves corresponding to each irreducible component. We
denote by $\pi_{x}\colon\mathfrak{C}\rightarrow\mathbb{C}$ and
$\pi_{y}\colon\mathfrak{C}\rightarrow\mathbb{C}$ the two projections
$\pi_{x}(x,y)=x$ and $\pi_{y}(x,y)=y$.

To define a spectral curve from this data, we need to obtain a Riemann surface
(or a disjoint union thereof). However, the affine curve $\mathfrak{C}$ is a
Riemann surface (or a disjoint union of Riemann surfaces) if and only if it is
smooth. Yet, we often need to deal with singular affine curves. How can we
proceed?

The singular locus

$\mathfrak{s}=\left\\{p\in\mathbb{C}^{2}\quad|\quad
P(p)=\partial_{x}P(p)=\partial_{y}P(p)=0\right\\}\subset\mathfrak{C}\,.$ (70)

is a finite subset. It contains the points at which some irreducible component
$\mathfrak{C}_{i}$ is singular, and the intersection points between the $N$
components (if the curve is reducible). If $\mathfrak{s}\neq\emptyset$,
$\mathfrak{C}$ is not a Riemann surface. However,
$\mathfrak{C}^{\circ}=\mathfrak{C}\setminus\pi_{x}^{-1}(\pi_{x}(\mathfrak{s}))$,
obtained by removing the whole $\pi_{x}$-fibres of singular points, is a
disjoint union of $N$ non-compact Riemann surfaces. As $\mathfrak{C}$ has no
factors in $\mathbb{C}[x]$, $\pi_{x}$ is non-constant on any of the components
of $\mathfrak{C}^{\circ}$. In fact, it defines a finite-degree branched
covering555If we had only removed the singular points from $\mathfrak{C}$,
instead of their whole $\pi_{x}$-fibre, we would have obtained a disjoint
union of Riemann surfaces, but $\pi_{x}$ may not have been a finite-degree
branched covering.. The curve $\mathfrak{C}^{\circ}$ is connected if and only
if $\mathfrak{C}$ is irreducible. Let

$\mathfrak{r}=\left\\{p\in\mathbb{C}^{2}\quad|\quad
P(p)=\partial_{y}P(p)=0\right\\},\quad\mathsf{Ram}\coloneqq\big{(}\mathfrak{r}\setminus\pi_{x}^{-1}(\pi_{x}(\mathfrak{s}))\big{)}\subset\mathfrak{C}^{\circ}\,.$
(71)

$\mathsf{Ram}$ is the set of ramification points of
$\pi_{x}\colon\mathfrak{C}^{\circ}\rightarrow\mathbb{C}$.

######  Definition 4.1 (Non-compact spectral curve associated to a reduced
affine curve).

Let $\mathfrak{C}=\\{P(x,y)=0\\}\subset\mathbb{C}^{2}$ be a reduced affine
curve that has no factors in $\mathbb{C}[x]$ and such that
$\\{y=0\\}\not\subseteq\mathfrak{C}$, $\pi_{x}$ and $\pi_{y}$ the two
canonical projections, $\mathfrak{s}$ the set of singular points and
$\mathfrak{C}^{\circ}=\mathfrak{C}\setminus\pi_{x}^{-1}(\pi_{x}(\mathfrak{s}))$.
Given a fundamental bidifferential $\omega_{0,2}$ on $\mathfrak{C}^{\circ}$,
we define the _non-compact spectral curve associated to $\mathfrak{C}$ and
$\omega_{0,2}$_ to be the quadruple
$(\mathfrak{C}^{\circ},\pi_{x},\pi_{y}\mathrm{d}\pi_{x},\omega_{0,2})$.

####  4.1.2. Local admissibility and globalisation

Spectral curves obtained by removing singularities from affine curves are
nice: the next lemma says that they are always locally admissible, and
topological recursion can always be fully globalised on them, regardless of
the type and location of the ramification points of $\pi_{x}$ (even if some
fibres contain several ramification points).

######  Lemma 4.2.

Let
$\mathcal{S}=(\mathfrak{C}^{\circ},\pi_{x},\pi_{y}\mathrm{d}\pi_{x},\omega_{0,2})$
be a non-compact spectral curve associated to a reduced affine curve
$\mathfrak{C}=\\{P(x,y)=0\\}\subset\mathbb{C}^{2}$ as in Definition 4.1. Then
$\mathcal{S}$ is locally admissible and topological recursion can be fully
globalised on $\mathcal{S}$.

###### Proof.

We first show that $\mathcal{S}$ is locally admissible. The ramification set
$\mathsf{Ram}\subset\mathfrak{C}^{\circ}$ is finite because $P$ is a
polynomial. Let $p\in\mathsf{Ram}\subset\mathfrak{C}^{\circ}$ and $r$ the
order of $\pi_{x}$ at $p$. Let $b=\pi_{y}(p)$. Then, near $p$, we have
$y=b+c\zeta+o(\zeta)$ with some $c\neq 0$ since $p$ is a smooth point. It
implies

$\pi_{y}\mathrm{d}\pi_{x}=\big{(}rb\zeta^{r-1}+rc_{1}\zeta^{r}+o(\zeta^{r})\big{)}\quad{\rm
near}\,\,p\,.$

If $b\neq 0$, the local parameters (Definition 2.3) at $p$ are
$(r_{p},s_{p},\bar{s}_{p})=(r,r+1,r)$. If $b=0$, the local parameters are
$r_{p}=r$ and $\bar{s}_{p}=s_{p}=r+1$. We conclude that the spectral curve is
locally admissible.

We move on to globalisation. Let
$q\in\pi_{x}(\mathfrak{C}^{\circ})\subseteq\mathbb{C}$. We need to show that
topological recursion can be globalised at $q$. Let
$\\{p_{1},\ldots,p_{d}\\}=\pi_{x}^{-1}(q)\subset\mathfrak{C}^{\circ}$ and
$b_{i}=\pi_{y}(p_{i})$. Then $b_{i}\neq b_{j}$ for $i\neq j$ as these
correspond to distinct points on the curve with same $\pi_{x}$-coordinate $q$.
If $r_{i}$ is the order of $\pi_{x}$ at $p_{i}$, we have in a standard
coordinate $y=b_{i}+c_{i}\zeta+o(\zeta)$ near $p_{i}$ for some $c_{i}\neq 0$.
It implies

$\pi_{y}\mathrm{d}\pi_{x}=\big{(}r_{i}b_{i}\zeta^{r_{i}-1}+r_{i}c_{i}\zeta^{r_{i}}+o(\zeta^{r_{i}})\big{)}\mathrm{d}\zeta\quad{\rm
near}\,\,p\,.$

As the $b_{i}$ are pairwise distinct, at most one of them can vanish. If this
happens, for this particular point $p_{i_{0}}$ we have
$\bar{s}_{i_{0}}>r_{i_{0}}$, and hence
$\nu_{i_{0}}=\frac{\bar{s}_{i_{0}}}{r_{i_{0}}}>1$, and for $j\neq i_{0}$ we
have $\bar{s}_{j}=r_{j}$ and $\tau_{j}=r_{j}b_{j}$, and hence
$\nu_{j}=\frac{\bar{s}_{j}}{r_{j}}=1$ and $\frac{\tau_{j}}{r_{j}}=b_{j}$. As a
result, for any pairs of points $\\{p_{j},p_{k}\\}\subseteq\pi_{x}^{-1}(q)$,
either $\nu_{j}=\nu_{k}=1$ and $\frac{\tau_{j}}{r_{j}}=b_{j}\neq
b_{k}=\frac{\tau_{k}}{r_{k}}$, or $\nu_{j}=1$ and $\nu_{k}>1$ (or vice versa).
Those satisfy the conditions of Theorem 3.14 ((C-i) with $m=1$), and
topological recursion can be globalised above $q$.

As $\pi_{x}$ is a finite-degree branched covering, this shows that topological
recursion can be fully globalised. ∎

###  4.2. Compact spectral curves after projectivisation and normalisation

Whenever possible it is preferable to work with compact spectral curves, which
is not the case for the spectral curves described in Subsection 4.1. This is
the setting in which the Hitchin integrable system on the Higgs moduli space
is usually studied, and we refer to [DM18a, BH19, CNST20] for the relevance of
topological recursion in this context. It is also interesting to have compact
spectral curves when studying deformation of curves and limits in families, as
we do in Section 5.

As a prototypical example, consider the affine curve
$\mathfrak{C}_{r,s}\subset\mathbb{C}^{2}$ defined as the vanishing locus of

$P_{r,s}(x,y)=x^{r-s}y^{r}-1\quad{\rm with}\quad r\geq 2\,\,{\rm
and}\,\,s\in[r-1]\,.$ (72)

We see that neither $\partial_{x}P$ nor $\partial_{y}P$ vanish on
$\mathfrak{C}_{r,s}$, since the lines $\\{x=0\\}$ and $\\{y=0\\}$ do not
intersect in $\mathfrak{C}_{r,s}$. Thus $\mathfrak{C}_{r,s}$ is smooth as an
affine curve: $\mathfrak{C}^{\circ}_{r,s}=\mathfrak{C}_{r,s}$. It is connected
if and only if $r$ and $s$ are coprime. However, since $\partial_{y}P$ never
vanishes on $\mathfrak{C}_{r,s}$, $\pi_{x}$ is unramified, hence topological
recursion on the non-compact spectral curve associated to $\mathfrak{C}_{r,s}$
as in Definition 4.1 produces correlators $\omega_{g,n}$ that vanish
identically for $2g-2+n>0$. This simply comes from the fact that ramification
points at $\infty$ were not taken into account. To get a non-trivial output
from topological recursion, a solution is to projectivise the affine curve and
resolve its singularities via normalisation. We note that taking into account
ramification points at $x=\infty$ in the topological recursion and working on
the normalisation first appeared in the work of Dumitrescu and Mulase [DM18a].

In this section we would like to understand systematically how this procedure
affects local admissibility (Subsection 4.2.2) and the possibility of
globalisation (Subsection 4.2.3). On the negative side, we find in Lemma 4.7
an obstruction to obey the criterion of globalisation, which pertains to the
location of the singularities of $C$. On the positive side, we introduce a
notion of global admissibility (Definition 4.9), tailored to show in Theorem
4.10 that topological recursion is fully globalisable on globally admissible
spectral curves.

####  4.2.1. Setting

Start with a reduced affine curve
$\mathfrak{C}=\\{(x,y)\in\mathbb{C}^{2}\,\,|\,\,P(x,y)=0\\}$, where $P(x,y)$
is a polynomial of positive degrees $d_{x}$ in the variable $x$ and $d_{y}$ in
the variable $y$, with no factors in $\mathbb{C}[x]$, and such that
$\\{y=0\\}\not\subseteq\mathfrak{C}$.

The first step is to compactify the curve by projectivising it. It is standard
to projectivise in $\mathbb{P}^{2}$, but this is not what we want to do. The
reason is that we want to keep the structure of the map given by projection on
the $x$-axis. It is therefore more natural to projectivise in
$\mathbb{P}^{1}\times\mathbb{P}^{1}$, so that the $x$-projection becomes
projection on the first $\mathbb{P}^{1}$ factor.

Let $([X_{0}:X_{1}],[Y_{0}:Y_{1}])$ be homogeneous coordinates on
$\mathbb{P}^{1}\times\mathbb{P}^{1}$. We use notations $\mathbf{0}=[0:1]$ and
$\boldsymbol{\infty}=[1:0]$ for the special points in $\mathbb{P}^{1}$ and
$p_{\epsilon\epsilon^{\prime}}=(\boldsymbol{\epsilon},\boldsymbol{\epsilon}^{\prime})$
with $\epsilon,\epsilon^{\prime}\in\\{0,\infty\\}$ for the special points in
$\mathbb{P}^{1}\times\mathbb{P}^{1}$. The bihomogeneisation of $P$ is

$F(X_{0},X_{1},Y_{0},Y_{1})=X_{1}^{d_{x}}Y_{1}^{d_{y}}P\left(\frac{X_{0}}{X_{1}},\frac{Y_{0}}{Y_{1}}\right)\,.$
(73)

This is a bihomogeneous polynomial of bidegree $(d_{x},d_{y})$: all terms in
$F$ have total degree $d_{x}$ in $(X_{0},X_{1})$ and total degree $d_{y}$ in
$(Y_{0},Y_{1})$. We write

$C=\\{p\in\mathbb{P}^{1}\times\mathbb{P}^{1}\quad|\quad F(p)=0\\}\,.$

The polynomial $F$ is reduced; we write $F=F_{1}\cdots F_{N}$ for a
decomposition in irreducible factors, and
$C_{i}=\\{F_{i}=0\\}\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ for its
irreducible components. Let $(d_{x}^{i},d_{y}^{i})$ be the bidegree of the
bihomogeneous irreducible polynomial $F_{i}$. From the assumptions on $P$ we
know that $F$ has no factor in $\mathbb{C}[X_{0},X_{1}]$,
$\\{Y_{0}=0\\}\not\subset C$, and $\\{Y_{1}=0\\}\not\subset C$.

The natural projection $\pi_{X}\colon C\rightarrow\mathbb{P}^{1}$, defined by
$\pi_{X}([X_{0}:X_{1}],[Y_{0}:Y_{1}])=[X_{0}:X_{1}]$ is non-constant on each
component of $C$ since $F$ has no factor in $\mathbb{C}[X_{0},X_{1}]$. The
bihomogeneisation of the 1-form $\pi_{y}\mathrm{d}\pi_{x}$ is

$\alpha=\frac{Y_{0}}{Y_{1}}\mathrm{d}\left(\frac{X_{0}}{X_{1}}\right)\,.$ (74)

Since $\\{Y_{0}=0\\}\not\subset C$ and $\\{Y_{1}=0\\}\not\subset C$, $\alpha$
is a well-defined meromorphic $1$-form on $C$, whose restriction to each
component is not identically zero.

In general, the curve $C$ may be singular. Let

$\mathsf{Sing}=\left\\{p\in\mathbb{P}^{1}\times\mathbb{P}^{1}\quad|\quad
F(p)=\partial_{X_{0}}F(p)=\partial_{X_{1}}F(p)=\partial_{Y_{0}}F(p)=\partial_{Y_{1}}F(p)=0\right\\}\subset
C$ (75)

be its singular locus. It is finite, and contains as before the points at
which some irreducible component $C_{i}$ is singular, and the intersection
points between the components.

If $\mathsf{Sing}\neq\emptyset$, we could proceed as in the previous section
and remove the singularities from $C$, but the result would not be compact.
Instead, we resolve the singularities by normalising the curve.

Given a projective curve $C$ as above, there exists a $\tilde{C}$, which is a
disjoint union of $N$ compact Riemann surfaces, and a holomorphic map
$\eta:\tilde{C}\rightarrow C$, such that $\eta^{-1}(\mathsf{Sing})$ is finite
and $\eta:\tilde{C}\setminus\eta^{-1}(\mathsf{Sing})\rightarrow
C\setminus\mathsf{Sing}$ is biholomorphic. $(\tilde{C},\eta)$ is called
_normalisation_ of $C$ — it is unique up to unique isomorphism. Roughly
speaking, each singularity $p\in C$ is replaced with $\mathsf{b}_{p}$ points
in $\tilde{C}$, where $\mathsf{b}_{p}$ is the number of branches meeting at
the singularity. The $\delta$-invariant of $p$ is by definition
$\delta_{p}=\dim_{\mathbb{C}}\big{(}\mathcal{O}_{\tilde{C},p}/\mathcal{O}_{C,p}\big{)}$.
It can be calculated using Milnor’s formula
$2\delta_{p}=\mathsf{j}_{p}+\mathsf{b}_{p}-1$, where $\mathsf{j}_{p}$ is the
dimension of the Jacobi algebra at $p$. In a local affine patch $f(u,v)=0$
where $p$ has coordinates $(u,v)=(0,0)$, this is
$\mathsf{j}_{p}=\dim_{\mathbb{C}}\big{(}\mathbb{C}[\\![u,v]\\!]/\langle\partial_{u}f,\partial_{v}f\rangle\big{)}$.

The normalisation procedure splits the components of $C$ and normalises each
component separately; we have
$\tilde{C}=\tilde{C}_{1}\sqcup\cdots\sqcup\tilde{C}_{N}$, where
$\tilde{C}_{i}$ is the normalisation of the irreducible curve $C_{i}$. The
_geometric genus_ of $C_{i}$, which we denote by $\mathsf{g}(C_{i})$, is the
genus of its normalisation $\tilde{C}_{i}$. The _arithmetic genus_ of $C_{i}$,
which we denote by $\mathsf{p}_{{\rm a}}(C_{i})$, is equal to
$\mathsf{p}_{{\rm a}}(C_{i})=(d_{x}^{i}-1)(d_{y}^{i}-1)$, where
$(d_{x}^{i},d_{y}^{i})$ is the bidegree of the $i$-th component. By the
adjunction formula, the difference between the arithmetic and the geometric
genus is the sum of the $\delta$-invariants at the singularities.

$\mathsf{g}(C_{i})=\mathsf{p}_{{\rm a}}(C_{i})-\sum_{p\in\mathsf{Sing}\cap
C_{i}}\delta_{p}\,.$ (76)

On $\tilde{C}_{i}$, the set of fundamental bidifferentials is an affine space
of dimension $\frac{1}{2}\mathsf{g}(C_{i})(\mathsf{g}(C_{i})+1)$: its
underlying vector space is the space of holomorphic bidifferentials on
$\tilde{C}_{i}$, symmetric in their two variables. In particular if
$\mathsf{g}(C_{i})=0$ we have a unique fundamental bidifferential
$\omega_{0,2}(z,z^{\prime})=\frac{\mathrm{d}z\,\mathrm{d}z^{\prime}}{(z-z^{\prime})^{2}}$
in terms of the uniformising coordinate $z$. This allows us to construct a
compact spectral curve.

######  Definition 4.3 (Compact spectral curve associated to a reduced affine
curve).

Let $\mathfrak{C}\subset\mathbb{C}^{2}$ be a reduced affine curve cut out by a
polynomial $P(x,y)$ that has no factors in $\mathbb{C}[x]$ and such that
$\\{y=0\\}\not\subseteq\mathfrak{C}$, and

$C=\Big{\\{}X_{1}^{d_{x}}Y_{1}^{d_{y}}P\left(\frac{X_{0}}{X_{1}},\frac{Y_{0}}{Y_{1}}\right)=0\Big{\\}}\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$
(77)

its bihomogeneisation. Let $\eta\colon\tilde{C}\rightarrow C$ be the
normalisation of $C$, $\pi_{X}\colon C\rightarrow\mathbb{P}^{1}$ be the
natural projection. Given a fundamental bidifferential $\omega_{0,2}$ on
$\tilde{C}$, we define the _compact spectral curve associated to $C$ and
$\omega_{0,2}$_ to be the quadruple $(\tilde{C},x,\omega_{0,1},\omega_{0,2})$,
where

$x=\pi_{X}\circ\eta\qquad\text{and}\qquad\omega_{0,1}=\eta^{*}\left(\frac{Y_{0}}{Y_{1}}\mathrm{d}\left(\frac{X_{0}}{X_{1}}\right)\right)\,.$

Note that $\tilde{C}$ is a disjoint union of $N$ compact Riemann surfaces and
$x\colon\tilde{C}\rightarrow\mathbb{P}^{1}$ is a finite-degree branched
covering. The spectral curve is connected if and only if $\mathfrak{C}$ is
irreducible.

This construction can be reversed.

######  Proposition 4.4.

Every compact spectral curve in which $\omega_{0,1}$ separates fibres (cf.
Theorem 3.15) can be obtained (up to the choice of a fundamental
bidifferential) from a unique reduced affine curve as in Definition 4.3.

###### Proof.

Given a compact spectral curve
$\mathcal{S}=(\Sigma,x,\omega_{0,1},\omega_{0,2})$, its image
$(x,\frac{\omega_{0,1}}{\mathrm{d}x})(\Sigma)\subseteq\mathbb{P}^{1}\times\mathbb{P}^{1}$
is a closed curve, and therefore it is cut out by a reduced bihomogeneous
polynomial $F$, unique up to rescaling. As $x$ is not allowed to be constant
on any component of $\Sigma$, $F$ cannot have factors in
$\mathbb{C}[X_{0},X_{1}]$. In particular it has no factor $X_{1}$, and
therefore comes from the homogeneisation of a polynomial in $x$.

As $\omega_{0,1}$ is a meromorphic $1$-form, not identically zero on any
component, it is also not identically $\infty$ on any component, and again as
$x$ is not allowed to be constant on a component, $\mathrm{d}x$ is not
identically zero, so $y=\frac{\omega_{0,1}}{\mathrm{d}x}$ is a nonzero
meromorphic function on any component. By the argument already used for $x$,
this shows that $F$ comes from homogeneisation of a polynomial in $y$.

Therefore, the zero locus of $F$ is a reduced curve obtained from the
procedure of Definition 4.3. By the universal property of its normalisation
$\eta\colon\tilde{C}\rightarrow C$, the map $(x,y)\colon\Sigma\rightarrow C$
factors through $\eta$. This is now a surjective map between smooth compact
curves. It is generically injective, as $\omega_{0,1}$ separates fibres.
Therefore, it must be injective. Thus it is an isomorphism. ∎

######  Example 4.5 (The $(r,s)$-spectral curve).

We illustrate the construction for the $(r,s)$-curve
$\mathfrak{C}^{(r,s)}=\\{x^{r-s}y^{r}-1=0\\}\subset\mathbb{C}^{2}$, where
$r\geq 2$ and $s\in[r-1]$ are coprime. Its bihomogeneisation is

$C^{(r,s)}=\\{X_{0}^{r-s}Y_{0}^{r}-X_{1}^{r-s}Y_{1}^{r}=0\\}\subset\mathbb{P}^{1}\times\mathbb{P}^{1}\,.$
(78)

If $s=r-1$, the curve is smooth and has genus $0$. If $s<r-1$, the curve has
arithmetic genus $(r-1)(r-s-1)$ but is singular at the two points
$p_{0\infty}$ and $p_{\infty 0}$. These singularities are unibranched as $r$
and $s$ are coprime: each of them is replaced by a single point in the
normalisation $\tilde{C}^{(r,s)}$. A local affine equation centered at any of
these singularities is $f(u,v)=u^{r-s}-v^{r}$, so the local $\mathsf{j}$ is
$(r-1)(r-s-1)$. This leads to a total $\delta=(r-1)(r-s-1)=\mathsf{p}_{{\rm
a}}(C^{(r,s)})$ and geometric genus is $\mathsf{g}(C^{(r,s)})=0$. In other
words $\tilde{C}^{(r,s)}\simeq\mathbb{P}^{1}$. We can write the normalisation
map explicitly by parametrising the equation

$\begin{array}[]{llcll}\eta&\colon&\tilde{C}^{(r,s)}&\longrightarrow&C^{(r,s)}\\\
&&[w_{0}:w_{1}]&\longmapsto&([w_{0}^{r}:w_{1}^{r}],[w_{1}^{r-s}:w_{0}^{r-s}])\end{array}\,.$
(79)

Then

$\begin{array}[]{llcll}x=\pi_{X}\circ\eta&\colon&\tilde{C}^{(r,s)}&\longrightarrow&\mathbb{P}^{1}\\\
&&[w_{0}:w_{1}]&\longmapsto&[w_{0}^{r}:w_{1}^{r}]\end{array}$ (80)

is a branched covering of degree $r$, with ramification points at $\mathbf{0}$
and $\boldsymbol{\infty}$, both of order $r$. The data of the spectral curve
is completed by giving the $1$-form

$\omega_{0,1}=\eta^{*}\alpha=\frac{w_{1}^{r-s}}{w_{0}^{r-s}}\mathrm{d}\left(\frac{w_{0}^{r}}{w_{1}^{r}}\right)\,,$

and the unique fundamental bidifferential existing on
$\tilde{C}^{(r,s)}=\mathbb{P}^{1}$.

$\omega_{0,2}(p,p^{\prime})=\frac{\mathrm{d}(w_{0}/w_{1})\mathrm{d}(w_{0}^{\prime}/w_{1}^{\prime})}{(w_{0}/w_{1}-w_{0}^{\prime}/w_{1}^{\prime})^{2}},\qquad
p=[w_{0}:w_{1}],\quad p^{\prime}=[w_{0}^{\prime}:w_{1}^{\prime}]\,.$

In the global coordinate $w=\frac{w_{0}}{w_{1}}$, the curve can be
parametrised as

$x(w)=w^{r},\qquad
y(w)=w^{s-r},\qquad\omega_{0,2}(w,w^{\prime})=\frac{\mathrm{d}w\,\mathrm{d}w^{\prime}}{(w-w^{\prime})^{2}}\,.$
(81)

At the ramification point $\mathbf{0}$ a standard coordinate is
$\zeta=\frac{w_{0}}{w_{1}}=w$. As $\omega_{0,1}=r\zeta^{s-1}\mathrm{d}\zeta$
the local parameters are

$r_{\mathbf{0}}=r,\qquad
s_{\mathbf{0}}=\bar{s}_{\mathbf{0}}=s,\qquad\tau_{\mathbf{0}}=r\,.$

At the ramification point $\boldsymbol{\infty}$, we rather have
$\zeta=\frac{w_{1}}{w_{0}}=\frac{1}{w}$ as standard coordinate. Then
$x=\zeta^{-r}$ and
$\omega_{0,1}=\zeta^{r-s}\mathrm{d}(\zeta^{-r})=-r\zeta^{-s-1}\mathrm{d}\zeta$.
From there we read the local parameters:

$r_{\boldsymbol{\infty}}=r,\qquad
s_{\boldsymbol{\infty}}=\bar{s}_{\boldsymbol{\infty}}=-s,\qquad\tau_{\boldsymbol{\infty}}=-r.$

Comparing with Definition 2.5, we see that the spectral curve
$\tilde{C}^{(r,s)}$ is locally admissible if and only if $r=\pm 1\,\,{\rm
mod}\,\,s$. Due to Theorem 2.17, the ramification point at
$\boldsymbol{\infty}$ does not contribute to the topological recursion.

This spectral curve in parametrised form (81) was first considered in
[BBCCN23], where it was proved that topological recursion is well-defined if
and only if $r=\pm 1\,\,{\rm mod}\,\,s$. This explains the origin of this
condition in Definition 2.5.

####  4.2.2. Local admissibility and globalisation

The next question we address is: when are the compact spectral curves obtained
in Definition 4.3 locally admissible, and when can topological recursion on
them be globalised? For this purpose, we need to analyse in detail the local
behavior of $\omega_{0,1}$.

######  Definition 4.6.

Let $L_{Y}=\\{\boldsymbol{\infty}\\}\times\mathbb{P}^{1}$ and
$L_{X}=\mathbb{P}^{1}\times\\{\boldsymbol{\infty}\\}$ be the two lines at
infinity in $\mathbb{P}^{1}\times\mathbb{P}^{1}$. We introduce the finite sets

$E=(L_{X}\cup L_{Y})\cap C,\qquad E^{\prime}=\big{(}L_{X}\cup\\{p_{\infty
0}\\}\big{)}\cap C\,.$

$E$ contains the points that are added when we go from the affine curve
$\mathfrak{C}$ to the projective curve $C$. That is: $C\setminus
E=\mathfrak{C}$. The set $E^{\prime}\subseteq E$ contains only those
additional points which are either poles of $y$, or poles of $x$ that are
zeros of $y$. The set $E^{\prime}$ is illustrated in Figure 2. It turns out
that, if we want $\tilde{C}$ to give a locally admissible spectral curve and
be suitable for globalisation, the singularities of $C$ are restricted to
appear in $E^{\prime}$, as we show in the next lemma. In particular,
$\mathfrak{C}$ must be smooth.

Figure 2. The locus $E^{\prime}$ is shown in blue. According to Lemma 4.7, the
possible singularities of $C$ can only be in $E^{\prime}$.

######  Lemma 4.7.

Let $\mathcal{S}=(\tilde{C},x,\omega_{0,1},\omega_{0,2})$ be the compact
spectral curve associated to a reduced affine curve $\mathfrak{C}$ as in
Definition 4.3. Suppose that $p\in C\setminus E^{\prime}$ is a singular point.

  * •

If $p$ is unibranched, then the spectral curve is not locally admissible.

  * •

If $p$ is multibranched, then either the spectral curve is not locally
admissible or the conditions of Theorem 3.14 for globalisation at
$q=\pi_{X}(p)\in\mathbb{P}^{1}$ are not satisfied.

###### Proof.

Let $p$ be a singular point in $\mathfrak{C}$. We write
$(\pi_{X}(p),\pi_{Y}(p))=(q,b)\in\mathbb{C}^{2}$.

Suppose that $p$ is unibranched. This means that $\eta^{-1}(p)$ contains only
one point $\tilde{p}$ in the normalisation $\tilde{C}$. Moreover, by the
definition of normalisation, in a standard coordinate near $\tilde{p}$ we have
$x=q+\zeta^{r}$ for some $r\geq 2$ and $y=b+c\zeta^{k}+o(\zeta^{k})$ for some
$k\geq 2$ (otherwise $p$ would not be singular) with $c\neq 0$. This implies:

$\omega_{0,1}=y\mathrm{d}x=r\zeta^{r-1}\big{(}b+c\zeta^{k}+o(\zeta^{k})\big{)}\mathrm{d}\zeta\quad{\rm
near}\,\,\tilde{p}\,.$

Regardless of whether $b=0$ or $b\neq 0$, the local parameter $s$ at
$\tilde{p}$ must satisfy $s>r+1$, since $k\geq 2$. Therefore the condition
(lA3) in Definition 2.5 is violated and the spectral curve is not locally
admissible at $\tilde{p}$.

Now suppose that $p$ is multibranched. Then there are at least two distinct
points $\tilde{p}_{1},\tilde{p}_{2}\in\eta^{-1}(p)$ in the normalisation
$\tilde{C}$. In a standard coordinate near $\tilde{p}_{i}$ we have
$x=q+\zeta^{r_{i}}$ for some $r_{i}\geq 1$ and
$y=b+c_{i}\zeta^{k_{i}}+o(\zeta^{k_{i}})$ for some $k_{i}\geq 1$ with
$c_{i}\neq 0$. This implies

$\omega_{0,1}=r_{i}\zeta^{r_{i}-1}\big{(}b+c_{i}\zeta^{k_{i}}+o(\zeta^{k_{i}})\big{)}\mathrm{d}\zeta\quad{\rm
near}\,\,\tilde{p}_{i}\,.$

The important fact is that $b$ is the same for both $\tilde{p}_{1}$ and
$\tilde{p}_{2}$, as they project to the same point $p\in C$. Thus, if $b\neq
0$, we have two points $\tilde{p}_{1}$ and $\tilde{p}_{2}$ in the fibre
$x^{-1}(q)$ with same aspect ratio
$\nu_{1}=\frac{\bar{s}_{1}}{r_{1}}=1=\frac{\bar{s}_{2}}{r_{2}}=\nu_{2}$ and
$\frac{\tau_{1}}{r_{1}}=b=\frac{\tau_{2}}{r_{2}}$, which contradicts the
sufficient conditions given in Theorem 3.14 for vertical globalisation above
$q=\pi_{X}(p)$. If $b=0$, either the curve is not locally admissible (if
$k_{1}\geq 2$ or $k_{2}\geq 2$, like in the unibranched case), or it does not
satisfy the conditions for globalisation of Theorem 3.14 (if $k_{1}=k_{2}=1$,
then $\bar{s}_{1}=r_{1}+1$ and $\bar{s}_{2}=r_{2}+1$, leading to $\nu_{1}>1$
and $\nu_{2}>1$).

Next, consider a singular point $p\in E\setminus E^{\prime}$. This means that
$\pi_{X}(p)=\boldsymbol{\infty}$ and $\pi_{Y}(p)=b\in\mathbb{C}^{*}$.

Suppose that $p$ is unibranched. $\eta^{-1}(p)$ contains only one point
$\tilde{p}$. In a standard coordinate near $\tilde{p}$ we have $x=\zeta^{-r}$
for some $r\geq 2$ and $y=b+c\zeta^{k}+o(\zeta^{k})$ for some $k\geq 2$ with
$c\neq 0$. This implies

$\omega_{0,1}=-r\zeta^{-r-1}\big{(}b+c\zeta^{k}+o(\zeta^{k})\big{)}\mathrm{d}\zeta\quad{\rm
near}\,\,\tilde{p}\,.$

Since $b\neq 0$, the local parameters at $\tilde{p}$ are $\bar{s}=-r$, but
$s>-r+1$, since $k\geq 2$. Therefore the spectral curve is not locally
admissible at $\tilde{p}$: it violates condition (lA3) in Definition 2.5.

Now suppose that $p$ is multibranched. Then there are at least two distinct
points $\tilde{p}_{1},\tilde{p}_{2}\in\eta^{-1}(p)$. In a standard coordinate
near $\tilde{p}_{i}$ we have $x=\zeta^{-r_{i}}$ for some $r_{i}\geq 1$ and
$y=b+c_{i}\zeta^{k_{i}}+o(\zeta^{k_{i}})$ for some $k_{i}\geq 1$ with
$c_{i}\neq 0$. This implies

$\omega_{0,1}=-r_{i}\zeta^{-r_{i}-1}\big{(}b+c_{i}\zeta^{k_{i}}+o(\zeta^{k_{i}})\big{)}\mathrm{d}\zeta\quad{\rm
near}\,\,\tilde{p}_{i}\,.$

The constant $b$ is the same for both $\tilde{p}_{1}$ and $\tilde{p}_{2}$.
Since $b\neq 0$, then we have two points $\tilde{p}_{1}$ and $\tilde{p}_{2}$
in the fibre $x^{-1}(q)$ with same aspect ratio
$\nu_{1}=\frac{\bar{s}_{1}}{r_{1}}=-1=\frac{\bar{s}_{2}}{r_{2}}=\nu_{2}$ and
$\frac{\tau_{1}}{r_{1}}=b=\frac{\tau_{2}}{r_{2}}$, which contradicts the
conditions of Theorem 3.14 for vertical globalisation above
$\pi_{X}(p)=\boldsymbol{\infty}$. ∎

We summarize the local parameters and whether the curve is locally admissible
at points $p\in C$ that are either smooth ramification points or unibranched
singularities in Table 1.

$p\in C$ | $x(p)$ | $y(p)$ | $\bar{s}$ | $s$ | $\nu$ | loc. adm.
---|---|---|---|---|---|---
smooth ramified | $\in\mathbb{C}$ | $0$ | $r+1$ | $r+1$ | $1+\frac{1}{r}$ | yes
$\in\mathbb{C}$ | $\in\mathbb{C}^{*}$ | $r$ | $r+1$ | $1$ | yes
$\in\mathbb{C}$ | $\infty$ | $r-1$ | $r-1$ | $1-\frac{1}{r}$ | yes
$\infty$ | $0$ | $-r+1$ | $-r+1$ | $-1+\frac{1}{r}$ | yes
$\infty$ | $\in\mathbb{C}^{*}$ | $-r$ | $-r+1$ | $-1$ | yes
$\infty$ | $\infty$ | $-r-1$ | $-r-1$ | $-1-\frac{1}{r}$ | yes
unibranched singular | $\in\mathbb{C}$ | $0$ | $>r+1$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}>r+1}$ | $>1$ | no
$\in\mathbb{C}$ | $\in\mathbb{C}^{*}$ | $r$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}>r+1}$ | $1$ | no
$\in\mathbb{C}$ | $\infty$ | $r-k$ |  | $<1$ | maybe
$\infty$ | $0$ | $-r+k$ |  | $>-1$ | maybe
$\infty$ | $\in\mathbb{C}^{*}$ | $-r$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}>-r+1}$ | $-1$ | no
| $\infty$ | $\infty$ | $-r-k$ |  | $<-1$ | maybe
Table 1.  Local parameters for smooth ramification points and unibranched
singularities of projectivised affine curves. In red we emphasise the reason
for not being locally admissible. In the table, $k$ is an integer $\geq 2$.

######  Remark 4.8.

We _did not prove_ that topological recursion can be globalised above
$q\in\mathbb{P}^{1}$ _if and only if_ the conditions of Theorem 3.14 hold; the
conditions of Theorem 3.14 are only sufficient conditions. Thus we cannot
conclude from the previous lemma that topological recursion cannot be
globalised if the affine curve $\mathfrak{C}$ contains multibranched
singularities. However, in this case we believe that the implication holds in
both directions, and that vertical globalisation does not work if the affine
curve has multibranched singularities. Henceforth we will require that
$\mathfrak{C}$ is smooth.

####  4.2.3. Global admissibility

Lemma 4.7 leads us to propose the following notion of‘global admissibility.

######  Definition 4.9 (Global admissibility).

Let $\mathcal{S}=(\tilde{C},x,\omega_{0,1},\omega_{0,2})$ be the compact
spectral curve associated to a reduced affine curve $\mathfrak{C}$ as in
Definition 4.3. We say that the spectral curve is _globally admissible_ if,
for any singular point $p$ of $C$:

  1. (gA1)

$p\in E^{\prime}$.

  2. (gA2)

$\mathcal{S}$ is locally admissible at all points in $\eta^{-1}(p)$.

  3. (gA3)

Every pair of points in $\eta^{-1}(p)$ is non-resonant and satisfies “(C-i) or
(C-ii)” from Theorem 3.14.

Global admissibility is all about the singular points of the projective curve
$C$. This definition is tailored for the following theorem.

######  Theorem 4.10 (Global admissibility implies local admissibility and
globalisation).

Let $\mathcal{S}=(\tilde{C},x,\omega_{0,1},\omega_{0,2})$ be the compact
spectral curve associated to a reduced affine curve $\mathfrak{C}$ as in
Definition 4.3. If $\mathcal{S}$ is globally admissible, then $\mathcal{S}$ is
locally admissible and topological recursion can be fully globalised on
$\mathcal{S}$.

###### Proof.

We start with local admissibility. First, since the affine curve
$\mathfrak{C}$ is smooth, by Lemma 4.2, we know that the spectral curve is
locally admissible at all ramification points in $\eta^{-1}(\mathfrak{C})$. By
global admissibility, we also know that it is locally admissible at all points
in $\eta^{-1}(p)$ for singular points $p\in\mathsf{Sing}\subset C$. It remains
to check local admissibility at the smooth points in $E$ which are ramified
for $x$. For such points $p$, $\eta^{-1}(p)$ contains a single point
$\tilde{p}$. Let $r\geq 2$ be the order of ramification.

(i) Case $\pi_{X}(p)=\boldsymbol{\infty}$ and $\pi_{Y}(p)=b\in\mathbb{C}$. In
a standard coordinate near $\tilde{p}$, we have $x=\zeta^{-r}$ and
$y=b+c\zeta+\mathcal{O}(\zeta^{2})$ for some $c\neq 0$. This implies

$\
\omega_{0,1}=-r\zeta^{-r-1}\big{(}b+c\zeta+\mathcal{O}(\zeta^{2})\big{)}\mathrm{d}\zeta\quad{\rm
near}\,\,\tilde{p}\,.$

If $b\neq 0$, the local parameters at $\tilde{p}$ are $\bar{s}=-r$ and
$s=-r+1$, since $c\neq 0$. If $b=0$, the local parameters are
$\bar{s}=s=-r+1$. In both cases the curve is locally admissible at
$\tilde{p}$.

(ii) Case $\pi_{X}(p)=q\in\mathbb{C}$ and $\pi_{Y}(p)=\boldsymbol{\infty}$. In
a standard coordinate near $\tilde{p}$, we have $x=q+\zeta^{r}$ and
$y\sim\frac{c}{\zeta}$ for some $c\neq 0$. This implies

$\omega_{0,1}\sim cr\zeta^{r-2}\mathrm{d}\zeta\quad{\rm near}\,\,\tilde{p}\,.$

The local parameters are $\bar{s}=s=r-1$, showing locally admissibility at
$p$.

(iii) Case $\pi_{X}(p)=\pi_{Y}(p)=\boldsymbol{\infty}$. In a standard
coordinate near $\tilde{p}$, we have $x=\zeta^{-r}$ and $y\sim\frac{c}{\zeta}$
for some $c\neq 0$. This implies

$\omega_{0,1}\sim-cr\zeta^{-r-1}\mathrm{d}\zeta\quad{\rm
near}\,\,\tilde{p}\,.$

The local parameters are $\bar{s}=s=-r-1$, and the curve is locally admissible
at $\tilde{p}$.

This concludes the proof of local admissibility, and we move on to
globalisation. Let $q\in\mathbb{P}^{1}$. We want to show that topological
recursion can be globalised above $q$. If $q\neq\boldsymbol{\infty}$ and
$\boldsymbol{\infty}\notin\pi_{Y}(\pi_{X}^{-1}(q))$, then all the preimages in
$\pi_{X}^{-1}(q)$ are in fact on the affine curve $\mathfrak{C}$ and by Lemma
4.2, we know that topological recursion can be globalised above $q$.

Suppose next that $q\neq\boldsymbol{\infty}$ but that there is one special
point $p_{0}\in\pi_{X}^{-1}(q)$ such that
$\pi_{Y}(p_{0})=\boldsymbol{\infty}$. All other points $p\in\pi_{X}^{-1}(q)$
are on $\mathfrak{C}$, and thus are smooth and have a unique preimage
$\tilde{p}$ in the normalisation. From the proof of Lemma 4.2 we know that all
such $\tilde{p}$ except potentially one have aspect ratio
$\nu=\frac{\bar{s}}{r}=1$ and $\omega_{0,1}$ separates fibres; there can be
one point $\tilde{p}$ with $\nu>1$ (this happens when there is a zero of
$\pi_{Y}$ in $\pi_{X}^{-1}(q)$). If $p_{0}$ is a smooth point in $C$, then it
has a unique preimage $\tilde{p}_{0}$ in $\tilde{C}$ and we have learned in
(ii) that $\bar{s}=s=r-1$ so $\nu<1$. If $p_{0}$ is singular, all preimages in
$\eta^{-1}(p_{0})$ have $\bar{s}<r-1$, so we still have $\nu<1$, and global
admissibility tells us that we can globalise over $\eta^{-1}(p_{0})$. Putting
all this together, if we pick a pair of points
$\\{\tilde{p}_{i},\tilde{p}_{j}\\}\subseteq x^{-1}(q)$, then:

  * •

either $\nu_{i}\leq 1$ and $\nu_{j}\geq 1$ (or vice-versa) and if
$\nu_{i}=\nu_{j}=1$ then $\omega_{0,1}$ separates fibres;

  * •

or $\nu_{i}<1$ and $\nu_{j}<1$ but both points are in $\eta^{-1}(p_{0})$ where
$p_{0}$ is a singular point, and by global admissibility the points satisfy
the conditions of Theorem 3.14.

Therefore topological recursion can be globalised above $q\in\mathbb{C}$.

It remains to consider the special point $q=\boldsymbol{\infty}$. If
$p\in\pi_{X}^{-1}(q)$ is such that $\pi_{Y}(p)\in\mathbb{C}^{*}$, then $p$
must be smooth (due to condition (gA1)) and have a unique preimage $\tilde{p}$
in the normalisation. By (i) it has $\bar{s}=-r$ and hence $\nu=-1$. Moreover,
such points $\tilde{p}$ map to different points $p$ in $C$, so $\omega_{0,1}$
separates fibres for all such points. If $p_{\infty 0}\in\pi_{X}^{-1}(q)$,
then there are two situations. If it is smooth, then it has a unique preimage
$\tilde{p}_{\infty 0}$, and by (i) the local parameter is $\bar{s}=-r+1$ and
hence $\nu=-1+\frac{1}{r}>-1$. If it is singular, we find $\bar{s}>-r+1$ and
thus $\nu>-1$ for all preimages in $\eta^{-1}(p_{\infty 0})$, and the
assumption of global admissibility guarantees that topological recursion can
be globalised over these preimages. Finally, if
$p_{\infty\infty}\in\pi_{X}^{-1}(q)$, then there are two situations. If it is
smooth, by (iii) the corresponding point in $\tilde{C}$ has $\bar{s}=-r-1$ and
hence $\nu=-1-\frac{1}{r}<-1$. If it is singular, we find $\bar{s}<-r-1$ and
thus $\nu<-1$ at all preimages in $\eta^{-1}(p_{\infty\infty})$, and by global
admissibility topological recursion can be globalised over these preimages.
Putting all this together, we conclude that if we pick a pair of points
$\\{\tilde{p}_{i},\tilde{p}_{j}\\}\subseteq x^{-1}(\boldsymbol{\infty})$, then

  * •

either $\nu_{i}\leq-1$ and $\nu_{j}\geq-1$ (or vice-versa) and if
$\nu_{i}=\nu_{j}=-1$ then $\omega_{0,1}$ separates fibres;

  * •

or $\nu_{i}<-1$ and $\nu_{j}<-1$ with both points in
$\eta^{-1}(p_{\infty\infty})$; by global admissibility the pair satisfy the
conditions of Theorem 3.14;

  * •

or $\nu_{i}>-1$ and $\nu_{j}>-1$ with both points in $\eta^{-1}(p_{\infty
0})$; by global admissibility the pair satisfy the conditions of Theorem 3.14.

Therefore topological recursion can be globalised as well above
$q=\boldsymbol{\infty}$. ∎

###  4.3. Global admissibility and Newton polygons

In this section we study how global admissibility of spectral curves can be
understood from the point of view of the Newton polygon of the original affine
curve. Recall from Definition 4.9 that for a compact spectral curve associated
to an affine curve to be globally admissible, the affine curve must be smooth,
and, in particular, irreducible. Thus from now we assume that we start with an
irreducible affine curve.

####  4.3.1. Generalities on Newton polygons

We first review classical facts about the resolution of singularities seen
from the point of view of the Newton polygon, see for instance [Kho77, BP00].
Let $\mathfrak{C}=\\{P(x,y)=0\\}\subset\mathbb{C}^{2}$ be an irreducible
affine curve of positive degree $d_{x}$ in $x$ and $d_{y}$ in $y$. Write

$P(x,y)=\sum_{(i,j)\in\mathbb{Z}_{\geq 0}^{2}}c_{ij}x^{i}y^{j}\,,$ (82)

and denote by ${\rm supp}(P)=\\{(i,j)\in\mathbb{Z}_{\geq
0}^{2}~{}|~{}c_{ij}\neq 0\\}$ the support of $P$.

######  Definition 4.11 (Newton polygon and diagram).

The _Newton polygon_ $\Delta(P)$ is the convex hull of ${\rm supp}(P)$ in
$\mathbb{R}^{2}_{\geq 0}$. Let $\Gamma^{+}(P)\subseteq\mathbb{R}_{\geq 0}^{2}$
be the convex hull of the union of the quadrants $(i,j)+\mathbb{R}_{\geq
0}^{2}$ over $(i,j)\in{\rm supp}(P)$. The _Newton diagram_ $\Gamma(P)$
associated to $P$ is the closure of the complement of $\Gamma^{+}(P)$ in
$\mathbb{R}_{\geq 0}^{2}$. We denote by
$\hat{\Gamma}(P)=\Gamma(P)\cap\mathbb{R}_{>0}^{2}$ the set of points in
$\Gamma(P)$ that are not on the $x$\- or $y$-axes.

It will be useful for us to distinguish between edges and maximal edges of
$\Delta(P)$, vertices and extremal vertices.

######  Definition 4.12 (Discrete geometry conventions).

An _integral point_ is a point in $\mathbb{Z}^{2}$. If
$A\subseteq\mathbb{R}^{2}$ is bounded, we denote
$|A|_{\mathbb{Z}}=|A\cap\mathbb{Z}^{2}|$ its number of integral points.

An _edge_ of a Newton polygon $\Delta(P)$ is a closed segment in the boundary
of $\Delta(P)$ whose only integral points are located at the endpoints. A
_maximal edge_ is a closed segment obtained by a maximal union of edges (cf.
Figure 3). Maximal edges are edges if and only if they do not contain any
integral interior points. An _extremal vertex_ is the endpoint of a maximal
edge (in particular, $\Delta(P)$ is the convex hull of its extremal vertices).
A _vertex_ is any integral point in the boundary of $\Delta(P)$ (it is not
required to be in the support of $P$).

The Newton diagram of $P$ characterises a potential singularity of
$\mathfrak{C}$ at the origin $(0,0)$. More precisely, the $\delta$-invariant
of the origin satisfies $\delta_{(0,0)}\geq|\hat{\Gamma}(P)|_{\mathbb{Z}}$,
see e.g. [BP00, Section 3].

Figure 3.  Left panel: the maximal edge with slope $1$ consists of 4 edges.
Right panel: the maximal edge of slope $\frac{4}{3}$ consists of a single
edge.

Let us return to the projectivised curve

$C=\bigg{\\{}F(X_{0},X_{1},Y_{0},Y_{1})=X_{1}^{d_{x}}Y_{1}^{d_{y}}P\left(\frac{X_{0}}{X_{1}},\frac{Y_{0}}{Y_{1}}\right)=0\bigg{\\}}\subset\mathbb{P}^{1}\times\mathbb{P}^{1}\,.$
(83)

Let $\square(d_{x},d_{y})$ be the rectangle in $\mathbb{R}^{2}$ with vertices
$(0,0)$, $(d_{x},0)$, $(0,d_{y})$ and $(d_{x},d_{y})$. Clearly,
$\Delta(P)\subseteq\square(d_{x},d_{y})$. Since $P$ is irreducible, we know
that $\Delta(P)$ intersects each of the four maximal edges of
$\square(d_{x},d_{y})$. The rectangle $\square(d_{x},d_{y})$ naturally splits
into five regions: the interior $\mathring{\Delta}(P)$ of the Newton polygon
and the four corners (cf. Figure 4):

  * •

$\Gamma_{00}(P)$ is the Newton diagram of $F(x,1,y,1)$;

  * •

$\Gamma_{0\infty}(P)$ is obtained from the Newton diagram of $F(x,1,1,y)$ by a
horizontal reflection;

  * •

$\Gamma_{\infty 0}(P)$ is obtained from the Newton diagram of $F(1,x,y,1)$ by
a vertical reflection;

  * •

$\Gamma_{\infty\infty}(P)$ is obtained from the Newton diagram of $F(1,x,1,y)$
by a rotation.

By the discussion above, the $\delta$-invariants of the singularities of $C$
at these four special points must satisfy

$\forall\epsilon,\epsilon^{\prime}\in\\{0,\infty\\}\qquad\delta_{p_{\epsilon\epsilon^{\prime}}}\geq|\hat{\Gamma}_{\epsilon\epsilon^{\prime}}(P)|_{\mathbb{Z}}\,,$
(84)

where by $\hat{\Gamma}_{\epsilon\epsilon^{\prime}}(P)$ we mean the points in
$\Gamma_{\epsilon\epsilon^{\prime}}(P)$ which are not on the boundary of
$\square(d_{x},d_{y})$.

Figure 4.  Example of a Newton polygon and its four corners. We have
$|\hat{\Gamma}_{\infty\infty}|=1$ and
$|\hat{\Gamma}_{00}|_{\mathbb{Z}}=|\hat{\Gamma}_{0\infty}|_{\mathbb{Z}}=|\hat{\Gamma}_{\infty
0}|_{\mathbb{Z}}=0$.

These five regions do not overlap in the interior
$\mathring{\square}(d_{x},d_{y})$ of the rectangle:

$\mathring{\square}(d_{x},d_{y})=\mathring{\Delta}(P)\sqcup\left(\bigsqcup_{\epsilon,\epsilon^{\prime}\in\\{0,\infty\\}}\hat{\Gamma}_{\epsilon\epsilon^{\prime}}(P)\right)\,.$

This implies for the number of integral points

$(d_{x}-1)(d_{y}-1)=|\mathring{\Delta}(P)|_{\mathbb{Z}}+\sum_{\epsilon,\epsilon^{\prime}\in\\{0,\infty\\}}|\hat{\Gamma}_{\epsilon\epsilon^{\prime}}(P)|_{\mathbb{Z}}\,.$
(85)

Since the geometric genus of $C$ is
$\mathsf{g}(C)=(d_{x}-1)(d_{y}-1)-\sum_{p\in\mathsf{Sing}}\delta_{p}$ where
$\mathsf{Sing}$ is the singular locus of $C$, (84) implies that
$\mathsf{g}(C)\leq|\mathring{\Delta}(P)|$ (this is known as Baker’s formula,
see e.g. [BP00, Theorem 4.2]).

####  4.3.2. Nondegenerate curves and properties of Puiseux series

For generic curves, the above bounds are equalities, and the geometric genus
and the type of singularities of the generic curve can be directly read off
its Newton polygon. More precisely, this is true for curves that are
nondegenerate, which is defined as follows.

Let $\Delta(P)\subset\mathbb{R}_{\geq 0}^{2}$ be the Newton polygon of an
irreducible affine curve
$\mathfrak{C}=\\{P(x,y)=\sum_{i,j}c_{ij}\,x^{i}y^{j}=0\\}$ as in the previous
section. Denoting by $\tau$ a maximal edge of $\Delta(P)$ or $\Delta(P)$
itself, we write

$P_{\tau}(x,y)=\sum_{(i,j)\in\tau\cap\mathbb{Z}_{\geq
0}^{2}}c_{ij}\,x^{i}y^{j}\,,$ (86)

######  Definition 4.13 (Nondegenerate curves).

$\mathfrak{C}=\\{P(x,y)=0\\}\subset\mathbb{C}^{2}$ is _nondegenerate_ if for
any $\tau$ which is a maximal edge of $\Delta(P)$ or $\Delta(P)$ itself, the
system $P_{\tau}(x,y)=\partial_{x}P_{\tau}(x,y)=\partial_{y}P_{\tau}(x,y)=0$
has no solution in $(\mathbb{C}^{*})^{2}$.

In the literature this is usually called “nondegenerate with respect to its
Newton polygon”. Some references, for instance [BP00, Definition 4.1], do not
include a condition on $\tau=\Delta(P)$ in the definition, which then refers
to a weaker notion. For a fixed Newton polygon, nondegenerate curves are
generic. Nondegenerate curves are nice as many properties of a nondegenerate
curve can be read off directly from its Newton polygon.

######  Remark 4.14.

If
$\mathfrak{C}=\\{(x,y)\in\mathbb{C}^{2}\,\,|\,\,P(x,y)=0\\}\subset\mathbb{C}^{2}$
is an irreducible and nondegenerate affine curve, then the polynomial $P$ has
positive degrees in $x$ and in $y$, does not have factors in $\mathbb{C}[x]$
and is such that $\\{y=0\\}\not\subseteq\mathfrak{C}$. Therefore, by
Definition 4.3 we obtain from it a connected and compact spectral curve.

######  Lemma 4.15.

Let $\mathfrak{C}=\\{P(x,y)=0\\}\subset\mathbb{C}^{2}$ be an irreducible and
nondegenerate affine curve. Let $C\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$
the projectivisation and $\mathsf{Sing}$ its singular locus. Then:

  * •

$\mathsf{g}(C)=|\mathring{\Delta}(P)|_{\mathbb{Z}}$;

  * •

$\mathsf{Sing}\subseteq\\{p_{00},p_{\infty
0},p_{0\infty},p_{\infty\infty}\\}$;

  * •

$\delta_{p_{\epsilon\epsilon^{\prime}}}=|\hat{\Gamma}_{\epsilon\epsilon^{\prime}}(P)|_{\mathbb{Z}}$
for $\epsilon,\epsilon^{\prime}\in\\{0,\infty\\}$.

###### Proof.

First, take $\tau=\Delta(P)$. Then nondegeneracy implies that
$\mathfrak{C}\cap(\mathbb{C}^{*})^{2}$ is smooth. Now suppose that
$\mathfrak{C}$ is singular at $(x,y)=(0,b)$ for some $b\in\mathbb{C}^{*}$.
Then $\Delta(P)$ contains a vertical maximal edge $e$ issuing from the origin.
If we write $P(x,y)=\sum_{i=0}^{d_{x}}c_{i}(y)x^{i}$, then we must have
$c_{0}(b)=c_{0}^{\prime}(b)=0$. But $P_{e}(x,y)=c_{0}(y)=0$. Thus,
$P_{e}(0,b)=c_{0}(b)=0$, $\partial_{x}P_{e}(0,b)=0$ and
$\partial_{y}P_{e}(0,b)=c^{\prime}_{0}(b)=0$. This contradicts the
nondegeneracy condition for this particular $e$. A similar argument applies to
$(x,y)=(a,0)$ with $a\in\mathbb{C}^{*}$, the role of $e$ being now played by
the horizontal maximal edge issuing from the origin. We conclude that
$\mathfrak{C}$ can only have a singularity at the origin.

The bihomogeneisation $F$ of $P$ has the property that, if $P(x,y)=F(x,1,y,1)$
is nondegenerate, then $F(x,1,1,y)$, $F(1,x,y,1)$ and $F(1,x,1,y)$ are also
nondegenerate. Therefore, the argument of the previous paragraph goes through
in the other affine patches, and we conclude that singularities can only be
found among the points $\\{p_{00},p_{0\infty},p_{\infty
0},p_{\infty\infty}\\}$.

Finally, from [BP00, Proposition 3.17] we know that
$\delta_{p_{\epsilon\epsilon^{\prime}}}=|\hat{\Gamma}_{\epsilon\epsilon^{\prime}}(P)|_{\mathbb{Z}}$
for all $\epsilon,\epsilon^{\prime}\in\\{0,\infty\\}$. Since there are no
other singularities, from (76) and (85) it follows that
$\mathsf{g}(C)=|\mathring{\Delta}(P)|_{\mathbb{Z}}$. ∎

Our goal is to extract from the Newton polygon information about the local
parameters for compact spectral curves obtained of Definition 4.3, for which
the affine curve is nondegenerate. To this end, we recall some basic facts
about the Newton–Puiseux algorithm (see e.g. [Tei95]). The algorithm
determines the Puiseux series $y(x)$ satisfying $P(x,y(x))=0$. They take the
form

$y(x)=\sum_{m\in\mathbb{Z}_{\geq 0}}c_{m}x^{\frac{m}{r}}\,,\quad
k=\min\\{m\,\,|\,\,c_{m}\neq 0\\}\in\mathbb{Z}_{\geq 0}$ (87)

for some minimal $r\in\mathbb{Z}_{>0}$, where there is at least one $m$ not
divisible by $r$ such that $c_{l}\neq 0$. We will refer to $(r,k)$ as the
parameters of the Puiseux series. In general, $k$ and $r$ may not be coprime.
The ratio $\frac{k}{r}$ is often called the valuation, $-\frac{r}{k}$
coincides with a slope of some maximal edge $e$ in the corner
$\Gamma_{00}(P)$, and $c_{k}$ is a nonzero root of the _slope polynomial_
$Q_{e}(w)=w^{-l_{e}}P_{e}(1,w)$, where $l_{e}$ is the degree of the monomial
of lowest degree in $P_{e}(1,w)$. There are $(r-1)$ other Galois-conjugate
Puiseux series, obtained by multiplying $x^{\frac{1}{r}}$ by a $r$-th root of
unity. A similar discussion applies in the affine patches at the other points
$p_{\epsilon\epsilon^{\prime}}$ with
$\epsilon,\epsilon^{\prime}\in\\{0,\infty\\}$, by considering maximal edges in
the corners $\Gamma_{\epsilon\epsilon^{\prime}}(P)$ and the corresponding
slope polynomials. Although we always take $k\in\mathbb{Z}_{\geq 0}$ and
$r\in\mathbb{Z}_{>0}$ in the definition, the sign of the slope and its
expression in terms of $k$ and $r$ depends on the corner.

######  Remark 4.16.

We shall soon see that the parameter of the Puiseux series named $r$ coincides
with the local parameters of the spectral curve named $r$ in Definition 2.3.

######  Remark 4.17.

Nondegeneracy is equivalent to the property that all slope polynomials have
only simple roots [BP00, Remark 3.16], and that the system
$P(x,y)=\partial_{x}P(x,y)=\partial_{y}P(x,y)=0$ has no roots in
$(\mathbb{C}^{*})^{2}$. Again, note that our definition of nondegeneracy has
the additional condition that $P(x,y)=\partial_{x}P(x,y)=\partial_{y}P(x,y)=0$
has no roots in $(\mathbb{C}^{*})^{2}$, as compared to [BP00].

We summarise some basic properties below.

######  Lemma 4.18.

Let $C$ be the projectivisation of a nondegenerate irreducible affine curve,
$\mathcal{S}$ the spectral curve and let
$\epsilon,\epsilon^{\prime}\in\\{0,\infty\\}$.

  * (a)

Every branch above $p_{\epsilon\epsilon^{\prime}}$ is associated to a maximal
edge in $\Delta(P)\cap\Gamma_{\epsilon\epsilon^{\prime}}(P)$ and a Puiseux
series with parameters $(r,k)$ as in (87). There are as many branches
associated to a maximal edge $e$ as there are edges in $e$.

  * (b)

At a branch $p\in\eta^{-1}(p_{\epsilon\epsilon^{\prime}})$, the local
parameters $(r_{p},\bar{s}_{p})$ in the sense of Definition 2.3 can be
expressed in terms of the parameters $(r,k)$ of the corresponding Puiseux
series. Namely, $r_{p}=r$ and

$\bar{s}_{p}=\left\\{\begin{array}[]{lll}r+k&&{\rm
if}\,\,(\epsilon,\epsilon^{\prime})=(0,0)\\\ r-k&&{\rm
if}\,\,(\epsilon,\epsilon^{\prime})=(0,\infty)\\\ -r+k&&{\rm
if}\,\,(\epsilon,\epsilon^{\prime})=(\infty,0)\\\ -r-k&&{\rm
if}\,\,(\epsilon,\epsilon^{\prime})=(\infty,\infty)\end{array}\right.$ (88)

In all cases, $r_{p}$ and $\bar{s}_{p}$ are coprime and if $r_{p}\geq 2$, we
have $\bar{s}_{p}=s_{p}$.

  * (c)

Any pair of branches associated to the same maximal edge is non-resonant in
the sense of Definition 3.11.

###### Proof.

Consider a maximal edge $e$ in the corner $\Gamma_{00}(P)$, of slope
$-\frac{n_{e}}{m_{e}}$, where $m_{e},n_{e}\in\mathbb{Z}_{>0}$ are coprime. It
is the union of $\ell_{e}$ edges. By the Newton–Puiseux algorithm, Puiseux
series associated to this maximal edge must have as leading term
$c_{e}x^{\frac{m_{e}}{n_{e}}}$ and $c_{e}$ is a nonzero root of
$Q_{e}(w)=w^{-l_{e}}P_{e}(1,w)$. The latter is a polynomial in the variable
$w^{n_{e}}$ of degree $\ell_{e}$. Due to nondegeneracy, roots of $Q_{e}$ come
in $\ell_{e}$ disjoint groups. Within a group, the roots differ by
multiplication by a $n_{e}$-th root of unity. With the Newton–Puiseux
algorithm we find at least $\ell_{e}n_{e}$ distinct Puiseux series (associated
to $e$) that are roots of $P(x,y)$, and for degree reasons there cannot be
more.

By [BP00, Remark 3.18], the sets of conjugate Puiseux series we have described
are in bijection with the branches of the singularity at $p_{00}$. For each
branch $\tilde{p}_{e}\in\eta^{-1}(p_{00})$, call $(r_{e},k_{e},c_{e})$ the
parameters of the corresponding the Puiseux series. By comparison we must have
$r_{e}=n_{e}$ representing the number of conjugate Puiseux series within the
corresponding group. As $\frac{m_{e}}{n_{e}}=\frac{k_{e}}{r_{e}}$, we deduce
that $k_{e}=m_{e}$. In particular this means that $k_{e},r_{e}$ are coprime,
and by the above description if $e^{\prime}$ is an edge in the same maximal
edge we have

$c_{e^{\prime}}^{r_{e}}\neq c_{e}^{r_{e}}\,.$ (89)

Clearly, $x$ has order $r_{e}$ at $\tilde{p}_{e}$. And in a standard
coordinate near $p$ we have $x=x(p)+\zeta^{r_{e}}$ and $y\sim
c_{e}\zeta^{\frac{k_{e}}{r_{e}}}$. This implies

$\omega_{0,1}\sim r_{e}c_{e}\zeta^{r_{e}+k_{e}-1}\mathrm{d}\zeta\qquad{\rm
near}\,\,p$

Therefore $\bar{s}_{e}=r_{e}+k_{e}$ and it is coprime with $r_{e}$. This
implies that $s_{e}=\bar{s}_{e}$ as soon as $r_{e}\geq 2$. Besides
$\frac{\tau_{e}}{r_{e}}=c_{e}$ and the fact that $r_{e}$ and $\bar{s}_{e}$ are
coprime together with (89) implies that all pairs of points above the same
branch of $p_{00}$ (corresponding to all edges in the same maximal edge) are
non-resonant.

For branches corresponding to edges at other corners, the argument is similar.
We only indicate how to compute the local parameters. For a branch
$\tilde{p}_{e}\in\eta^{-1}(p_{0\infty})$ corresponding to an edge $e$, the
Puiseux series gives the series expansion of $y^{-1}$ near $\tilde{p}_{e}$ in
the variable $x$. Then, in a standard coordinate near $p$ we have
$x=x(p)+\zeta^{r_{e}}$ and $y^{-1}\sim c_{e}\zeta^{\frac{k_{e}}{r_{e}}}$,
therefore

$\omega_{0,1}\sim r_{e}c_{e}\zeta^{r_{e}-k_{e}-1}\mathrm{d}\zeta\quad{\rm
near}\,\,p\,.$

and we read $\bar{s}_{e}=r_{e}-k_{e}$. For a branch $p\in\eta^{-1}(p_{\infty
0})$, the Puiseux series expresses the series expansion of $y$ in the variable
$x^{-1}$. Then, in a standard coordinate near $p$ we have $x=\zeta^{-r_{e}}$
and $y\sim c_{e}\zeta^{\frac{k_{e}}{r_{e}}}$, therefore

$\omega_{0,1}\sim-r_{e}c_{e}\zeta^{-r_{e}+k_{e}-1}\mathrm{d}\zeta\quad{\rm
near}\,\,p\,.$

For a branch $p\in\eta^{-1}(p_{\infty\infty})$, the Puiseux series expresses
the series expansion of $y^{-1}$ in the variable $x^{-1}$. Then, in standard
coordinate near $p$ we have $x=\zeta^{-r_{e}}$ and $y^{-1}\sim
c_{e}\zeta^{\frac{k_{e}}{r_{e}}}$, therefore

$\omega_{0,1}\sim-r_{e}c_{e}\zeta^{-r_{e}-k_{e}-1}\mathrm{d}\zeta\quad{\rm
near}\,\,p\,.$

∎

####  4.3.3. Global admissibility for nondegenerate curves

We can now formulate a characterisation of global admissibility for compact
spectral curves coming from nondegenerate curves.

######  Proposition 4.19 (Global admissibility for nondegenerate spectral
curves).

Let $\mathfrak{C}=\\{P(x,y)=0\\}\subset\mathbb{C}^{2}$ be an irreducible and
nondegenerate affine curve, and
$\mathcal{S}=(\tilde{C},x,\omega_{0,1},\omega_{0,2})$ be a compact spectral
curve as in Definition 4.3. Then $\mathcal{S}$ is globally admissible if and
only if the following conditions are satisfied:

  1. (GA1)

$|\hat{\Gamma}_{00}(P)|_{\mathbb{Z}}=0$, i.e. $\mathfrak{C}$ is smooth at the
origin;

  2. (GA2)

$\mathcal{S}$ is locally admissible at all points in $\eta^{-1}(p_{0\infty})$
and any pair of points in $\eta^{-1}(p_{0\infty})$ satisfies “(C-i) or (C-ii)”
from Theorem 3.14;

  3. (GA3)

$\mathcal{S}$ is locally admissible at all points in $\eta^{-1}(p_{\infty 0})$
and any pair of points in $\eta^{-1}(p_{\infty 0})$ satisfies “(C-i) or
(C-ii)” from Theorem 3.14.

###### Proof.

The direct implication is clear. Conversely, in view of Theorem 4.10 and
assuming (GA1)-(GA2)-(GA3), we need to check that if $p_{\infty\infty}$ is a
singular point of $C$, $\mathcal{S}$ is locally admissible at all points
$\eta^{-1}(p_{\infty\infty})$, and every pair of points in
$\eta^{-1}(p_{\infty\infty})$ is non-resonant and satisfies “(C-i) or (C-ii)”.
We also need to check that all pairs of points in $\eta^{-1}(p_{0\infty})$
(resp. in $\eta^{-1}(p_{\infty 0})$) are non-resonant (cf. Definition 3.11).

The absence of resonance is given by Lemma 4.18. Besides, if
$p_{\infty\infty}$ is a singular point of $C$, take
$\tilde{p}\in\eta^{-1}(p_{\infty\infty})$. If $p$ is unramified, the curve is
locally admissible at $\tilde{p}$. Otherwise, $r_{\tilde{p}}\geq 2$ and in
Lemma 4.18 we have learned that
$s_{\tilde{p}}=\bar{s}_{\tilde{p}}=-r_{\tilde{p}}-k_{\tilde{p}}$ in terms of
the parameters of a corresponding Puiseux series. Since $k_{\tilde{p}}>0$ by
definition, this gives $\nu_{p}=-1-\frac{k_{p}}{r_{p}}<-1$ and thus the curve
is locally admissible at $p$. So the property of global admissibility
concerning $p_{0\infty}$ (resp. $p_{\infty 0}$) reduces to checking “(C-i) or
(C-ii)” of Theorem 3.14. ∎

####  4.3.4. Global admissibility in terms of slopes

Consider a nondegenerate spectral curve. We now reformulate (GA2) and (GA3) in
Proposition 4.19 in terms of slopes of the edges in

$E_{0\infty}=\Gamma_{0\infty}(P)\cap\Delta(P)\qquad{\rm and}\qquad E_{\infty
0}=\Gamma_{\infty 0}(P)\cap\Delta(P)\,.$ (90)

For this we make a choice of bijection between $\eta^{-1}(p_{0\infty})$ (set
of branches) and the set of edges in $E_{0\infty}$, so that each branch is
mapped to an edge contained in the maximal edge associated with this branch.
We do the same for $\eta^{-1}(p_{\infty 0})$ and the set of edges in
$E_{\infty 0}$.

$E_{0\infty}$ is a point or a union of edges. In the latter situation, let
$0<\mu_{1}\leq\cdots\leq\mu_{\ell}<\infty$ be the slopes of the edges, which
appear in weakly increasing order from right to left in $E_{0\infty}$. When a
slope appear several times in this list, it means that we have a maximal edge
made of several edges. By our choice of bijection, to an edge $e$ we have
associated a point $\tilde{p}_{e}\in\eta^{-1}(p_{0\infty})$. We have local
parameters $(r_{e},\bar{s}_{e},s_{e})$ for the spectral curve at
$\tilde{p}_{e}$, and local parameters $(k_{e},r_{e})$ of the group of Puiseux
series associated with $\tilde{p}_{e}$. From Lemma 4.18 we can read them from
the slope of $e$. More precisely, we have $\bar{s}_{e}=r_{e}-k_{e}$ with
$\bar{s}_{e}<r_{e}$ and $\bar{s}_{e},r_{e}$ coprime, and the relation between
the aspect ratio and the slope reads

$\mu_{e}=\frac{r_{e}}{k_{e}}=\frac{r_{e}}{r_{e}-\bar{s}_{e}},\qquad\nu_{e}=\frac{\bar{s}_{e}}{r_{e}}=1-\frac{k_{e}}{r_{e}}=1-\frac{1}{\mu_{e}}\,.$
(91)

Likewise, for $E_{\infty 0}$ the slopes
$0<\mu_{1}\leq\ldots\leq\mu_{\ell}<\infty$ appear in weakly increasing order
from left to right. For each edge $e$ we have a point $\tilde{p}$ in
$\eta^{-1}(p_{\infty 0})$ and the local parameters satisfy
$\bar{s}_{e}=k_{e}-r_{e}$ with $\bar{s}_{e}>-r_{e}$ and $\bar{s}_{e},r_{e}$
coprime, and

$\mu_{e}=\frac{r_{e}}{k_{e}}=\frac{r_{e}}{\bar{s}_{e}+r_{e}},\qquad\nu_{e}=-1+\frac{k_{e}}{r_{e}}=-1+\frac{1}{\mu_{e}}\,.$
(92)

In both corners, the slope faithfully encodes the local parameters, we have
$s_{e}=\infty$ whenever $r_{e}=1$, and $\bar{s}_{e}=s_{e}$ whenever $r_{e}\geq
2$. For an edge in $E_{0\infty}$ we note that $\mu_{e}>1$ implies $r_{e}\geq
2$.

######  Definition 4.20 (Local and global admissibility for edges in
$E_{0\infty}$).

Consider an edge of slope $\mu$ in $E_{0\infty}$ with local parameters $(r,s)$
as above. We say that this edge is _locally admissible_ if $\mu\leq 1$ or
$r=\pm 1\,\,{\rm mod}\,\,s$.

A pair of edges in $E_{0\infty}$ with slopes $\mu_{i}\leq\mu_{j}$ is
_globalisable_ if

  1. (G1)

$1<\mu_{i}\leq\mu_{j}$ and there exists $m\in(r_{ij}-1]$ such that
$\frac{1}{\mu_{j}}\leq\frac{m-1}{m}\leq\frac{1}{\mu_{i}}$; or

  2. (G2)

$1<\mu_{i}\leq\mu_{j}$ and $\frac{1}{\mu_{i}}\geq\frac{r_{ij}-1}{r_{ij}}$; or

  3. (G3)

$\mu_{i}\leq 1$.

A sequence of edges in $E_{0\infty}$ is _globally admissible_ if any edge in
the sequence if locally admissible and any pair of edges in the sequence is
globalisable.

######  Lemma 4.21.

A pair of edges in $E_{0\infty}$ having the same slope $\mu$ is globalisable
if and only if $\mu\leq 1$ or $\mu=\frac{r}{r-1}$ for some $r\geq 2$.

###### Proof.

Assume we have a pair of edges in $E_{0\infty}$ have the same slope
$\mu_{i}=\mu_{j}=\mu$. (G3) is the condition $\mu\leq 1$. If $\mu>1$, we have
$r_{i},k_{i}$ coprime and likewise for $r_{j},k_{j}$. Then
$\mu=\frac{r_{i}}{k_{i}}=\frac{r_{j}}{k_{j}}$ implies
$(k_{i},r_{i})=(k_{j},r_{j})$ and we simply denote it $(k,r)$. (G1) is then
equivalent to the existence of $m\in(r-1]$ such that
$\frac{1}{\mu}=\frac{m-1}{m}$, but this would force $m=r$ and is thus
impossible. (G2) states that $\frac{1}{\mu}\geq\frac{r-1}{r}$, but as
$\frac{1}{\mu}<1$ has denominator $r$ the only possibility is
$\frac{1}{\mu}=\frac{r-1}{r}$. ∎

######  Definition 4.22 (Local and global admissibility for edges in
$E_{\infty 0}$).

Consider an edge of slope $\mu$ in $E_{\infty 0}$ with local parameters
$(r,s)$ as above. We say that this edge is _locally admissible_ if $\mu\geq 1$
or $r=\pm 1\,\,{\rm mod}\,\,s$. A pair of edges in $E_{\infty 0}$ with slopes
$\mu_{i}\leq\mu_{j}$ is _globalisable_ if

  1. (G1’)

$0<\mu_{i}\leq\mu_{j}<1$ and there exists $m\in[r_{ij}-1]$ such that
$\frac{1}{\mu_{j}}\leq\frac{m+1}{m}\leq\frac{1}{\mu_{i}}$, or

  2. (G2’)

$0<\mu_{i}\leq\mu_{j}<1$ and $\frac{1}{\mu_{j}}\geq\frac{r_{ij}+1}{r_{ij}}$,
or

  3. (G3’)

$\mu_{j}\geq 1$.

An edge $e$ in $E_{0\infty}$ is _globally admissible_ if it is locally
admissible and for any other edge $e^{\prime}$ in $E_{\infty 0}$, the pair
$e,e^{\prime}$ is globalisable.

######  Lemma 4.23.

A pair of edges in $E_{\infty 0}$ having the same slope $\mu$ is globalisable
if and only if $\mu\geq 1$ or $\mu=\frac{r-1}{r}$ for some $r\geq 1$.

###### Proof.

Similar to Lemma 4.21 and thus omitted. ∎

######  Theorem 4.24 (Conditions for global admissibility in terms of Newton
polygon).

Let $\mathfrak{C}=\\{P(x,y)=0\\}\subset\mathbb{C}^{2}$ be an irreducible and
nondegenerate affine curve, and
$\mathcal{S}=(\tilde{C},x,\omega_{0,1},\omega_{0,2})$ be a compact spectral
curve as in Definition 4.3. Then $\mathcal{S}$ is globally admissible if and
only if the following three conditions are satisfied:

  1. ($\Gamma$A1)

$|\hat{\Gamma}_{00}(P)|_{\mathbb{Z}}=0$, i.e. $\mathfrak{C}$ is smooth at the
origin.

  2. ($\Gamma$A2)

The edges of $\Delta(P)$ in $E_{0\infty}$ are globally admissible.

  3. ($\Gamma$A3)

The edges of $\Delta(P)$ in $E_{\infty 0}$ are globally admissible.

###### Proof.

This is merely a reformulation of Proposition 4.19, taking into account the
coding of points in $\eta^{-1}(p_{\epsilon\epsilon^{\prime}})$ by edges and
the discussion preceding Definition 4.20, especially formulas (91)-(92)
relating aspect ratios and slopes. Condition ($\Gamma$A1) is (GA1) verbatim.
We recall that condition (GA2) from Proposition 4.19 requires local
admissibility of all points $\eta^{-1}(p_{0\infty})$ and that every pair in
$\eta^{-1}(p_{0\infty})$ satisfies “(C-i) or (C-ii)”.

We recall that edges with $\mu>1$ have $r\geq 2$ coprime to $\bar{s}=s$. So,
in the Definition 2.5 of local admissibility, (lA1) and (lA3) are automatic
and in (lA2) the condition $s\in[r+1]$ is equivalent to $\mu>1$ while the
congruence condition is repeated in Definition 4.20. Edges with $\mu=1$ have
$r=1$ (unramified point) and they automatically correspond to locally
admissible points. Edges with $\mu<1$ have either $r=1$, or $\bar{s}=s\leq-1$
coprime to $r$, so automatically correspond to locally admissible points. This
identifies local admissibility of edges with local admissibility of the
corresponding point. Condition (C-i) for a pair of points matches (G1) for the
corresponding pair of edges, once we notice that slopes of edges in
$E_{0\infty}$ are finite, so $m=1$ can be excluded. Condition (C-ii) can be
split in two cases: $\nu_{1}\in(0,\frac{1}{r_{12}})$, which amounts to
$\frac{1}{\mu_{1}}\geq\frac{r_{12}-1}{r_{12}}$ and matches (G2); and
$\nu_{1}\leq 0$, which corresponds to $\mu_{1}\leq 1$ and matches (G3).

Checking that condition (GA3) is equivalent to ($\Gamma$A3) can be carried out
similarly. One should only be careful that $\nu$ is now a decreasing function
of $\mu$ by (92): the choice $\mu_{1}\leq\mu_{2}$ in Theorem 3.14 leads to
$\nu_{2}\leq\nu_{1}$, hence the conditions (G2’)-(G3’) translating (C-ii)
involve $\mu_{2}$ instead of $\mu_{1}$.

∎

###  4.4. Global admissibility and deformations of the Newton polygon

It turns out that the property of global admissibility is preserved under
certain types of deformations, which will be useful in the next section.

Given a Newton polygon $\Delta(P)$, we call _interior point_ an integral point
in the interior of $\Delta(P)$, and _edge interior point_ an integral point
belonging to a maximal edge of the polygon $\Delta(P)$ and which is not an
endpoint of this maximal edge. The context will not allow any confusion
between this denomination and points in the interior of a set.

####  4.4.1. Farey sequences and triangles

Let us recall the notion of a Farey sequence and some of its properties. Given
an integer $r>1$, the Farey sequence of order $r$ is the sequence
$\mathcal{F}_{r}$ of irreducible nonnegative fractions with denominator $\leq
r$, considered in increasing order. Unlike some references, we allow elements
larger than $1$ and thus our $\mathcal{F}_{r}$ is infinite to the right. For
example:

$\mathcal{F}_{5}=0,\frac{1}{5},\frac{1}{4},\frac{1}{3},\frac{2}{5},\frac{1}{2},\frac{3}{5},\frac{2}{3},\frac{3}{4},\frac{4}{5},1,\frac{6}{5},\frac{5}{4},\frac{4}{3},\frac{7}{5},\cdots$
(93)

Farey neighbours of a fraction $\frac{c}{d}>0$ in the Farey sequence
$\mathcal{F}_{r}$ are elements $\frac{a}{b},\frac{e}{f}$ in $\mathcal{F}_{r}$
that are adjacent to $\frac{c}{d}$ and such that
$\frac{a}{b}<\frac{c}{d}<\frac{e}{f}$. They are known to satisfy the relation

$\frac{c}{d}=\frac{a+e}{b+f}\,.$ (94)

In addition, two neighbouring elements $\frac{a}{b}<\frac{c}{d}$ in
$\mathcal{F}_{r}$ must satisfy Bézout’s identity:

$bc-ad=1\,.$ (95)

We will also need to consider the trivial Farey sequence

$\mathcal{F}_{1}=0,1,2,3,\ldots,\infty.$

By convention, we define Farey neighbours in this sequence to be triples
$(p-1,p,\infty)$ for $p\geq 1$. Viewing $\infty=\frac{1}{0}$ and
$p=\frac{p}{1}$, we see that (94) and (95) remain valid. This convention may
seem strange, but it will make some of the results below easier to state.

In this section Newton polygons with the shape of a triangle will play a
crucial role. Given rational numbers $\alpha,\mu,\beta$ (with $\beta$ possibly
infinite) such that $0\leq\alpha<\mu<\beta\leq\infty$, we consider a triangle
$T\subseteq\mathbb{R}_{\geq 0}^{2}$ having integral extremal vertices, and
slopes $\alpha,\mu,\beta$. We write

$\frac{1}{\mu}=\frac{r-\bar{s}}{r},\quad\frac{1}{\alpha}=\frac{r_{\alpha}-\bar{s}_{\alpha}}{r_{\alpha}},\quad\frac{1}{\beta}=\frac{r_{\beta}-\bar{s}_{\beta}}{r_{\beta}}\,.$
(96)

Here $r\in\mathbb{Z}_{>0}$ and $\bar{s}\in\mathbb{Z}$ are coprime such that
$\bar{s}\leq r$; in particular if $\bar{s}=0$ we must have $r=1$, representing
the inverse slope $1$. Analogous conditions hold for
$(r_{\alpha},\bar{s}_{\alpha})$ and $(r_{\beta},\bar{s}_{\beta})$. In
particular, if $r_{\alpha}=0$, we must have $\bar{s}_{\alpha}=-1$,
representing the slope $\alpha=0$, and if $r_{\beta}=\bar{s}_{\beta}$, we must
have $r_{\beta}=\bar{s}_{\beta}=1$, representing the slope $\beta=\infty$.

######  Definition 4.25 (Elementary triangle).

$T$ is an _elementary triangle_ if its only integral points are the extremal
vertices. Equivalently, if it does not have edge interior points nor interior
points.

Elementary triangles can be characterised in terms of the slopes of the edges.

######  Lemma 4.26.

Let $T$ be a triangle with no edge interior points. $T$ is elementary if and
only if $\frac{1}{\beta}<\frac{1}{\mu}<\frac{1}{\alpha}$ are Farey neighbours
in $\mathcal{F}_{r}$, where $r$ is the numerator of $\mu$.

###### Proof.

Up to global translation, the endpoints of the edge of slope $\mu$ must be of
the form $(0,0)$ and $(r-\bar{s},r)$ as $r,\bar{s}$ are coprime, and the edge
has no interior points. Without loss of generality we can assume that the edge
with slope $\beta$ has $(0,0)$ as endpoint. Since $r_{\beta},\bar{s}_{\beta}$
are coprime, the third vertex of the triangle must have coordinates
$(r_{\beta}-\bar{s}_{\beta},r_{\beta})$ and the edge with slope $\beta$ has no
interior points. Calculating the slope of the remaining edge yields

$\frac{r_{\alpha}}{r_{\alpha}-\bar{s}_{\alpha}}=\alpha=\frac{r-r_{\beta}}{r-r_{\beta}-\bar{s}+\bar{s}_{\beta}}\,.$
(97)

However, the edge of slope $\alpha$ is also assumed to have no interior point,
and hence we must have $r_{\alpha}=r-r_{\beta}$ and
$\bar{s}_{\alpha}=\bar{s}-\bar{s}_{\beta}$. In particular $r_{\alpha\beta}\leq
r$ so $\frac{1}{\beta},\frac{1}{\mu},\frac{1}{\alpha}$ belong to
$\mathcal{F}_{r}$. We shall first treat the case $r>1$, which implies
$\frac{1}{\beta}<\frac{1}{\mu}<\frac{1}{\alpha}$, and deal with the case $r=1$
later.

If $T$ had an interior point, there would exist edges of slopes
$\beta^{\prime}$ and $\alpha^{\prime}$ satisfying
$\alpha<\alpha^{\prime}<\mu<\beta^{\prime}<\beta$, such that
$\frac{1}{\beta^{\prime}}<\frac{1}{\mu}<\frac{1}{\alpha^{\prime}}$ belong to
$\mathcal{F}_{r}$. Then $\frac{1}{\beta}$ and $\frac{1}{\alpha}$ would not be
Farey neighbours of $\frac{1}{\mu}$ in $\mathcal{F}_{r}$. Therefore, if
$\frac{1}{\beta}$ and $\frac{1}{\alpha}$ are Farey neighbours of
$\frac{1}{\mu}$ in $\mathcal{F}_{r}$, then $T$ has no interior points.

For the converse statement, assume that $T$ has no interior point. Suppose
that $\frac{1}{\beta}$ is not a left Farey neighbour of $\frac{1}{\mu}$ in
$\mathcal{F}_{r}$, and denote $\frac{1}{\beta^{\prime}}>\frac{1}{\beta}$ be
this left Farey neighbour. We write as before
$\frac{1}{\beta^{\prime}}=\frac{r_{\beta^{\prime}}-\bar{s}_{\beta^{\prime}}}{r_{\beta^{\prime}}}$.
Defining $r_{\alpha^{\prime}}=r-r_{\beta^{\prime}}$ and
$\bar{s}_{\alpha^{\prime}}=\bar{s}-\bar{s}_{\beta^{\prime}}$, we see from (94)
that
$\frac{1}{\alpha^{\prime}}=\frac{r_{\alpha^{\prime}}-\bar{s}_{\alpha^{\prime}}}{r_{\alpha^{\prime}}}$
is the right Farey neighbour of $\frac{1}{\mu}$ in $\mathcal{F}_{r}$. In
particular,

$\frac{1}{\beta}<\frac{1}{\beta^{\prime}}<\frac{1}{\mu}<\frac{1}{\alpha^{\prime}}\leq\frac{1}{\alpha}\,.$
(98)

Following the first paragraph of the proof, we obtain a triangle $T^{\prime}$
with integral vertices: two of them are $(0,0)$ and $(r-\bar{s},r)$ joined by
the edge of slope $\mu$, the second edge issuing from $(0,0)$ has slope
$\beta^{\prime}$, the third edge has slope $\alpha^{\prime}$, and none of the
three edges has interior points. Besides, (98) implies that $T^{\prime}\subset
T$, in particular the third vertex of $T^{\prime}$ must be interior to $T$
(cf. Figure 5), which is a contradiction. The same argument goes through if
$\frac{1}{\alpha}$ is not a right Farey neighbour of $\frac{1}{\mu}$ in
$\mathcal{F}_{r}$. We conclude that $\frac{1}{\beta}$ and $\frac{1}{\alpha}$
must be Farey neighbours of $\frac{1}{\mu}$ in $\mathcal{F}_{r}$, which
completes the proof for $r>1$.

Figure 5.  In yellow: a triangle $T$ with inverse slopes $\frac{1}{3}$,
$\frac{4}{5}$ and $\frac{3}{2}$ in $\mathcal{F}_{5}$, in which edges have no
interior points. Note that $\frac{1}{3}<\frac{4}{5}<\frac{3}{2}$ are not Farey
neighbours in $\mathcal{F}_{5}$, and $T$ has interior points. In blue: a
triangle $T^{\prime}\subset T$ with inverse slopes $\frac{3}{4}$,
$\frac{4}{5}$ and $1$ in $\mathcal{F}_{5}$. Note that
$\frac{3}{4}<\frac{4}{5}<1$ are Farey neighbours in $\mathcal{F}_{5}$, and
$T^{\prime}$ has no interior point.

If $r=1$, since $r=r_{\alpha}+r_{\beta}$ we must have $r_{\beta}=1$ and
$r_{\alpha}=0$ so that the edge with slope $\alpha$ has no interior point.
Hence $\bar{s}_{\beta}=\bar{s}-\bar{s}_{\alpha}=\bar{s}+1$ and
$\bar{s}_{\alpha}=-1$. We obtain the sequence
$\frac{-\bar{s}}{1}<\frac{1-\bar{s}}{1}<\infty$, where $\bar{s}<r=1$. We
recognise a triple of Farey neighbours in $\mathcal{F}_{1}$ according to our
convention. In this case, the triangle never has interior points. Thus, the
equivalence holds as well for $r=1$. ∎

####  4.4.2. Deformations of Newton polygons

The main result of this section is that the global admissibility of spectral
curves is invariant under enlarging (or conversely, shrinking) Newton polygons
while preserving its set of interior points. We start by studying general
properties of these deformations.

######  Definition 4.27 (Polygons and equivalence).

For us, a _polygon_ is a non-empty subset of $\mathbb{R}^{2}$ which is the
convex hull of finitely many points in $\mathbb{Z}_{\geq 0}^{2}$. It is
_inscribed_ if there exists $(d_{x},d_{y})\in\mathbb{Z}_{>0}^{2}$ (its
_bidegree_) such that $\Delta\subseteq\square(d_{x},d_{y})$ and $\Delta$
touches each four closed segments forming the boundary of
$\square(d_{x},d_{y})$. We say that $\Delta$ is a _long diagonal_ if
$d_{x},d_{y}$ are not coprime and $\Delta=[(0,0),(d_{x},d_{y})]$ or
$\Delta=[(d_{x},0),(0,d_{y})]$. Corners of $\Delta$ can be defined as in
Subsection 4.3.1 and are denoted $\Gamma_{\epsilon\epsilon^{\prime}}(\Delta)$.
We say that two inscribed polygons are _equivalent_ if they have the same
bidegree and the same interior points.

We can move within an equivalence class of inscribed polygons by adding or
removing certain vertices.

######  Definition 4.28 (Adding and removing vertices).

Let $\Delta$ be an inscribed polygon in $\square(d_{x},d_{y})$ and
$v\in\square(d_{x},d_{y})\cap\mathbb{Z}^{2}$. We say that $v\notin\Delta$ is
_addible_ to $\Delta$ if the convex hull of $\Delta\cup\\{v\\}$, denoted by
$\Delta_{+}[v]$, is an inscribed polygon equivalent to $\Delta$. We say that
$v$ is _removable_ from $\Delta$ if there exists an inscribed polygon
$\Delta^{\prime}$ equivalent to $\Delta$ and such that
$\Delta=\Delta^{\prime}_{+}[v]$ (we then write
$\Delta^{\prime}=\Delta_{-}[v]$). We say that two inscribed polygons in
$\square(d_{x},d_{y})$ are _strongly equivalent_ if one can transform one into
the other by a finite sequence of addition/removal of addible/removable
vertices.

We note that equivalent inscribed polygons may not be strongly equivalent, for
instance $\Delta=[(0,0),(d_{x},d_{y})]$ and
$\Delta^{\prime}=[(d_{x},0),(0,d_{y})]$.

An interesting question is whether (strong) equivalence classes of Newton
polygons have minimal or maximal polygons under inclusion. It is clear that a
given equivalence class of Newton polygons may not have a minimal polygon (cf.
Figure 7) nor a maximum (cf. Figure 6). It turns out that strong equivalence
classes may still not have minimal polygons; cf. Figure 7. However, up to a
pathological case, they do have a maximum, as we show next.

Figure 6.  The two Newton polygons depicted in the figure are inscribed in
$\square(5,7)$ and have no interior points, and thus are equivalent.
Furthermore, they are both maximal under inclusion. However, they are not
strongly equivalent: one cannot transform one into the other by
addition/removal of vertices without changing the bidegree, keeping them
inscribed or creating an interior point in an intermediate step. Figure 7.  A
Newton polygon $\Delta$ is depicted in green. An elementary triangle $T_{v}$
is shown in blue (with slopes $2<\frac{5}{2}<3$), and another elementary
triangle $T_{w}$ is shown in orange (with slopes $\frac{3}{2}<\frac{5}{3}<2$).
$v$ and $w$ are both separately removable from $\Delta$, aligning with the
fact that the inverse slopes of $T_{v}$ and $T_{w}$ are separately Farey
neighbours in $\mathcal{F}_{5}$ (the numerator of the middle slope is $5$).
However, $w$ is not removable from $\Delta_{-}[v]$ (the $\circ$ which were
interior points become boundary points), aligning with the fact that
$\frac{2}{5}$ and $\frac{2}{3}$ are not Farey neighbours in $\mathcal{F}_{5}$.
Likewise, $v$ is not removable from $\Delta_{-}[w]$. This shows that
equivalence classes, and even strong equivalence classes, may not have minimal
polygons under inclusion.

######  Proposition 4.29.

Let $\Delta$ be an inscribed polygon which is not a long diagonal. Then the
following properties hold:

  1. (1)

If $v$ is addible to $\Delta$, there exists a sequence of
$v^{1},\ldots,v^{k}\in\mathbb{Z}^{2}_{\geq 0}$ and polygons
$\Delta^{0}\ldots,\Delta^{k}$ equivalent to $\Delta$ such that
$\Delta^{0}=\Delta$, $\Delta^{k}=\Delta_{+}[v]$, and for each $i\in[k]$,
$v_{i}$ is addible to $\Delta^{i-1}$ and $\Delta^{i}=\Delta^{i-1}_{+}[v_{i}]$
differs from $\Delta^{i-1}$ by the addition of an elementary triangle.

  2. (2)

If $v,\tilde{v}$ are addible to $\Delta$, then $\tilde{v}$ is addible to
$\Delta_{+}[v]$.

Each irreducible strong equivalence class which does not contain a long
diagonal admits a maximal polygon.

###### Proof.

We start with claim (1). Let $v\notin\Delta$ be addible to $\Delta$. As
$\Delta_{+}[v]$ has the same interior points as $\Delta$, $\Delta_{+}[v]$
differs from $\Delta$ by the addition of a triangle $T_{v}$ with vertices $v$
and $u,w\in\Delta$, with no interior points. The edge $(u,w)$ cannot have edge
interior points, unless $\Delta$ consists of the single maximal edge $(u,w)$;
otherwise, the edge interior points of $(u,w)$ would become interior points in
$\Delta_{+}[v]$, and $\Delta$ and $\Delta_{+}[v]$ would not be equivalent.
However, if we assume that $\Delta$ is not a long diagonal, it cannot consist
of a single edge $(u,w)$ with edge interior points. We conclude that $(u,w)$
has no edge interior point.

As for the other two edges of the triangle $T_{v}$, if there were edge
interior points in $(u,v)\cup(v,w)$, we could add them to $\Delta$ one by one,
so that at each step one adds only an elementary triangle (cf. Figure 8). This
justifies claim (1).

Figure 8.  The triangle $T_{v}$ (in orange) can be added to the polygon
$\Delta$ (in green) by a succession of addition of elementary triangles.

We now proceed to claim (2). Taking into account claim (1), it is easy to see
that it is sufficient to prove claim (2) when $v,\tilde{v}$ are addible to a
polygon $\Delta$ which is not a long diagonal and such that $T_{v}$ and
$T_{\tilde{v}}$ are elementary triangles, which we now assume. We denote
$\Delta_{+}[v,\tilde{v}]=\Delta_{v}\cup T_{v}\cup T_{\tilde{v}}$.

First assume that $v$ and $\tilde{v}$ lie in different corners of $\Delta$. If
$\Delta$ consists of a single maximal edge, then among the two vertices $v$
and $\tilde{v}$, one lies below $\Delta$ and the other above. As $\Delta$ is
not a long diagonal, the maximal edge in $\Delta$ does not contain an edge
interior point, and $\tilde{v}$ is clearly addible to $\Delta_{+}[v]$. If
$\Delta$ does not consist of a single maximal edge, then it is also clear that
$\tilde{v}$ is addible to $\Delta_{+}[v]$.

We now assume that $v$ and $\tilde{v}$ belong to the same corner
$\Gamma_{0\infty}(\Delta)$ (the discussion at the other corners would be
similar). Let $u,w\in\Delta$ be the other two vertices of the elementary
triangle $T_{v}$ as in the beginning of the proof. As $\Delta$ is inscribed,
it must touch both the closure of the left vertical side and the upper
horizontal side of $\square(d_{x},d_{y})$. This forces $u$ and $w$ to belong
to the same corner $\Gamma_{0\infty}(\Delta)$. Then, denoting $\mu>0$ the
slope of the base $[u,w]\subseteq\Delta$, and $\alpha,\beta>0$ the slopes of
$[u,v]$ and $[v,w]$, we have (up to renaming $u$ and $w$ so that $u$ is below
$w$) the inequality $0\leq\frac{1}{\alpha}<\frac{1}{\mu}<\frac{1}{\beta}$. By
Lemma 4.26, this is a triple of neighbours in $\mathcal{F}_{r}$, where $r$ is
the numerator of $\mu$. The same construction applied to $\tilde{v}$ yields an
elementary triangle $T_{\tilde{v}}$ with vertices $\tilde{v}$ and
$\tilde{u},\tilde{w}\in\Delta$ in $\Gamma_{0\infty}(\Delta)$ such that
$\tilde{u}$ is below $\tilde{v}$. Up to exchanging the role of $v$ and
$\tilde{v}$, we can assume that $v$ is below $\tilde{v}$.

If $[u,w]\cap[\tilde{u},\tilde{w}]=\emptyset$, then $\Delta_{+}[v,\tilde{v}]$
is convex and has the same interior points as $\Delta$, so $\tilde{v}$ is
addible to $\Delta_{+}[v]$.

If $w=\tilde{u}$, $\Delta_{+}[v,\tilde{v}]$ contains no other interior points
than those already present in $\Delta$. Then, $\tilde{v}$ is addible to
$\Delta_{+}[v]$ if and only if $\Delta_{+}[v,\tilde{v}]$ is convex. As $T_{v}$
and $T_{\tilde{v}}$ are elementary triangles, we have two triples of Farey
neighbours $0\leq\frac{1}{\alpha}<\frac{1}{\mu}<\frac{1}{\beta}$ in
$\mathcal{F}_{r}$ and
$0\leq\frac{1}{\tilde{\alpha}}\leq\frac{1}{\tilde{\mu}}\leq\frac{1}{\tilde{\beta}}$
in $\mathcal{F}_{\tilde{r}}$. Since $v$ and $\tilde{v}$ are addible to
$\Delta$, we know that $\Delta_{+}[v]$ and $\Delta_{+}[\tilde{v}]$ are convex.
This implies $\frac{1}{\tilde{\beta}}\leq\frac{1}{\mu}$ and
$\frac{1}{\tilde{\mu}}\leq\frac{1}{\alpha}$. The aforementioned Farey
neighbour property forces $\frac{1}{\tilde{\beta}}\leq\frac{1}{\alpha}$, which
implies that $\Delta_{+}[v,\tilde{v}]$ is convex.

Figure 9.  $\tilde{v}$ is addible to $\Delta_{+}[v]$ if and only if there are
no interior points in the orange region $(vq\tilde{v})$. We justify this is
the case by an area argument. Note that contrarily to $u,v,\tilde{v},w$, the
intersection $q$ may not be integral.

It remains to analyse the case of common bases: $[u,w]=[\tilde{u},\tilde{w}]$
(cf. Figure 9). We first introduce three notations.

  * •

If $a_{1},a_{2},\ldots,a_{k}$ are points in $\mathbb{R}^{2}$, we denote
$[a_{1},\ldots,a_{k}]$ their convex hull and $(a_{1},\ldots,a_{k})$ their
strict convex hull (all coefficients of the convex combination are positive);

  * •

$\ell(a_{1}a_{2})$ is the length of the segment $[a_{1},a_{2}]$;

In our situation, the convex hull of $\Delta_{+}[v,\tilde{v}]$ is obtained by
adding to $\Delta$ the quadrilateral $Q=[u,v,\tilde{v},w]$. Let $q$ be the
point at which $[u,\tilde{v}]$ and $[v,w]$ intersect (this may not be an
integral point). Showing that $\tilde{v}$ is addible to $\Delta_{+}[v]$
amounts to checking that $|(v,q,\tilde{v})|_{\mathbb{Z}}=0$. Pick’s formula
(see e.g. [AZ18, p 89]) tells us that, if $D$ is a polygon

${\rm Area}(D)=\tfrac{1}{2}|\partial
D|_{\mathbb{Z}}+|\mathring{D}|_{\mathbb{Z}}-1\,.$ (99)

Given that $(u,w)$, $(u,v)$, $(\tilde{v},w)$, $(v,w)$ have no integral points,
(99) yields

${\rm
Area}(Q)=\tfrac{1}{2}|(v,\tilde{v})|_{\mathbb{Z}}+|(v,q,\tilde{v})|_{\mathbb{Z}}+1\,,$

and we seek to prove ${\rm Area}(Q)<\tfrac{3}{2}$. This would imply that the
two first terms vanish in the right-hand side vanish, which implies that
$\tilde{v}$ is addible to $\Delta_{+}[v]$.

We first observe that by Pick’s formula, the elementary triangles
$T_{v}=[u,v,w]$ and $T_{\tilde{v}}=[u,\tilde{v},w]$ have area $\frac{1}{2}$.
By superposition

$\begin{split}{\rm Area}(Q)&<{\rm Area}(T_{v})+{\rm Area}(T_{\tilde{v}})+{\rm
Area}[v,q,\tilde{v}]=1+{\rm Area}[v,q,\tilde{v}]\,,\\\ {\rm Area}[u,q,w]&<{\rm
Area}(T_{v})=\frac{1}{2}\,.\end{split}$ (100)

Denoting by $\theta\in(0,\pi)$ the angle $\widehat{u,q,w}$, which is also the
angle $\widehat{v,q,\tilde{v}}$, we have

${\rm Area}[u,q,w]=\tfrac{1}{2}\,\ell(uq)\,\ell(qw)\,\sin\theta\,,\qquad{\rm
Area}[v,q,\tilde{v}]=\tfrac{1}{2}\,\ell(vq)\,\ell(q\tilde{v})\sin\theta\,,$

from which we conclude that

${\rm
Area}[v,q,\tilde{v}]<\frac{1}{2}\,\frac{\ell(vq)}{\ell(uq)}\,\frac{\ell(q\tilde{v})}{\ell(qw)}\,.$

Since $v,q,w$ (resp. $u,q,\tilde{v}$) lie on a straight line, we can express
the right-hand side in terms of the horizontal coordinates (denoted by $h$) of
the points involved:

${\rm
Area}[v,q,\tilde{v}]<\frac{1}{2}\,\frac{h_{q}-h_{v}}{h_{q}-h_{u}}\,\frac{h_{q}-h_{\tilde{v}}}{h_{w}-h_{q}}\,.$

In our situation the ordering $h_{u}<h_{v}<h_{q}<h_{\tilde{v}}<h_{w}$ implies
that the two ratios are strictly smaller than $1$, hence ${\rm
Area}[v,q,\tilde{v}]<\frac{1}{2}$. Together with (100) we get the sought-for
inequality ${\rm Area}(Q)<\frac{3}{2}$. This concludes the proof of claim (2).

The last claim then follows by adding simultaneously all addible vertices to
$\Delta$, one obtains an inscribed polygon strongly equivalent to $\Delta$,
which by construction is the maximum (for the inclusion) among all inscribed
polygons obtained from $\Delta$ by addition of vertices. ∎

We now return to the main subject of this section: whether global
admissibility of spectral curves is preserved within strong equivalence
classes. Our main result is the following theorem:

######  Theorem 4.30 (Global admissibility is preserved in strong equivalence
classes).

Let $\mathfrak{C}=\\{P(x,y)=0\\}\subset\mathbb{C}^{2}$ and
$\mathfrak{C}^{\prime}=\\{P^{\prime}(x,y)=0\\}\subset\mathbb{C}^{2}$ be two
irreducible nondegenerate affine curves whose Newton polygon belong to the
same strong equivalence class, and we assume that the latter does not contain
a long diagonal. Let $\mathcal{S}$ and $\mathcal{S}^{\prime}$ be the
corresponding compact spectral curves as in Definition 4.3. Then $\mathcal{S}$
is globally admissible if and only if $\mathcal{S}^{\prime}$ is globally
admissible.

######  Remark 4.31.

If $\mathfrak{C}$ is irreducible, $\Delta(P)$ must be inscribed in
$\square(d_{x},d_{y})$ where $(d_{x},d_{y})$ is the bidegree. If $\Delta(P)$
and $\Delta(P^{\prime})$ are equivalent, $P$ and $P^{\prime}$ must have the
same bidegree. Although the Newton polygon of an irreducible affine curve
cannot be a long diagonal, it could still be strongly equivalent to a long
diagonal, and such a case is excluded in Theorem 4.30.

Given $P(x,y)=0$ defining a globally admissible spectral curve $\mathcal{S}$,
these results tell us that we can perform two types of operations preserving
global admissibility:

  * •

_Enlarging the Newton polygon/Deforming the curve._ We can add to $P$ linear
combinations of monomials $x^{i}y^{j}$ for which $(i,j)$ is addible to
$\Delta(P)$. Due to Proposition 4.29, we can in fact add all addible vertices
at the same time, and include $\mathcal{S}$ in a largest possible family of
globally admissible spectral curves.

  * •

_Squeezing the Newton polygon/Taking limits in a family of curves._ We can
send to $0$ the coefficient in $P$ of one monomial $x^{i}y^{j}$ where $(i,j)$
is removable from $\Delta(P)$. Contrarily to enlargements, we can kill
monomials one by one, but not necessarily simultaneously (as it may remove
interior points). The possible existence of several minimal elements in the
strong equivalence class of $\Delta(P)$ means that different types of spectral
curves may be obtained by taking limits from $\mathcal{S}$.

In the two operations, for generic choice of coefficients, the resulting
affine curve will remain nondegenerate, and we can also deform the
coefficients of monomials whose exponents $(i,j)$ are already in $\Delta(P)$.

###### Proof of Theorem 4.30.

As we can transform both $\Delta(P)$ and $\Delta(P^{\prime})$ to the maximal
polygon in their common strong equivalence classes, it is enough to consider
the situation where $\Delta(P^{\prime})$ is obtained from $\Delta(P)$ by
addition of an addible vertex, and prove that global admissibility of
$\Delta(P)$ is then equivalent to global admissibility of
$\Delta(P^{\prime})$. Let us analyse what happens corner by corner. Global
admissibility does not impose any condition on $\Gamma_{\infty\infty}$.
Condition ($\Gamma$A1) for global admissibility in Theorem 4.24 requires that
$|\hat{\Gamma}_{00}|_{\mathbb{Z}}=0$. This is preserved by addition/removal of
vertices which does not create/remove interior points — in particular these
vertices must be on the horizontal or vertical axes. We need to take a closer
look at $\Gamma_{0\infty}$ where global admissibility requires ($\Gamma$A2),
and $\Gamma_{\infty 0}$ where global admissibility requires ($\Gamma$A3).

We proceed to the top left corner. The edges in $E_{0\infty}$ have slopes
$0<\mu_{1}<\mu_{2}<\ldots<\mu_{\ell}$, appearing from right to left. Let us
add $\mu_{0}=0$ and $\mu_{\ell+1}=\infty$ to this sequence, although they do
not represent edges in $E_{0\infty}$. We claim that a vertex $v=(p,q)$ added
in the corner $\Gamma_{0\infty}(P)$ to produce $\Delta(P^{\prime})$ must
create exactly two new edges in $E_{0\infty}$ with slopes $\alpha,\beta$ such
that $\mu_{i-1}\leq\alpha<\mu_{i}<\beta\leq\mu_{i+1}$ for some $i\in[\ell]$.
Indeed, we must keep the endpoints of the edges in $E_{0\infty}$ (except
perhaps the leftmost and the rightmost one) as vertices, otherwise it would
result in new interior points in $\Delta(P^{\prime})$ compared to $\Delta(P)$.
Adding the vertex $v$ can then only result in adding to the convex hull of a
triangle with slopes $\alpha<\mu_{i}<\beta$. If we add such a $v$, the edge of
slope $\mu_{i}$ cannot have an interior point, unless $\Delta(P)$ consists of
a single edge from $(0,0)$ to $(d_{x},d_{y})$. In the latter case, any
interior point of that edge would remain on the boundary of
$\Delta(P^{\prime})$. However, in such a case, the assumption that $\Delta(P)$
is not a long diagonal forces $d_{x},d_{y}$ to be coprime, implying the
absence of interior point on this edge.

Figure 10.  Allowed addition of a vertex $v$ to $\Delta(P)$.

Denote by $u$ (resp. $w$) the left (resp. right) endpoint of $\mu_{i}$, and
consider the triangle $T$ formed by the vertices $u,v$ and $w$ (cf. Figure
10). Assume that the edge with slope $\alpha$ has interior points
$v_{1},\ldots,v_{q}$ when going from $u$ to $v$. The segment joining $v_{i}$
to $w$ cannot have an interior point as $T$ does not have interior points.
Then, instead of adding $v$ as a vertex, we could have added the $v_{j}$ one
by one, with increasing $j$. The $j$-th step amounts to adding to the so-far
obtained Newton polygon a triangle $T_{j}$ whose edges do not have interior
points. Therefore, it is enough to study these elementary steps where the
added triangle is an elementary triangle.

For such a triangle, still denoting by $\alpha,\mu_{i},\beta$ the slopes of
its edges, as in Figure 10:

  * •

We claim that the sequence
$\mu_{1},\dots,\mu_{i-1},\alpha,\beta,\mu_{i+1},\ldots,\mu_{\ell}$ is globally
admissible provided that the sequence
$\mu_{1},\ldots,\mu_{i-1},\mu_{i},\mu_{i+1},\ldots,\mu_{\ell}$ is globally
admissible. If $\mu_{i}>1$ this is proved in Lemma 4.33; if $\mu_{i}\geq 1$
this is proved in Lemma 4.34, showing that if $\alpha$ and/or $\beta$ are in
$E_{0\infty}$, they are $\leq 1$ and thus are automatically globally
admissible.

  * •

Conversely, we claim that the sequence
$\mu_{1},\ldots,\mu_{i-1},\mu_{i},\mu_{i+1},\ldots,\mu_{\ell}$ is globally
admissible provided that the sequence
$\mu_{1},\dots,\mu_{i-1},\alpha,\beta,\mu_{i+1},\ldots,\mu_{\ell}$ is globally
admissible. This is proved in Lemma 4.36.

The proof for the corner $\Gamma_{\infty 0}$ and condition ($\Gamma$A3) can be
obtained by reflection of these arguments over the axis $y=x$ and is thus
omitted. ∎

###  4.5. Proof of the lemmata

We are going to complete the proof of Theorem 4.30 in several steps. We
consider as in this proof a situation where an elementary triangle with edges
of slopes $\alpha<\mu_{i}<\beta$ is added to $\Delta(P)$ (or removed from
$\Delta(P^{\prime})$). Recall the encoding of slopes by parameters
$(r_{\alpha},\bar{s}_{\alpha})$, $(r_{i},\bar{s}_{i})$ and
$(r_{\beta},\bar{s}_{\beta})$ from (96). From Lemma 4.26 we know that
$\frac{1}{\beta},\frac{1}{\mu_{i}},\frac{1}{\alpha}$ are Farey neighbours in
$\mathcal{F}_{r_{i}}$. We decide to call edges by their slopes, when it does
not lead to confusion.

######  Lemma 4.32.

Assume that $\mu_{i}$ is locally admissible and $\mu_{i}>1$. Then $r_{i}\geq
2$, $\bar{s}_{i}=s_{i}>0$ and we have the following alternative:

  * •

If $r_{i}=+1\,\,{\rm mod}\,\,s_{i}$, then $\bar{s}_{\beta}=1$ and
$\bar{s}_{\alpha}=s_{i}-1$;

  * •

If $r_{i}=-1\,\,{\rm mod}\,\,s_{i}$ and $s_{i}\geq 2$, then
$\bar{s}_{\alpha}=1$ and $\bar{s}_{\beta}=s_{i}-1$.

In both cases, $\alpha$ and $\beta$ are locally admissible.

###### Proof.

The assumption $\mu_{i}>1$ implies that $r_{i}\geq 2$ and $\bar{s}_{i}\geq 1$.
It also implies $\alpha\geq 1$ as $\frac{1}{\alpha}$ is a right Farey
neighbour of $\frac{1}{\mu_{i}}<1$.

If $\bar{s}_{i}=1$, we have $\alpha=1$ and must be in the situation
$(\frac{1}{\beta},\frac{1}{\mu_{i}},\frac{1}{\alpha})=(\frac{r_{i}-2}{r_{i}-1},\frac{r_{i}-1}{r_{i}},1)$,
in particular $\bar{s}_{i}=\bar{s}_{\beta}=1$ and $\bar{s}_{\alpha}=0$. In
this case the first alternative is satisfied, $\alpha=1$ is locally admissible
because $r_{\alpha}=1$, and $\beta$ is locally admissible because either
$r_{\beta}=1$ (if $r_{i}=2$) or $r_{\beta}\geq 2$ is congruent to $1$ modulo
$s_{\beta}=\bar{s}_{\beta}=r_{\beta}-1$.

Now assume $\bar{s}_{i}\geq 2$. This forces $\beta>\alpha>1$, in particular
$r_{\alpha},r_{\beta}\geq 2$ and
$(s_{i},s_{\alpha},s_{\beta})=(\bar{s}_{i},\bar{s}_{\alpha},\bar{s}_{\beta})$
is a triple of positive numbers, implying that $s_{\alpha}+s_{\beta}=s_{i}\geq
2$ and $\max(s_{\alpha},s_{\beta})<s_{i}$. Furthermore, Bézout’s identity (95)
for the two pairs of neighbours yields

$r_{i}s_{\alpha}-r_{\alpha}s_{i}=1,\qquad r_{\beta}s_{i}-r_{i}s_{\beta}=1\,.$
(101)

Taking the reduction modulo $s_{i}$ yields:

$r_{i}s_{\alpha}=1\,\,{\rm mod}\,\,s_{i},\qquad r_{i}s_{\beta}=-1\,\,{\rm
mod}\,\,s_{i}\,.$ (102)

Recall from (91) and the discussion above it that the edge
$\mu_{i}=\frac{r_{i}}{r_{i}-s_{i}}>1$ has local parameter $s_{i}\in[r_{i}-1]$.
As we assumed it is locally admissible, we have $r_{i}=\pm 1\,\,{\rm
mod}\,\,s_{i}$. Together with (102) we deduce that $s_{\alpha}=\pm 1\,\,{\rm
mod}\,\,s_{i}$ and $s_{\beta}=\mp 1\,\,{\rm mod}\,\,s_{i}$. Recalling that
$1\leq s_{\alpha}<s_{i}$ and $1\leq s_{\beta}<s_{i}$ we conclude that
$(s_{\alpha},s_{\beta})=(1,s_{i}-1)$ in the case $r_{i}=-1\,\,{\rm
mod}\,\,s_{i}$ and $(s_{\alpha},s_{\beta})=(s_{i}-1,1)$ in the case
$r_{i}=1\,\,{\rm mod}\,\,s_{i}$, as announced.

We now turn to the local admissibility of $\alpha$ and $\beta$. When
$s_{\alpha}=1$, the edge $\alpha$ is clearly locally admissible. We are also
in the situation $s_{\beta}=s_{i}-1$ due to the previous discussion, while the
reduction of (101) modulo $s_{\beta}$ yields $r_{\beta}s_{i}=1\,\,{\rm
mod}\,\,s_{\beta}$. This implies $r_{\beta}=1\,\,{\rm mod}\,\,s_{\beta}$, thus
$\beta$ is locally admissible as well. When $s_{\beta}=1$, the edge $\beta$ is
clearly admissible, while $s_{\alpha}=s_{i}-1$ and the reduction of (101)
modulo $r_{\alpha}$ leads to $r_{\alpha}=-1\,\,{\rm mod}\,\,s_{\alpha}$, thus
$\alpha$ is locally admissible as well. ∎

######  Lemma 4.33.

Assume that $\mu_{1},\ldots,\mu_{i},\ldots,\mu_{\ell}$ is a sequence of
globally admissible edges, and that $\mu_{i}>1$. Then,
$\mu_{1},\ldots,\mu_{i-1},\alpha,\beta,\mu_{i+1},\ldots,\mu_{\ell}$ is a
sequence of globally admissible edges.

###### Proof.

As $\mu_{i}$ is assumed locally admissible, Lemma 4.32 tells us that
$r_{i}\geq 2$ and $\bar{s}_{i}=s_{i}$, that $r_{i}=\pm 1\,\,{\rm
mod}\,\,\bar{s}_{i}$, and that $\alpha,\beta$ are locally admissible.
According to Definition 4.22, we only need to check that the pairs
$\alpha,\mu_{j}$, the pairs $\beta,\mu_{j}$ (for each $j\neq i$) and the pair
$\alpha,\beta$ are globalisable. We shall examine the conditions
(G1)-(G2)-(G3) of Definition 4.22. Since $\frac{1}{\alpha}$ is the right Farey
neighbour of $\frac{1}{\mu_{i}}$ in $\mathcal{F}_{r_{i}}$, we have $\alpha\geq
1$.

We treat in detail the case $r_{i}=1\,\,{\rm mod}\,\,s_{i}$, meaning
$\bar{s}_{\beta}=1$ from Lemma 4.32 and thus
$\frac{1}{\beta}=\frac{r_{\beta}-1}{r_{\beta}}$, with $r_{\beta}\geq 2$ since
$\beta>\mu_{i}>1$.

Case $j>i$. We have $1<\mu_{i}<\mu_{j}$ and thus

$1\leq\alpha<\mu_{i}<\beta\leq\mu_{j}\,.$ (103)

First, we check that the pair $\beta,\mu_{j}$ is globalisable. When
$\beta=\mu_{j}$, the pair $\beta,\mu_{j}$ is globalisable due to Lemma 4.21 as
$\frac{1}{\beta}=\frac{r_{\beta}-1}{r_{\beta}}$. We now assume
$\beta<\mu_{j}$, and observe that
$\frac{1}{\mu_{j}}\leq\frac{r_{\beta}-1}{r_{\beta}}<\frac{1}{\mu_{i}}$ with
$0<r_{\beta}<r_{i}$. If $r_{\beta}<r_{j}$, the pair $\beta,\mu_{j}$ is
globalisable by choosing $m=r_{\beta}\leq r_{j}-1=r_{\beta j}-1$ in (G1). If
we rather have $r_{\beta}\geq r_{j}$, then
$\frac{r_{\beta}-1}{r_{\beta}}=\frac{r_{\beta j}-1}{r_{\beta j}}$ and thus
$\beta,\mu_{j}$ is globalisable by (G2).

Now, we check that $\alpha,\mu_{i}$ is globalisable. When $\alpha=1$ the pair
$\alpha,\mu_{i}$ satisfies (G3), so we move on to the case $\alpha>1$. As
$\mu_{i},\mu_{j}$ is globalisable by assumption and $1<\mu_{i}<\mu_{j}$, it
satisfies either (G1) or (G2).

  * •

If $\mu_{i},\mu_{j}$ satisfies (G1), then there exists $m\in(r_{ij}-1]$ such
that $\frac{1}{\mu_{j}}\leq\frac{m-1}{m}\leq\frac{1}{\mu_{i}}$. When
$m\in(r_{\alpha j}-1]$, since $\frac{1}{\mu_{i}}<\frac{1}{\alpha}$ we
recognise that $\alpha,\mu_{j}$ is globalisable by (G1). When $m\geq r_{\alpha
j}$, we write $\frac{r_{\alpha j}-1}{r_{\alpha
j}}\leq\frac{m-1}{m}<\frac{1}{\alpha}$ and deduce that $\alpha,\mu_{j}$ is
globalisable by (G2).

  * •

If $\mu_{i},\mu_{j}$ satisfies (G2), we have
$\frac{1}{\mu_{i}}\geq\frac{r_{ij}-1}{r_{ij}}$, and since $r_{i}>r_{\alpha}$
we get $\frac{1}{\alpha}>\frac{1}{\mu_{i}}\geq\frac{r_{\alpha i}-1}{r_{\alpha
i}}\geq\frac{r_{\alpha j}-1}{r_{\alpha j}}$, thus $\alpha,\mu_{j}$ is
globalisable by (G2).

Case $j<i$. We have $1<\mu_{j}\leq\alpha<\mu_{i}<\beta$. First, we check that
the pair $\mu_{j},\beta$ is globalisable. We have
$\frac{r_{\beta}-1}{r_{\beta}}<\frac{1}{\mu_{i}}<\frac{1}{\alpha}\leq\frac{1}{\mu_{j}}$
with $0<r_{\beta}<r_{i}$. When $r_{\beta}<r_{j}$, the pair $\mu_{j},\beta$ is
globalisable by choosing $m=r_{\beta}\leq r_{j}-1=r_{\beta j}-1$ in (G1). When
$r_{\beta}\geq r_{j}$, we have $\frac{r_{\beta}-1}{r_{\beta}}=\frac{r_{\beta
j}-1}{r_{\beta j}}$, hence $\mu_{j},\beta$ is globalisable by (G2).

Second we check that $\mu_{j},\alpha$ is globalisable. We use that
$\mu_{j},\mu_{i}$ is globalisable with $\mu_{j}>1$, so satisfies either (G1)
or (G2).

  * •

If $\mu_{j},\mu_{i}$ satisfies (G1), then there exists $m\in(r_{ij}-1]$
satisfying $\frac{1}{\mu_{i}}\leq\frac{m-1}{m}\leq\frac{1}{\mu_{j}}$. When
$m<r_{\alpha}$, since $r_{\alpha}<r_{i}$ the first inequality must be strict
and $m\in(r_{\alpha j}-1]$. As $\frac{1}{\alpha}$ is the right Farey neighbour
of $\frac{1}{\mu_{i}}$ in $\mathcal{F}_{r_{i}}$, we deduce that
$\frac{1}{\mu_{i}}<\frac{1}{\alpha}\leq\frac{m-1}{m}\leq\frac{1}{\mu_{j}}$,
thus $\mu_{j},\alpha$ is globalisable by (G1). When $m\geq r_{\alpha j}$, we
have $\frac{r_{\alpha j}-1}{r_{\alpha
j}}\leq\frac{m-1}{m}\leq\frac{1}{\mu_{j}}$ and thus $\mu_{j},\alpha$ is
globalisable by (G2). When $r_{\alpha}\leq m<r_{j}$, recalling that $\alpha>1$
implies $\bar{s}_{\alpha}\geq 1$, we have
$\frac{1}{\alpha}=\frac{r_{\alpha}-\bar{s}_{\alpha}}{r_{\alpha}}\leq\frac{r_{\alpha}-1}{r_{\alpha}}\leq\frac{m-1}{m}\leq\frac{1}{\mu_{j}}$
with $m\in(r_{\alpha j}-1]$, hence $\mu_{j},\alpha$ is globalisable by (G1).

  * •

If $\mu_{j},\mu_{i}$ satisfies (G2), we have
$\frac{1}{\mu_{j}}\geq\frac{r_{ij}-1}{r_{ij}}$. Since $r_{i}>r_{\alpha}$ we
deduce $\frac{1}{\mu_{j}}\geq\frac{r_{\alpha j}-1}{r_{\alpha j}}$ and thus
$\mu_{j},\alpha$ is globalisable by (G2).

This establishes that the sequence
$\mu_{1},\ldots,\mu_{i-1},\alpha,\beta,\mu_{i+1},\ldots,\mu_{l}$ is globally
admissible in the case $r_{i}=1\,\,{\rm mod}\,\,\bar{s}_{i}$. The case
$r_{i}=-1\,\,{\rm mod}\,\,\bar{s}_{i}$ can be handled similarly. ∎

######  Lemma 4.34.

Assume $\mu_{i}\leq 1$. Then:

  * •

either $\alpha=0$ (the corresponding edge does not belong to $E_{0\infty}$) or
$\alpha\in(0,1)$.

  * •

either $\beta=\infty$ (the corresponding edge does not belong to
$E_{0\infty}$) or $\beta\leq 1$.

###### Proof.

As $\mu_{i}\leq 1$, the corresponding edge is locally admissible.

When $r_{i}=1$, we have $\mu_{i}=\frac{1}{m}$ for some $m\in\mathbb{Z}_{>0}$.
Due to our convention, the right Farey neighbour of $\frac{1}{\mu_{i}}$ in
$\mathcal{F}_{1}$ is $\infty$, leading to $\alpha=0$. This edge is horizontal,
so is not in $E_{0\infty}$, does not correspond to a branch of the singularity
at $p_{0\infty}$ and can be ignored. Considering the left Farey neighbour of
$\frac{1}{\mu_{i}}=m$ in $\mathcal{F}_{1}$ we obtain $\beta=\frac{1}{m-1}$. If
$m\geq 2$ we have $\beta\leq 1$. If $m=1$, we rather have $\beta=\infty$: the
corresponding edge is vertical, so not in $E_{0\infty}$ and can be ignored.

Since $\mu_{i}=1$ has $(r_{i},\bar{s}_{i})=(1,0)$, if $r_{i}\geq 2$ we must
have $\mu_{i}<1$. Then, the right Farey neighbour $\frac{1}{\alpha}$ of
$\frac{1}{\mu_{i}}$ in $\mathcal{F}_{r_{i}}$ is finite and $>1$, while the
left Farey neighbour $\frac{1}{\beta}$ is $\geq 1$. ∎

######  Lemma 4.35.

Assume that $\alpha,\beta$ is globalisable. Then $\mu_{i}$ is locally
admissible.

###### Proof.

We will rely on the properties of $\alpha,\mu_{i},\beta$ seen in the course of
the proof of Lemma 4.32. By assumption, $\alpha,\beta$ satisfies one of the
conditions (G1), (G2) or (G3).

  * •

If $\alpha,\beta$ satisfies (G1), then there exists $m\in(r_{\alpha\beta}-1]$
such that
$\frac{1}{\beta}=\frac{r_{\beta}-\bar{s}_{\beta}}{r_{\beta}}\leq\frac{m-1}{m}\leq\frac{1}{\alpha}=\frac{r_{\alpha}-\bar{s}_{\alpha}}{r_{\alpha}}$.
As $r_{\alpha\beta}<r_{i}$, all the elements in the inequality are in the
Farey sequence $\mathcal{F}_{r_{i}}$. Since
$\frac{1}{\beta},\frac{1}{\mu_{i}},\frac{1}{\alpha}$ are Farey neighbours in
$\mathcal{F}_{r_{i}}$, we must have
$\frac{r_{i}-\bar{s}_{i}}{r_{i}}=\frac{1}{\mu_{i}}=\frac{m-1}{m}$. As
$r_{i},\bar{s}_{i}$ are coprime, we deduce $r_{i}=m$ and $\bar{s}_{i}=1$. Thus
$\mu_{i}$ is locally admissible.

  * •

If $\alpha,\beta$ satisfies (G2), we have
$1>\frac{1}{\alpha}=\frac{r_{\alpha}-\bar{s}_{\alpha}}{r_{\alpha}}\geq\frac{r_{\alpha\beta}-1}{r_{\alpha\beta}}$.
From this we infer $\bar{s}_{\alpha}=1$, hence
$\bar{s}_{\beta}=\bar{s}_{i}-1$. As $\mu_{i}>\alpha>1$, we have $r_{i}\geq 2$
and $\bar{s}_{i}=s_{i}$. The second equation of (102) then yields
$r_{i}=-1\,\,{\rm mod}\,\,s_{i}$, and we deduce that $\mu_{i}$ is locally
admissible.

  * •

If $\alpha,\beta$ satisfies (G3), we have $\alpha\leq 1$. Let us first
consider the case $r_{i}\geq 2$. Then
$\frac{1}{\beta},\frac{1}{\mu_{i}},\frac{1}{\alpha}$ are Farey neighbours in
$\mathcal{F}_{r_{i}}$. When $\alpha<1$, this implies $\mu_{i}\leq 1$, hence
$\mu_{i}$ is locally admissible. When $\alpha=1$, we have
$(r_{\alpha},r_{j})=(1,r_{i}-1)$ and
$\frac{1}{\mu_{i}}=\frac{r_{i}-1}{r_{i}}$, leading to $\bar{s}_{i}=s_{i}=1$,
so $\mu_{i}$ is locally admissible. In the case $r_{i}=1$, we have a triple
$\frac{1}{\beta},\frac{1}{\mu_{i}},\frac{1}{\alpha}$ of Farey neighbours in
$\mathcal{F}_{1}$, implying that $\frac{1}{\mu_{i}}\in\mathbb{Z}_{>0}$, hence
$\mu_{i}\leq 1$ is again locally admissible.

∎

######  Lemma 4.36.

Assume that the sequence
$\mu_{i},\ldots,\mu_{i-1},\alpha,\beta,\mu_{i+1},\ldots,\mu_{\ell}$ is
globally admissible. Then the sequence
$\mu_{1},\ldots,\mu_{i},\ldots,\mu_{\ell}$ is also globally admissible.

###### Proof.

The assumption implies that the pair $\alpha,\beta$ is globalisable, so Lemma
4.35 already says that $\mu_{i}$ is locally admissible. It remains to show
that for any $j\neq i$, the pair $\mu_{j},\mu_{i}$ is globalisable.

Case $\mu_{j}<\alpha<\mu_{i}<\beta$. By assumption $\mu_{j},\alpha$ is
globalisable.

  * •

If $\mu_{j},\alpha$ satisfies (G1), we have $1<\mu_{j}<\alpha$ and there
exists $m\in(r_{j\alpha}-1]$ such that
$\frac{1}{\alpha}\leq\frac{m-1}{m}\leq\frac{1}{\mu_{j}}$. Taking into account
$\alpha<\mu_{i}$ and $r_{\alpha}<r_{i}$ we deduce
$\frac{1}{\mu_{i}}\leq\frac{m-1}{m}\leq\frac{1}{\mu_{j}}$ with
$m\in(r_{ij}-1]$. Since $1<\mu_{j}<\mu_{i}$, we conclude that
$\mu_{i},\mu_{j}$ is globalisable by (G1).

  * •

If $\mu_{j},\alpha$ satisfies (G2), we have $1<\mu_{j}<\alpha$ and
$\frac{1}{\mu_{j}}\geq\frac{r_{j\alpha}-1}{r_{j\alpha}}$. When $r_{j}\geq
r_{i}$, since $r_{i}>r_{\alpha}$ we have
$\frac{1}{\mu_{i}}\geq\frac{r_{ji}-1}{r_{ij}}$ so $\mu_{j},\mu_{i}$ is
globalisable by (G2). When $r_{j}<r_{i}$, recalling that $\alpha>1$ guarantees
$\bar{s}_{\alpha}\geq 1$, we write
$\frac{1}{\mu_{i}}<\frac{1}{\alpha}=\frac{r_{\alpha}-\bar{s}_{\alpha}}{r_{\alpha}}\leq\frac{r_{\alpha}-1}{r_{\alpha}}\leq\frac{r_{j\alpha}-1}{r_{j\alpha}}\leq\frac{1}{\mu_{j}}$.
Taking $m=r_{\alpha}\leq r_{j}-1$ we see that the pair $\mu_{j},\mu_{i}$ is
globalisable by (G2).

  * •

If $\mu_{j},\alpha$ satisfies (G3), then $\mu_{j}\leq 1$ and thus
$\mu_{j},\mu_{i}$ is globalisable by (G3).

Limit case $\mu_{j}=\alpha<\mu_{i}<\beta$. By assumption $\mu_{j},\alpha$ is
globalisable, and Lemma 4.21 imposes $\mu_{j}\leq 1$ (only relevant for (G3))
or $s_{j}=1$.

  * •

If $\mu_{j},\beta$ satisfies (G1), we have $1<\mu_{j}$ and there exists
$m\in(r_{j\beta}-1]$ such that
$\frac{1}{\beta}\leq\frac{m-1}{m}\leq\frac{1}{\mu_{j}}$. As
$r_{j}=r_{\alpha}<r_{i}$ and $r_{\beta}<r_{i}$, the fraction $\frac{m-1}{m}$
is in the Farey sequence $\mathcal{F}_{r_{i}}$. We recall that
$\frac{1}{\beta},\frac{1}{\mu_{i}},\frac{1}{\alpha}=\frac{1}{\mu_{j}}$ are
Farey neighbours of $\mathcal{F}_{r_{i}}$. If
$\frac{1}{\mu_{i}}\leq\frac{m-1}{m}\leq\frac{1}{\mu_{j}}$ with
$m\in(r_{ij}-1]$, then $\mu_{j},\mu_{i}$ is globalisable by (G1). If
$\frac{1}{\mu_{i}}>\frac{m-1}{m}$, we must have
$\frac{1}{\beta}=\frac{m-1}{m}$ and then $r_{\beta}=m$ and
$\bar{s}_{\beta}=1$. Recalling that $s_{j}=1$, we obtain $\bar{s}_{i}=2$.
Bézout’s identities for the triple of Farey neighbours
$\frac{m-1}{m},\frac{r_{i}-2}{r_{i}},\frac{r_{j}-1}{r_{j}}$ then yield
$r_{i}=2m+1$ and $r_{j}=m+1<r_{i}=r_{ij}$, that is
$(\frac{1}{\beta},\frac{1}{\mu_{i}},\frac{1}{\mu_{j}})=(\frac{m-1}{m},\frac{2m-1}{2m+1},\frac{m}{m+1})$,
and we conclude that $\mu_{j},\mu_{i}$ still satisfies (G1) using $m+1$.

  * •

If $\mu_{j},\beta$ satisfies (G2), we have $1<\mu_{j}$ and
$\frac{1}{\mu_{j}}\geq\frac{r_{j\beta}-1}{r_{j\beta}}$, meaning in fact
$\frac{1}{\mu_{j}}=\frac{r_{j}-1}{r_{j}}$ and in particular $\bar{s}_{j}=1$.
When $r_{i}\leq r_{j}$, the latter is also $\frac{r_{ij}-1}{r_{ij}}$, hence
the pair $\mu_{j},\mu_{i}$ is globalisable by (G2). When $r_{i}>r_{j}$, we
have clearly $\frac{1}{\mu_{i}}\leq\frac{r_{j}-1}{r_{j}}\leq\frac{1}{\mu_{j}}$
hence the pair $\mu_{j},\mu_{i}$ is globalisable by (G1) with
$m=r_{j}\in(r_{i}-1]$.

  * •

If $\mu_{j},\beta$ satisfies (G3), we have $\alpha\leq 1$. So the pair
consisting of $\alpha=\mu_{j}$ and $\mu_{i}$ also satisfies (G3).

Case $\alpha<\mu_{i}<\beta\leq\mu_{j}$. The analysis is similar to the two
previous cases and is omitted. ∎

##  5\. Analyticity of topological recursion in families

In this section, we investigate how topological recursion behaves in families.
We will consider families of spectral curves over a base $T$, and analyse the
behavior of the topological recursion correlators as we move in the family. In
particular, we are interested in knowing when we can take limits, i.e., when
the correlators $\omega_{g,n}$ at a certain point in the family can be
recovered by taking an appropriate limit of the correlators at nearby points.

In the local topological recursion, the integrand in Definition 2.11 of the
recursion is only locally defined on the curve: it is different for each
ramification point $p\in\Sigma$ that we take the residue at, and the structure
of the neighbourhoods where they are locally defined as well as the behavior
of the branched covering $x$ there could change when we deform the spectral
curve. For instance, if we deform a locally admissible spectral curve we could
end up with a spectral curve that is not locally admissible or vice versa, and
in such cases the local topological recursion typically cannot hold or does
not make sense for all deformation parameters.

The global topological recursion is a tool to bypass these difficulties and it
enables studying the behaviour of topological recursion under deformations
and/or limits of spectral curves. More precisely, we take the following
approach. Given a spectral curve, we first use the results of Sections 3 and 4
to globalise the topological recursion as defined in Definition 3.1. Recall
that horizontal globalisation is a rewriting of the recursion formula as a sum
of contour integrals containing clusters of ramification points. Now, we can
deform the spectral curve while keeping the $x$-projection of the contours
fixed so that the ramification points remain inside the contour (and perhaps
collide there), and finally transform back from the global to the local
topological recursion on the deformed spectral curve of interest. In summary,
the strategy is

$\text{local to global}\quad\longrightarrow\quad\text{take limit or
deform}\quad\longrightarrow\qquad\text{global to local}\,.$

This procedure can be made to work under various conditions: the local
admissibility conditions, the sufficient conditions we found for
globalisation, and conditions on the type of deformations of spectral curves
that are allowed. In this section, we will formulate conditions on families of
spectral curves guaranteeing that the correlators behave nicely in families.
One important request is that one can find adapted disc collections so that
the branch points do not escape the discs as we vary the spectral curve. Our
main result is then Theorem 5.8 which says that the topological recursion
correlators are analytic over the base of the family.

This nevertheless raises an interesting question: why not start directly with
global topological recursion and avoid the discussion of whether local
topological recursion is equivalent to global topological recursion? That is,
why not take any family of spectral curves and simply study global topological
recursion directly for this family? The difficulty with this approach is that
one would need to show that global topological recursion is well-defined in
the first place, i.e. that it produces symmetric differentials, and this is
far from obvious a priori (see the discussion in Subsection 3.6). By starting
with local topological recursion, we avoid this issue, as we know that the
correlators produced by local topological recursion are well-defined symmetric
differentials, assuming that the spectral curve is locally admissible (Theorem
2.17). The price to pay is that, to study limits and deformations, we need to
assume that local topological recursion is equivalent to global topological
recursion, for instance by requiring that the sufficient conditions of Theorem
3.14 are satisfied.

###  5.1. Admissible families of spectral curves

######  Definition 5.1 (Family of spectral curves).

A _family of spectral curves_ over a connected complex manifold $T$ is a
quadruple  $\mathcal{S}_{T}=(f\colon\Sigma_{T}\rightarrow
T,x_{T},(\omega_{0,1})_{T},(\omega_{0,2})_{T})$ such that

  1. (1)

$f$ is a surjective submersion of complex manifolds with $1$-dimensional
fibres $\Sigma_{t}=f^{-1}(\\{t\\})$ having finitely many connected components.
We denote by $\iota_{t}\colon\Sigma_{t}\hookrightarrow\Sigma_{T}$ the natural
inclusion.

  2. (2)

$x_{T}\colon\Sigma_{T}\rightarrow\mathbb{P}^{1}$ is a holomorphic map whose
restriction to each connected component of the fibres of $f$ is not constant:

  3. (3)

$(\omega_{0,1})_{T}$ is a meromorphic section of the relative cotangent bundle
$\Omega_{f}$, whose divisor of poles and zeros intersects each fibre on a
discrete set;

  4. (4)

$(\omega_{0,2})_{T}\in
H^{0}(\Sigma_{T}\times_{T}\Sigma_{T};\Omega_{f}^{\boxtimes 2}(2\Delta))$ with
biresidue $1$ along the diagonal
$\Delta_{T}\subseteq\Sigma_{T}\times_{T}\Sigma_{T}$.

We say that the family of spectral curves is proper if the map
$f:\Sigma_{T}\rightarrow T$ is proper. We call _partial spectral curve_ the
data of $(f:\Sigma_{T}\rightarrow T,x_{T})$ satisfying in (1) and (2).

In practice, the base $T$ of the family can be taken to be a open subset of
$\mathbb{C}^{D}$. We do not necessarily restrict to proper families, as there
are various situations in topological recursion (most notably, in applications
to Hurwitz theory) where the family happens not to be proper. In lack of
properness there will be additional subtleties that we should treat carefully.
For instance, we could have situations where ramification points move off the
curve, and we will want to exclude such pathologies. Nevertheless, proper
families form an important class of families of spectral curves that we treat
in detail based on Section 4. In this case, $T$ is the parameter space for the
coefficients of the bihomogeneous polynomial equation cutting out the spectral
curve.

Given a family of spectral curves as above, the fibre over $t\in T$ is the
spectral curve

$\mathcal{S}_{t}\coloneqq\big{(}\Sigma_{t},x_{t}\coloneqq
x_{T}\circ\iota_{t},(\omega_{0,1})_{t}\coloneqq\iota_{t}^{*}(\omega_{0,1})_{T},(\omega_{0,2})_{t}\coloneqq(\iota_{t},\iota_{t})^{*}(\omega_{0,2})_{T}\big{)}\,.$

Note that the fibre of a family of proper spectral curves is a compact
spectral curve as defined in Definition 2.1. We define the map

$X\coloneqq(x_{T},f):\Sigma_{T}\rightarrow\mathbb{P}^{1}\times T\,,$

and denote $\mathsf{Ram}_{T}$ the zero locus of $\mathrm{d}X$, which is a
closed complex submanifold of $\Sigma_{T}$ intersecting each fibre in a
discrete set, and $\mathsf{Br}=x_{T}(\mathsf{Ram}_{T})\subset\mathbb{P}^{1}$.
For later use, we introduce $f_{n}\colon\Sigma^{n}_{T}\rightarrow T$, defined
as the cartesian product over $T$ of $n$ copies of $f:\Sigma_{T}\rightarrow
T$. We set $\mathsf{Ram}^{n}_{T}=\bigsqcup_{i=1}^{n}{\rm
pr}_{i}^{*}(\mathsf{Ram}_{T})$, where ${\rm
pr}_{i}\colon\Sigma_{T}^{n}\rightarrow\Sigma_{T}$ is the projection onto the
$i$-th factor.

We shall add conditions on our families of spectral curves in order to
establish an analytic dependence of the correlators on points $t\in T$. The
most basic ones are finiteness conditions on $x_{t}$ (useful for
globalisation) and conditions preventing ramification points from “escaping
the curve” as we vary in family. If a ramification point were escaping the