# A family of integrable maps associated with the Volterra lattice

A.N.W. Hone and J.A.G. Roberts
School of Mathematics & Statistics
University of New South Wales
Sydney, NSW 2052, Australia.
<EMAIL_ADDRESS><EMAIL_ADDRESS>

P. Vanhaecke
Laboratoire de Mathématiques et Applications
UMR 7348 CNRS-Université de Poitiers
86360 Chasseneuil-du-Poitou, France.
<EMAIL_ADDRESS>Work begun while on leave from School of
Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury
CT2 7NF, UK.

###### Abstract

Recently Gubbiotti, Joshi, Tran and Viallet classified birational maps in four
dimensions admitting two invariants (first integrals) with a particular degree
structure, by considering recurrences of fourth order with a certain symmetry.
The last three of the maps so obtained were shown to be Liouville integrable,
in the sense of admitting a non-degenerate Poisson bracket with two first
integrals in involution. Here we show how the first of these three Liouville
integrable maps corresponds to genus 2 solutions of the infinite Volterra
lattice, being the $g=2$ case of a family of maps associated with the
Stieltjes continued fraction expansion of a certain function on a
hyperelliptic curve of genus $g\geqslant 1$. The continued fraction method
provides explicit Hankel determinant formulae for tau functions of the
solutions, together with an algebro-geometric description via a Lax
representation for each member of the family, associating it with an algebraic
completely integrable system. In particular, in the elliptic case ($g=1$), as
a byproduct we obtain Hankel determinant expressions for the solutions of the
Somos-5 recurrence, but different to those previously derived by Chang, Hu and
Xin. By applying contraction to the Stieltjes fraction, we recover integrable
maps associated with Jacobi continued fractions on hyperelliptic curves, that
one of us considered previously, as well as the Miura-type transformation
between the Volterra and Toda lattices.

##  1 Introduction

In classical mechanics, the study of integrable Hamiltonian systems, given by
Hamiltonian vector fields with a sufficient number of independent first
integrals in involution with respect to a Poisson bracket, has a long history
that goes back to the origins of calculus. It was further enriched in the
latter half of the last century by the discovery of the method of inverse
scattering for solving certain Hamiltonian partial differential equations,
which gave new perspectives and new techniques for deriving finite-dimensional
integrable systems obtained as reductions of the latter. The case of discrete
integrable systems, in the form of difference equations or maps preserving a
symplectic (or Poisson) structure and satisfying the conditions for a discrete
analogue of Liouville’s theorem, soon began to attract attention [7, 34, 54],
but it is fair to say that, despite the fact that many examples are now known,
the theory of discrete integrability is much less well developed. For
integrable birational maps in the plane, the archetypal example is provided by
the QRT family of maps [44], whose level sets are biquadratic curves
(generically, of genus one), which are associated with elliptic fibrations
[15]. If one imposes a requirement of subexponential degree growth (zero
algebraic entropy, in the terminology of [6]), then in two dimensions the only
possibilities are maps that preserve a pencil of genus one curves (like QRT),
maps that preserve a pencil of rational curves, or completely periodic maps
[13]. This fits in with an observation of Veselov [54], that for an infinite
order birational map of the plane with an algebraic invariant, the level
curves can have genus at most one (as a consequence of the Hurwitz theorem on
the automorphism group of a Riemann surface).

Poisson maps in three dimensions with two first integrals, of which one is a
Casimir, can be reduced to the two-dimensional case by restricting to
symplectic leaves, and the common level sets are curves, so in an algebro-
geometric setting this will typically lead to elliptic fibrations. Thus, in
order to see new geometrical features, with invariant tori of dimension
greater than one, it is necessary to look to integrable maps in four
dimensions. Building on the work [30] and [20], which was based on considering
autonomous versions of the fourth-order members of hierarchies of discrete
Painlevé I/II equations from [11], in [21] Gubbiotti et al. presented a
classification of four-dimensional birational maps of recurrence type, that is

$\varphi:\qquad(w_{0},w_{1},w_{2},w_{3})\mapsto\Big{(}w_{1},w_{2},w_{3},\mathrm{F}(w_{0},w_{1},w_{2},w_{3})\Big{)},$
(1.1)

for a suitable rational function $\mathrm{F}$ of affine coordinates
$(w_{0},w_{1},w_{2},w_{3})\in{\mathbb{C}}^{4}$, with $\varphi$ being invariant
under the involution
$\iota:\,(w_{0},w_{1},w_{2},w_{3})\mapsto(w_{3},w_{2},w_{1},w_{0})$ and having
two independent polynomial invariants, $K_{1}$, $K_{2}$ say, with specific
degree patterns
$(\deg_{w_{0}}K_{j},\deg_{w_{1}}K_{j},\deg_{w_{2}}K_{j},\deg_{w_{3}}K_{j})=(1,3,3,1)$
and $(2,4,4,2)$ for $j=1,2$, respectively. The result of this classification
was six maps with parameters, labelled (P.i-vi), together with six associated
maps, labelled (Q.i-vi), dual to them in the sense of [45], meaning that they
arise as discrete integrating factors for linear combinations of the first
integrals. (Note that, aside from the original connection with [11], the
letter P in this nomenclature has nothing to do with the usual labelling of
continuous Painlevé equations.)

###  1.1 The map (P.iv): an integrable map in 4D

From our point of view, the most interesting examples among those presented in
[21] are the maps labelled (P.iv), (P.v) and (P.vi). According to Table 1 in
[21], these are the only ones arising from a discrete variational principle
(Lagrangian), which leads to a non-degenerate Poisson bracket in four
dimensions, such that the two first integrals $K_{1}$, $K_{2}$ are in
involution, and this means that in the real case the Liouville tori are two-
dimensional (cf. Fig.1). In this paper, our main concern will be the case of
(P.iv), which is the birational map given in affine form by the recurrence

$\begin{array}[]{l}w_{n+4}w_{n+3}w_{n+2}+w_{n+2}w_{n+1}w_{n}+2w_{n+2}^{2}(w_{n+3}+w_{n+1})\\\
+w_{n+2}(w_{n+3}^{2}+w_{n+3}w_{n+1}+w_{n+1}^{2})+w_{n+2}^{3}+\nu
w_{n+2}(w_{n+3}+w_{n+2}+w_{n+1})+bw_{n+2}+a=0.\end{array}$ (1.2)

The above map depends on three essential parameters $a,b,\nu$ (compared with
[21], by rescaling we have set the parameter $d=1$), and it can be written in
the form (1.1), with

$\mathrm{F}=-\frac{w_{0}w_{1}w_{2}+w_{1}w_{2}w_{3}+w_{1}^{2}w_{2}+w_{2}w_{3}^{2}+2w_{1}w_{2}^{2}+2w_{2}^{2}w_{3}+w_{2}^{3}+\nu(w_{1}w_{2}+w_{2}w_{3}+w_{2}^{2})+bw_{2}+a}{w_{2}w_{3}},$

which is the rational function of $w_{0},w_{1},w_{2},w_{3}$ obtained by
solving for $w_{4}$ in the recurrence (1.2) with $n=0$. More recently,
Gubbiotti, showed how the equation (1.2) also arises from a classification of
additive fourth-order difference equations, based on the requirement of a
discrete Lagrangian structure alone [22].

The first integral denoted $I_{\mathrm{low}}^{\mathrm{P}.\mathrm{iv}}$ in [21]
is given in affine coordinates by

$K_{1}=w_{1}w_{2}\Big{(}w_{2}w_{3}+w_{0}w_{1}-w_{0}w_{3}+(w_{1}+w_{2})^{2}+\nu(w_{1}+w_{2})+b\Big{)}+a(w_{1}+w_{2}).$
(1.3)

The latter has the degree pattern $(1,3,3,1)$. In particular, it is linear in
$w_{3}$, which implies that, on each three-dimensional level set
$K_{1}=k_{1}=\,\,$const, the map (1.2) reduces to a birational map in three
dimensions, given by the recurrence

$\begin{array}[]{rcl}w_{n+3}w_{n+2}w_{n+1}(w_{n+2}-w_{n})+w_{n+2}w_{n+1}^{2}w_{n}+w_{n+2}w_{n+1}(w_{n+1}+w_{n+2})^{2}&&\\\
+\nu\,w_{n+2}w_{n+1}(w_{n+1}+w_{n+2})+b\,w_{n+2}w_{n+1}+a\,(w_{n+1}+w_{n+2})&=&k_{1}.\end{array}$

A second independent invariant for (1.2), with degree pattern $(2,4,4,2)$, is
given by

$\begin{array}[]{rcr}K_{2}&=&w_{1}w_{2}\left(\begin{array}[]{c}w_{0}^{2}w_{1}+w_{3}^{2}w_{2}+w_{0}w_{3}(w_{1}+w_{2})+w_{0}(w_{2}^{2}+2w_{1}^{2})+w_{3}(w_{1}^{2}+2w_{2}^{2})\\\
+\,3(w_{0}+w_{3})w_{1}w_{2}+(w_{1}+w_{2})^{3}\\\
+\nu\,\,\Big{(}w_{0}w_{3}+(w_{0}+w_{3})(w_{1}+w_{2})+(w_{1}+w_{2})^{2}\Big{)}+b\,(w_{0}+w_{1}+w_{2}+w_{3})\end{array}\right)\\\
&&+a\,\Big{(}w_{0}w_{1}+w_{3}w_{2}+(w_{1}+w_{2})^{2}\Big{)}\,\,.\end{array}$
(1.4)

This differs slightly from the second invariant presented in [21], which is
$I_{\mathrm{high}}^{\mathrm{P}.\mathrm{iv}}=K_{2}-\nu K_{1}$.

The non-degenerate Poisson bracket between the coordinates, which was obtained
in [21] by making use of a discrete Lagrangian for (1.2), is given by

$\\{\,w_{n},w_{n+1}\,\\}=0,\,\\{\,w_{n},w_{n+2}\,\\}=\frac{1}{w_{n+1}},\,\\{\,w_{n},w_{n+3}\,\\}=-\frac{w_{n}+2w_{n+1}+2w_{n+2}+w_{n+3}+\nu}{w_{n+1}w_{n+2}},$
(1.5)

for all $n$. So the 4D map of the form (1.1) defined by (1.2) is a Poisson
map, in the sense that
$\\{\,\varphi^{*}G,\varphi^{*}H\,\\}=\varphi^{*}\\{\,G,H\,\\}$ for all
functions $G,H$ on ${\mathbb{C}}^{4}$. Equivalently, the fact that $\varphi$
is Poisson with respect to a non-degenerate bracket means that it preserves a
symplectic form ${\omega}$, such that $\varphi^{*}{\omega}={\omega}$. The two
independent invariants given in [21] are in involution with respect to the
Poisson bracket, which is equivalent to the involutivity of functions (1.3)
and (1.4), that is

$\\{\,K_{1},K_{2}\,\\}=0.$

Hence the four-dimensional map defined by (1.2) is integrable in the Liouville
sense.

Computing the Hamiltonian vector field for the first flow, generated by
$K_{1}$, we find that on the phase space ${\mathbb{C}}^{4}$ with coordinates
$(w_{0},w_{1},w_{2},w_{3})$ this takes the form

$\frac{\mathrm{d}w_{n}}{\mathrm{d}t}=w_{n}(w_{n+1}-w_{n-1})\qquad\mathrm{for}\quad
n=1,2,$ (1.6)

while the components of the vector field for $n=0,3$ appear to be more
complicated rational functions of these 4 coordinates and the parameters
$a,b,\nu$. However, since (1.2) is a Poisson map it commutes with this flow,
so it follows that the relation (1.6) extends to all $n\in{\mathbb{Z}}$. To
see this, note that the vecor field
$\tfrac{\mathrm{d}}{\mathrm{d}t}=\\{\cdot,K_{1}\\}$ commutes with the action
of $\varphi$, and $\varphi^{*}(w_{n})=w_{n+1}$; hence, if (1.6) holds for some
particular $n$, then

$\frac{\mathrm{d}w_{n+1}}{\mathrm{d}t}=\varphi^{*}\left(\frac{\mathrm{d}w_{n}}{\mathrm{d}t}\right)=\varphi^{*}\big{(}w_{n}(w_{n+1}-w_{n-1})\big{)}=w_{n+1}(w_{n+2}-w_{n}),$

which is just the same equation with $n\to n+1$. Thus the combined solutions
of the iterated map and the flow, which are compatible with one another,
generate a sequence of functions $\big{(}w_{n}(t)\big{)}_{n\in{\mathbb{Z}}}$
satisfying (1.6), which is the Volterra lattice equation, first considered by
Kac and van Moerbeke [31]. We will see that, in a certain sense to be made
precise, these are genus 2 solutions of this lattice equation.

Figure 1: Plot of the 3D projection of 10000 points on the orbit of (1.2) with
initial values
$(\tfrac{21}{16},\tfrac{21}{16},\tfrac{452}{343},\tfrac{3124}{2373})$ and
parameters $a=-9$, $b=29$, $\nu=-10$.

A wide variety of difference equations admitting Lax pairs and explicit
formulae for first integrals have been presented by Svinin [49, 50], including
a family that arises as reductions of the hierarchy of symmetries of the
Volterra lattice. By eliminating the parameter $b$ from (1.2) we get an
equation of fifth order, that is

$w_{n+4}\left(\sum_{j=2}^{5}w_{n+j}+\nu\right)+\frac{a}{w_{n+3}}=w_{n+1}\left(\sum_{j=0}^{3}w_{n+j}+\nu\right)+\frac{a}{w_{n+2}},$
(1.7)

and upon setting $a=0$ this reduces to equation (1) in [51] when $s=4$ (cf.
also the case $N=4$ in [27], where an equivalent equation is obtained via a
periodic reduction of the lattice KdV equation); thus the map (1.2) can be
viewed as a 1-parameter generalization of one of Svinin’s symmetry reductions
of the Volterra lattice hierarchy, which in turn is a generalization of one of
the maps considered in [12].

###  1.2 Outline of the paper

The purpose of this article is to give a complete description of the complex
geometry of the solutions of the map defined by (1.2). In reaching this goal,
we found that all of the structures we obtained could naturally be extended to
analogous constructions associated with a family of curves of arbitrary genus
$g$ (elliptic for $g=1$, hyperelliptic for $g\geqslant 2$).

Section 2 presents a set of empirical observations, numerical examples and
standalone results about the (P.iv) map. Originally, these were the specific
clues that led us to uncover the geometrical structure of the solutions of
(1.2). To begin with, we use a $p$-adic method to identify the singularity
pattern of the solutions, leading us to introduce a tau function $\tau_{n}$,
which lifts (1.2) to a recurrence of order 7 with the Laurent property; this
is a Laurentification of the original map, in the sense of [23]. By
considering a pattern of initial values that approaches a singularity, and
substituting this set of initial data into the expressions (1.3) and (1.4) on
the level set $K_{j}=k_{j}$ for $j=1,2$, in the limit that the singularity is
reached we find a hyperelliptic curve of genus 2, isomorphic to the
Weierstrass quintic

$y^{2}=(1+\nu x+bx^{2})^{2}+4a(1+\nu x+bx^{2})x^{3}+4k_{1}x^{4}+4(k_{2}+\nu
k_{1})x^{5}.$ (1.8)

We also show that the tau function $\tau_{n}$ satisfies a Somos-9 recurrence
with coefficients that depend on $a,b,\nu$ and the values of $K_{1},K_{2}$
along each orbit of (1.2). It turns out that both the singularity pattern, and
the corresponding tau function substitution
$w_{n}={\tau_{n}\tau_{n+3}}/({\tau_{n+1}\tau_{n+2}})$ found for (1.2), are the
same as for the QRT map associated with the Somos-5 recurrence [24], which is
associated with a family of elliptic curves; so this was a strong initial hint
that analogues of the (P.iv) map should exist for any genus $g$. For
enthusiasts of detective stories, the results in this section provide
motivation and insight into how we made the first steps on the trail that led
to the rest of the paper. However, a reader who is not particularly fond of
experimental mathematics can safely omit Section 2 on first reading, since the
subsequent sections are not logically dependent on it, and are written in a
more linear, deductive style.

In Section 3, we start from a hyperelliptic curve $\Gamma_{f}$ of arbitrary
genus $g\geqslant 1$, given by a Weierstrass equation $y^{2}=f(x)$ where
$f\in{\mathbb{C}}[x]$ is of odd degree $2g+1$, analogous to (1.8), together
with a particular choice of rational function $F_{0}$ on the curve, and show
how a Stieltjes continued fraction (S-fraction) expansion of this function, of
the form

$F_{0}=1-\frac{w_{1}x}{F_{1}}=1-\cfrac{w_{1}x}{1-\cfrac{w_{2}x}{F_{2}}}=1-\cfrac{w_{1}x}{1-\cfrac{w_{2}x}{1-\cfrac{w_{3}x}{1-\cdots}}}\;,$
(1.9)

leads to a birational map on the coefficients $w_{j}$ of the fraction, in
dimension $3g+1$, which we refer to as the Volterra map ${\cal V}_{g}$. As we
shall see, iterating ${\cal V}_{g}$ for generic initial data produces the
infinite sequence of coefficients $w_{n}$ for $n\geqslant 1$ that appear in
the fraction (1.9), while applying the inverse map ${\cal V}_{g}^{-1}$ extends
this sequence to $n\leqslant 0$. Furthermore, the recursion for the S-fraction
can be rewritten in the form of a discrete Lax equation. In this setting, the
hyperelliptic curve $\Gamma_{f}$ is the spectral curve, and the polynomial $f$
has $2g+1$ non-trivial coefficients which provide conserved quantities (first
integrals) for the map. In particular, for $g=1$ it reduces to a QRT map whose
tau functions satisfy the Somos-5 recurrence, while when $g=2$ we find that,
by fixing the values of three of the first integrals to reduce it to four
dimensions, the map is precisely (1.2). We also show from the S-fraction that,
for any $g$, the solutions of the map can be written explicitly in terms of
tau functions that (up to gauge transformations) are expressed as Hankel
determinants.

Next, in Section 4, we introduce a family of compatible Poisson brackets for
the map ${\cal V}_{g}$: it is a Poisson map with respect to any of these
brackets, and the conserved quantities provide a sufficient number of
invariants in involution, so we have a Liouville integrable map for any
positive integer $g$. There is a considerable literature on algebraic
completely integrable (a.c.i.) Hamiltonian systems [1, 53], and in this vein
we prove a stronger result on the integrability of the Volterra maps, namely
that they can be considered as discrete analogues of a.c.i. systems: for each
$g$, the generic level set of the invariants is an affine part of an Abelian
variety, namely the Jacobian $\operatorname{Jac}(\bar{\Gamma}_{f})$ of the
(completion of the) corresponding spectral curve, and the restriction of the
map to any of these complex tori is given by translation over a fixed vector
which we describe explicitly. In particular, for the map (P.iv) given by
(1.2), each generic level set defined by fixing $K_{j}=k_{j}$ for $j=1,2$ is
an affine part of an Abelian surface, that is the Jacobian of the curve (1.8).

The whole basis of our construction is the S-fraction expansion (1.9), which
may appear to be a deus ex machina in Section 3, but in fact has many
antecedents in the literature on integrable systems, and especially in the
development of van der Poorten’s results on Jacobi fraction (J-fraction)
expansions in elliptic [41] and hyperelliptic function fields [40, 42, 43], as
presented in recent work by one of us [28]. The latter revealed the integrable
structure of maps generated by J-fractions of the form

$Y_{0}=\alpha_{0}(X)+\frac{1}{Y_{1}}={\alpha}_{0}(X)+\cfrac{1}{{\alpha}_{1}(X)+\cfrac{1}{Y_{2}}}={\alpha}_{0}(X)+\cfrac{1}{{\alpha}_{1}(X)+\cfrac{1}{{\alpha}_{2}(X)+\cfrac{1}{\ddots}}}\;,$
(1.10)

where $Y_{0}=\big{(}Y+P_{0}(X)\big{)}/Q_{0}(X)$ is a rational function on a
hyperelliptic curve ${\cal C}$ defined by a polynomial of even degree $2g+2$,
that is ${\cal C}:\,Y^{2}=P_{0}^{2}+Q_{0}Q_{-1}$, for polynomials
$P_{j},Q_{j}$ of degrees $g+1,g$ in $X$, respectively, with the coefficents
${\alpha}_{j}={\alpha}_{j}(X)$ in (1.10) being linear in $X$. It was shown in
[28] that, for appropriate such $Y_{0}$, the shift from one line of the
J-fraction to the next defines a Liouville integrable map on a phase space of
dimension $3g+1$, which (on a generic level set of the first integrals)
corresponds to a fixed translation on the Jacobian of (the completion of)
${\cal C}$.

There are classical results going back to Abel on the continued fraction
expansion of the square root of an even degree polynomial (i.e. the function
$Y$ on an even hyperelliptic curve ${\cal C}$), although the fact that the
sequence of degrees of the coefficients ${\alpha}_{j}(X)$ in such an expansion
is eventually periodic was proved only very recently [56] (they need not all
be linear in $X$, as per the above assumption about the function $Y_{0}$ in
(1.10)). This is intimately related to elliptic [2] and hyperelliptic
analogues of orthogonal polynomials [3], as well as more general types of Padé
approximation problems connected with integrable systems [5, 14]. In fact,
Stieltjes continued fractions (of finite type) were already used in the
solution of the finite Volterra lattice by Moser [37], and such fractions were
applied to obtain Hankel determinant solutions for non-isospectral extensions
more recently [9].

Section 5 of the paper starts by considering the continuous Hamiltonian system
that shares the same phase space with the Volterra map ${\cal V}_{g}$. After
proving that this continuous system is a.c.i., we show that iteration of
${\cal V}_{g}$ (and its inverse) leads to an infinite sequence of meromorphic
functions $\big{(}w_{n}(t)\big{)}_{n\in{\mathbb{Z}}}$ of $t$, the time
associated with one of the commuting Hamiltonian flows, providing a
meromorphic solution of the Volterra lattice equation (1.6) (hence reproducing
the above observation about (P.iv) in the particular case $g=2$). Next we show
that this also produces a meromorphic solution of the Toda lattice, taken in
the form

$\frac{\mathrm{d}d_{n}}{\mathrm{d}t}=d_{n}(v_{n-1}-v_{n})\;,\qquad\frac{\mathrm{d}v_{n}}{\mathrm{d}t}=d_{n}-d_{n+1}\;,\qquad
n\in{\mathbb{Z}},$ (1.11)

by applying the well-known Miura transformation between the Volterra and Toda
lattices. We further show that the latter transformation arises naturally via
the contraction procedure for continued fractions, due to Stieltjes [48],
which combines successive pairs of lines in an S-fraction into a single line
in a J-fraction, and thereby maps a generic solution of the Volterra map
${\cal V}_{g}$ to an associated solution of the map generated by van der
Poorten’s construction in genus $g$. The paper ends in Section 6, with a few
conclusions and observations concerning transformations relating solutions of
(P.v) and (P.vi) to solutions of the map (P.iv), which we plan to discuss in
detail elsewhere. There is also an appendix (Section 7), in which a birational
Poisson isomorphism is established between the genus $g$ even Mumford system
(see [53]) and the Hamiltonian system associated with the Volterra map ${\cal
V}_{g}$.

##  2 Laurentification and tau functions for the map (P.iv)

In this section we exhibit certain phenomena displayed by the iterates of the
map (P.iv), which are related to its discrete integrability. Firstly, we
describe the singularity pattern of the iterates, which is found from an
empirical $p$-adic approach, and leads to the introduction of a sequence of
tau functions $\tau_{n}$ for these iterates. On the one hand, these tau
functions satisfy a homogeneous recursion relation of order 7 with the Laurent
property; so this is a Laurentification of (P.iv), as we state here and prove
in Section 3.4. On the other hand, these tau functions are also shown to
satisfy a Somos-9 relation, with invariants of (P.iv) as coefficients.
Secondly, by considering the limit where a solution of (P.iv) approaches a
singularity, we are led to a family of genus two curves which will turn out to
be at the core of the Stieltjes continued fractions (Section 3) and the
algebraic integrability of (P.iv) (Section 4).

The Laurent property is a very special feature of certain birational
transformations, appearing in cluster algebras and their generalizations [17,
33], which a priori is unrelated to integrability [25]. However, it turns out
that the solutions of discrete integrable systems are often encoded by tau
functions satisfying relations that have the Laurent property, such as
bilinear equations of discrete Hirota type [35]. Despite the fact that
integrable maps occurring “in the wild” typically do not exhibit the Laurent
phenomenon, it nevertheless seems to be a common feature of such maps that
they admit Laurentification, that is, a lift to a higher-dimensional relation
that does have the Laurent property [23]. For some time, singularity analysis
has been used as a tool to detect integrability of maps (see [36] and
references), and when the pattern of places where the solutions have a zero or
pole is sufficiently simple, this can further suggest an appropriate way to
introduce tau functions and perform Laurentification [27].

To start with, we apply the $p$-adic approach described in [27] (see also
[32]) to the map (P.iv) defined by (1.2), and derive a singularity pattern
from it. This empirical approach is based on examining the prime factorization
of the terms of specific orbits $(w_{n})_{n\in{\mathbb{N}}}$ defined in
${\mathbb{Q}}$, chosen arbitrarily, and considering the behaviour of the
$p$-adic norms $|w_{n}|_{p}$ for particular primes $p$. As a concrete example,
upon picking the specific parameter values $\nu=3$, $a=5$, $b=7$ and setting
all four initial values to be 1, we then find a sequence of rational numbers
given by

$-30,\tfrac{743}{30},\tfrac{10541}{22290},\tfrac{3819540}{7831963},-\tfrac{4315187227}{1342059038},\tfrac{6624290612327}{739436079902},-\tfrac{23965197528782842}{3649794341246183},-\tfrac{304709076970269230792}{118290200741883010693},\ldots,$
(2.1)

where the latter terms factorize as $-2\cdot 3\cdot 5$, $\tfrac{743}{2\cdot
3\cdot 5}$, $\tfrac{83\cdot 127}{2\cdot 3\cdot 5\cdot 743}$,
$\tfrac{2^{2}\cdot 3\cdot 5\cdot 63659}{83\cdot 127\cdot 743}$,
$-\tfrac{13\cdot 743\cdot 446753}{2\cdot 83\cdot 127\cdot 63659}$,
$\tfrac{19\cdot 83\cdot 127\cdot 1579\cdot 20947}{2\cdot 13\cdot 63659\cdot
446753}$, $-\tfrac{2\cdot 59\cdot 51593\cdot 61837\cdot 63659}{13\cdot 19\cdot
1579\cdot 20947\cdot 446753}$, $-\tfrac{2^{3}\cdot 13\cdot 967\cdot
446753\cdot 6782004923}{19\cdot 59\cdot 1579\cdot 20947\cdot 51593\cdot
61837},\ldots$ For several different primes, e.g. $p=3$, $5$, $83$, $127$,
$743,\ldots$, this reveals a common pattern whereby, for some $n$,

$|w_{n}|_{p}=p^{-1}\;,\quad|w_{n+1}|_{p}=p\;,\quad|w_{n+2}|_{p}=p\;,\quad|w_{n+3}|_{p}=p^{-1}\;,$
(2.2)

with the prime $p$ being absent from the factorization on the previous and on
the next terms: $|w_{n-1}|_{p}=|w_{n+4}|_{p}=1$. The $p$-adic norms (2.2)
identify places where the orbit of the map over the finite field
${\mathbb{F}}_{p}$ has a zero or pole, as well as the order of these [32].
Since the recurrence (1.2) defines a birational map, any orbit defined for
$n\geqslant 0$ can be extended to $n<0$ (at least, provided that it does not
reach a singularity, where $w_{n}=0$ for some $n$; but see Corollary 2.6
below). Hence, the pattern (2.2) suggests that for $n\in{\mathbb{Z}}$ one
should make the tau function substitution

$w_{n}=\frac{\tau_{n}\tau_{n+3}}{\tau_{n+1}\tau_{n+2}}\;,$ (2.3)

so that the places where a prime factor $p$ appears in the numerators or
denominators of the sequence $(w_{n})_{n\in{\mathbb{Z}}}$ can be encoded by
the appearance of the factor $p$ in the terms of the tau function sequence
$(\tau_{n})_{n\in{\mathbb{Z}}}$. These tau functions can be defined
recursively, in two quite different ways:

######  Proposition 2.1.

Suppose that $(w_{n})_{n\in{\mathbb{Z}}}$ is a solution of (1.2). Then the
corresponding sequence $(\tau_{n})_{n\in{\mathbb{Z}}}$ satisfies

  1. (1)

A homogeneous recurrence of order 7 and degree 8:

$\begin{array}[]{ccl}\tau_{n+7}\tau_{n+4}^{2}\tau_{n+3}^{3}\tau_{n+2}^{2}+\tau_{n+6}^{2}\tau_{n+3}^{4}\tau_{n+2}^{2}+2\tau_{n+6}\tau_{n+5}^{2}\tau_{n+3}^{2}\tau_{n+2}^{3}+\tau_{n+6}\tau_{n+5}\tau_{n+4}^{2}\tau_{n+3}^{2}\tau_{n+2}\tau_{n+1}&&\\\
+\,\tau_{n+5}^{4}\tau_{n+2}^{4}+2\tau_{n+5}^{3}\tau_{n+4}^{2}\tau_{n+2}^{2}\tau_{n+1}+\tau_{n+5}^{2}\tau_{n+4}^{4}\tau_{n+1}^{2}+\tau_{n+5}^{2}\tau_{n+4}^{3}\tau_{n+3}^{2}\tau_{n}&&\\\
+\,\nu(\tau_{n+6}\tau_{n+5}\tau_{n+4}\tau_{n+3}^{3}\tau_{n+2}^{2}+\tau_{n+5}^{3}\tau_{n+4}\tau_{n+3}\tau_{n+2}^{3}+\tau_{n+5}^{2}\tau_{n+4}^{3}\tau_{n+3}\tau_{n+2}\tau_{n+1})&&\\\
+\,b\,\tau_{n+5}^{2}\tau_{n+4}^{2}\tau_{n+3}^{2}\tau_{n+2}^{2}+a\,\tau_{n+5}\tau_{n+4}^{3}\tau_{n+3}^{3}\tau_{n+2}=0\;.\end{array}$
(2.4)

  2. (2)

A (generalized) Somos-9 recurrence:

${\alpha}_{1}\,\tau_{n+9}\tau_{n}+{\alpha}_{2}\,\tau_{n+8}\tau_{n+1}+{\alpha}_{3}\,\tau_{n+7}\tau_{n+2}+{\alpha}_{4}\,\tau_{n+6}\tau_{n+3}+{\alpha}_{5}\,\tau_{n+5}\tau_{n+4}=0\;.$
(2.5)

The coefficients ${\alpha}_{i}$ are polynomial functions of the parameters of
the map and of the values $k_{1},k_{2}$ of the invariants $K_{1},K_{2}$, hence
are constant along each orbit $(w_{n})_{n\in{\mathbb{Z}}}$. They are given by

$\begin{array}[]{c}{\alpha}_{1}=k_{1},\qquad{\alpha}_{2}=ak_{2}-k_{1}^{2},\qquad{\alpha}_{3}=a\Big{(}ak_{2}-2k_{1}^{2}\Big{)},\\\
{\alpha}_{4}=a\Big{(}k_{2}^{2}+\nu
k_{1}k_{2}+bk_{1}^{2}+a^{2}k_{1}\Big{)},\qquad{\alpha}_{5}=-k_{1}\Big{(}k_{2}^{2}+\nu
k_{1}k_{2}+bk_{1}^{2}+a^{2}k_{1}\Big{)}\;.\end{array}$

###### Proof.

The proof of _(1)_ is by a direct substitution of (2.3) in (1.2), while _(2)_
is derived directly by implementing the method from [26], which involves
computing determinants of matrices with entries that are of degree two in tau
functions. More precisely, for _(2)_ one should write five copies of the
Somos-9 relation as a matrix equation
$\mathrm{M}_{n}\,{\bm{\alpha}}=\mathbf{0}$, and then verify that there is a
non-zero vector of coefficients ${\bm{\alpha}}$, lying in the kernel of
$\mathrm{M}_{n}$, that is independent of $n$; the latter calculations require
extensive use of computer algebra. ∎

The recursion defined by (1.2) requires 4 initial values, while (2.4) requires
7, and the discrepancy between the two is described by the three-parameter
group $({\mathbb{C}}^{*})^{3}$ of gauge transformations, with action given by

$\tau_{n}\mapsto A_{\pm}\,B^{n}\,\tau_{n}\;,\qquad
A_{+},A_{-},B\in{\mathbb{C}}^{*}\;,$ (2.6)

corresponding to the freedom to rescale even/odd terms by a different factor
$A_{\pm}$, and apply a rescaling $B^{n}$ that is exponential in $n$ to all
terms. Any 4 non-zero initial values $w_{0},\dots,w_{3}$ of (1.2) allow a
corresponding set of non-zero initial data to be determined for (2.4), up to
this gauge freedom; for example, we may take as corresponding initial data
$\tau_{n}=1$ for $n=0,1,2$ and $\tau_{3}=w_{0}$, $\tau_{4}=w_{0}w_{1}$,
$\tau_{5}=w_{0}^{2}w_{1}w_{2}$ and $\tau_{6}=w_{0}^{2}w_{1}^{2}w_{2}w_{3}$,
which are polynomials (in fact, monomials) in $w_{0},w_{1},w_{2},w_{3}$.
Notice that (2.4) can also be solved rationally for $\tau_{n}$ in terms of
$\tau_{n+1},\dots,\tau_{n+7}$ so that the sequence $(\tau_{n})$ is actually
defined for all $n\in{\mathbb{Z}}$.

######  Example 2.2.

In the case $\nu=3$, $a=5$, $b=7$, taking the initial values $\tau_{j}=1$ for
$0\leqslant j\leqslant 6$ in (2.4) yields the sequence of tau functions
beginning

$1,1,1,1,1,1,1,-30,-743,10541,127318,5807789,628430947,-188231024119,52465590084328,\ldots,$
(2.7)

which consists of integers, and the terms after the initial 1s have prime
factorizations given by $-2\cdot 3\cdot 5,-743,83\cdot 127,2\cdot
63659,13\cdot 446753,19\cdot 1579\cdot 20947,-59\cdot 61837\cdot
51593,2^{3}\cdot 967\cdot 6782004923$, etc. These correspond to the prime
factors appearing in the numerators and denominators in the particular
sequence of rational values of $w_{n}$ illustrated in (2.1) above. Due to a
reversing symmetry of the recurrence (2.4), this sequence extends backwards to
$n<0$ in such a way that the property $\tau_{6-n}=\tau_{n}$ holds for all
$n\in{\mathbb{Z}}$, since the 7 initial data have this symmetry. Furthermore,
these tau functions also satisfy the Somos-9 recurrence

$28\,\tau_{n+9}\tau_{n}-239\,\tau_{n+8}\tau_{n+1}-5115\,\tau_{n+7}\tau_{n+2}+136125\,\tau_{n+6}\tau_{n+3}-762300\,\tau_{n+5}\tau_{n+4}=0,$
(2.8)

which corresponds to (2.5) with the coefficients ${\alpha}_{j}$ being fixed
(up to overall rescaling) by the specified choices of $\nu,a,b$, together with
the fact that the first integrals take the values $K_{1}=28$, $K_{2}=109$. For
any solution of (2.4), the subsequences consisting of even/odd index terms,
that is $\hat{\tau}_{n}=\tau_{2n}$ or $\tau_{2n+1}$, respectively, also
satisfy a Somos-8 relation, of the form

$\hat{{\alpha}}_{1}\,\hat{\tau}_{n+8}\hat{\tau}_{n}+\hat{{\alpha}}_{2}\,\hat{\tau}_{n+7}\hat{\tau}_{n+1}+\hat{{\alpha}}_{3}\,\hat{\tau}_{n+6}\hat{\tau}_{n+2}+\hat{{\alpha}}_{4}\,\hat{\tau}_{n+5}\hat{\tau}_{n+3}+\hat{{\alpha}}_{5}\,\hat{\tau}_{n+4}^{2}=0\;.$
(2.9)

For the particular integer sequence above, up to overall scale the
coefficients are given by

$\begin{array}[]{c}\hat{{\alpha}}_{1}=195848,\qquad\hat{{\alpha}}_{2}=-61660241775,\qquad\hat{{\alpha}}_{3}=13236763233189375,\\\
\hat{{\alpha}}_{4}=-8064076031989579800,\qquad\hat{{\alpha}}_{5}=-3603810041796109733\;.\end{array}$

The relation (2.9) can be regarded as an ordinary difference reduction of a
constraint for a tau function defined on a multidimensional lattice, which
arises from a Hermite-Padé approximation problem (cf. equation (2.10) in
[14]). An explanation for why this Somos-8 relation must hold will be provided
in Section 5, via the connection with the Toda lattice and the results in
[28].

The following proposition shows that the recurrence (2.4) is a
Laurentification of (P.iv). In particular, this explains why the tau functions
in the preceding example are all integers.

######  Proposition 2.3.

The recurrence (2.4) has the Laurent property. More precisely, for all
$n\in{\mathbb{Z}}$,

$\tau_{n}\in{\mathbb{Z}}[a,b,\nu,\tau_{0},\tau_{1},\tau_{2}^{\pm
1},\tau_{3}^{\pm 1},\tau_{4}^{\pm 1},\tau_{5},\tau_{6}]\;.$

In principle, for $n\geqslant 0$ the proof of the proposition is a direct
application of Theorem 2 in [23], and the same method of proof applies for
$n<0$, because the map defined by (2.4) is birational. However, the formal
verification to be done by this method quickly gets out of hand, since the
rational functions obtained as the formulae for the first few iterations of
the map defined by (2.4) soon become too complicated for the checks to be
carried out by a simple computer algebra program. A general proof of
Proposition 2.3 that does not require any computer algebra will be given in
Section 3.4.

######  Remark 2.4.

As mentioned in the introduction, through the relation (1.7), the special case
$a=0$ of the (P.iv) map is closely linked to a difference equation appearing
in the work of Svinin, and to a reduction of the lattice KdV equation. Thus,
when the parameter $a=0$, the result of Proposition 2.3 is subtly related to
the fact that the order 7, degree 6 recurrence

$\begin{array}[]{ccl}\tau_{n+7}\tau_{n+4}^{2}\tau_{n+3}\tau_{n+2}\tau_{n+1}+\tau_{n+6}\tau_{n+5}\tau_{n+4}\tau_{n+3}^{2}\tau_{n}+\tau_{n+6}^{2}\tau_{n+3}^{2}\tau_{n+2}\tau_{n+1}+\tau_{n+6}\tau_{n+5}\tau_{n+4}^{2}\tau_{n+1}^{2}&&\\\
+\tau_{n+6}\tau_{n+5}^{2}\tau_{n+2}^{2}\tau_{n+1}-{\alpha}\tau_{n+5}\tau_{n+4}^{2}\tau_{n+3}^{2}\tau_{n+2}+{\beta}\tau_{n+6}\tau_{n+5}\tau_{n+4}\tau_{n+3}\tau_{n+2}\tau_{n+1}=0\end{array}$
(2.10)

has the Laurent property, in the sense that it generates Laurent polynomials
in $\tau_{0},\ldots,\tau_{6}$ with coefficients in
${\mathbb{Z}}[{\alpha},{\beta}]$ (the case $N=4$ of Proposition 2.3 in [27]).
The point is that (2.3) coincides with the tau function substitution found for
maps related with lattice KdV reductions in [27], and the relation (2.10)
holds for solutions of (2.4) obtained by setting $a=0$, $\nu={\beta}$, and
taking an orbit for which the value of the first integral $K_{1}$ is fixed to
be $k_{1}=-{\alpha}$.

######  Example 2.5.

Upon taking $a=0$, $b=-17$, $\nu=-11$ and making a specific choice of 7
integer initial values for (2.4), with the 3 central values fixed to be 1, we
generate an integer sequence that begins as follows:

$3,2,1,1,1,4,5,699,-25626,453024,-112570254,23354432973,61327997061471,-35520663450983076,\ldots.$

Then we see that this sequence also satisfies the relation (2.10) with
${\beta}=-11$ and ${\alpha}=-k_{1}=327$. Also, since three of the coefficients
in (2.5) vanish when $a=0$, we see that the Somos-9 relation for this sequence
takes the shorter (three-term) Gale-Robinson form

$\tau_{n+9}\tau_{n}+327\,\tau_{n+8}\tau_{n+1}+3850083\,\tau_{n+5}\tau_{n+4}=0,$

which is a reduction of the discrete Hirota equation, and is in agreement with
the $N=4$ case of Theorem 1.1 in [27].

The Laurent property for (2.4), together with the formula (2.3), immediately
implies that the generic orbit $(w_{n})$ of (P.iv) is well-defined, as stated
in the following corollary.

######  Corollary 2.6.

For generic non-zero initial values
$(w_{0},w_{1},w_{2},w_{3})\in({\mathbb{C}}^{*})^{4}$, the orbit
$(w_{n})_{n\in{\mathbb{Z}}}$ exists, with
$w_{n}\in{\mathbb{P}}^{1}={\mathbb{C}}\cup\\{\infty\\}$.

###### Proof.

As we shall see in Section 3.4, the initial tau functions can be chosen so
that all $\tau_{n}$ are polynomials in the initial data for (1.2). For a fixed
index $n$ it is then clear from (2.3) that $w_{n}$ is an indeterminate element
of ${\mathbb{P}}^{1}$ only when at least two out of three successive tau
functions in (2.3) vanish, i.e. belong to a certain proper Zariski closed
subset of the space of (non-zero) initial data for (1.2). Considering this
condition for all $n$ yields a subset of this space of initial data, which is
the intersection of a countable family of Zariski open subsets. Such an
intersection is a residual, hence dense, subset so that for generic initial
data, the orbit is well-defined. ∎

Note that (P.iv) was originally defined as a birational affine map in
${\mathbb{C}}^{4}$, but the above corollary allows the existence of certain
orbits defined in $({\mathbb{P}}^{1})^{4}$. Notice also that this corollary
does not state that the subset of initial data for which the orbit exists is
open. This stronger statement will follow from algebraic integrability,
without use of the tau functions (see Section 4).

As an initial foray into the geometry of the map, defined by (1.2), we now
consider the singularity pattern in more detail, by taking three non-zero
initial values $w_{0},w_{1},w_{2}\in{\mathbb{C}}^{*}$ followed by a fourth
value proportional to a small parameter ${\epsilon}\in{\mathbb{C}}$, and
consider the behaviour of the solution in the limit ${\epsilon}\to 0$. To
reformulate this in terms of tau functions, we set

$\tau_{0}=Z\;,\;\tau_{1}=\tau_{2}=\tau_{3}=1,\,\tau_{4}=X\;,\;\tau_{5}=Y\;,\;\tau_{6}={\epsilon}\;,\;\quad
XYZ\neq 0\;,$

where three adjacent values have been set equal to 1 by a choice of gauge.
This gives, using (2.3), four non-zero initial values

$w_{0}=Z\;,\quad w_{1}=X\;,\quad w_{2}=\frac{Y}{X}\;,\quad
w_{3}=\frac{{\epsilon}}{XY}\;,$ (2.11)

for the map (1.2), such that the fourth value $w_{3}\to 0$ as ${\epsilon}\to
0$. Upon substituting these values in (1.2) we find as subsequent values

$w_{4}=C_{4}\,{\epsilon}^{-1}+O(1)\;,\quad
w_{5}=C_{5}\,{\epsilon}^{-1}+O(1)\;,\quad
w_{6}=C_{6}\,{\epsilon}+O({\epsilon}^{2})\;,\quad
w_{7}=C_{7}+O({\epsilon})\;,$

for certain coefficients $C_{j}$ which are rational functions of $X,Y,Z$.
Notice that the leading order behaviours of $w_{3},w_{4},w_{5},w_{6}$ are
${\epsilon},{\epsilon}^{-1},{\epsilon}^{-1},{\epsilon}$, respectively, with
terms of $O(1)$ on either side, which corresponds to the singularity pattern
(2.2) obtained above by the $p$-adic method. Now if we substitute the initial
values (2.11) into $K_{1}=k_{1}$, $K_{2}=k_{2}$, where $K_{1}$ and $K_{2}$ are
the invariants (1.3) and (1.4), and take the limit ${\epsilon}\to 0$, then
(after clearing denominators) we get two polynomial relations between $X,Y,Z$,
which define an affine algebraic curve. Upon taking resultants with respect to
$Z$, this yields a single relation between $X$ and $Y$, namely

$\displaystyle\big{(}aY^{2}-(\nu
k_{1}+k_{2})Y-ak_{1}\big{)}X^{4}+\big{(}(a\nu-
k_{1})Y^{2}+(a^{2}-bk_{1})Y+k_{1}^{2}\big{)}X^{3}$ (2.12)
$\displaystyle+\big{(}2aY^{3}+(ab-\nu
k_{1})Y^{2}-2ak_{1}Y\big{)}X^{2}+Y^{2}\big{(}(a\nu-
k_{1})Y+a^{2}\big{)}X+aY^{4}=0\;.$

This plane curve is birationally equivalent to the Weierstrass quintic (1.8),
which can be seen from the transformation

$x=\frac{X(aY-k_{1}X)}{Y(aX^{2}-k_{1}X+aY)}\;,$ (2.13)

together with a corresponding formula for $y=y(X,Y)$, which is rather unwieldy
and so is omitted. For generic values of $k_{1},k_{2}$, the curve (2.12) is a
smooth hyperelliptic curve of genus 2. As already said, in its Weierstrass
form (1.8), this family of curves will play an important role in all that
follows.

##  3 S-fractions on hyperelliptic curves and Volterra maps

In this section, for a fixed integer $g>0$, we introduce an affine space of
triplets of polynomials, reminiscent of the phase space of the Mumford system
[38, 53] and use Stieltjes continued fractions (S-fractions) to construct a
series of birational automorphisms of the affine space, indexed by $g$, which
we describe in several ways. When $g=1$ we recover several known integrable
maps, and for $g=2$ we recover the map (P.iv), which was the primary
motivation for this study, while the maps for $g>2$ appear to be new. We also
give solutions in terms of Hankel determinants of the iterates of these maps,
i.e., of the corresponding recursion relations.

For a fixed $g>0$ we consider the affine space

$M_{g}:=\left\\{({\cal P}(x),{\cal Q}(x),{\cal
R}(x))\in{\mathbb{C}}[x]^{3}~{}\Bigl{|}~{}\begin{array}[]{ll}\deg{\cal
P}(x)\leqslant g\;,&{\cal P}(0)=1\\\ \deg{\cal Q}(x)\leqslant g\;,&{\cal
Q}(0)=2\\\ \deg{\cal R}(x)\leqslant g+1\;,&{\cal
R}(0)=0\end{array}\right\\}\;.$ (3.1)

It is clear that $M_{g}$ is an affine space of dimension $3g+1$: writing

${\cal P}(x)=1+\sum_{i=1}^{g}p_{i}x^{i}\;,\qquad{\cal
Q}(x)=2+\sum_{i=1}^{g}q_{i}x^{i}\;,\qquad{\cal
R}(x)=\sum_{i=1}^{g+1}r_{i}x^{i}\;,$ (3.2)

the coefficients $p_{1},\dots,p_{g},q_{1},\dots,q_{g},r_{1},\dots,r_{g+1}$
provide a natural system of linear coordinates on $M_{g}$. We will often write
an element $({\cal P},{\cal Q},{\cal R})$ of $M_{g}$ as a traceless $2\times
2$ matrix

${\bf L}(x):=\left(\begin{array}[]{lr}{\cal P}(x)&{\cal R}(x)\\\ {\cal
Q}(x)&-{\cal P}(x)\end{array}\right)\;,$ (3.3)

which will later serve as a Lax operator, and think of $M_{g}$ as an affine
space of matrices (Lax operators). It is then natural to consider the map
$\mu$ defined by

$\begin{array}[]{lcrcl}\mu&:&M_{g}&\to&{\mathbb{C}}[x]\\\ &&{\bf
L}(x)=\left(\begin{array}[]{lr}{\cal P}(x)&{\cal R}(x)\\\ {\cal Q}(x)&-{\cal
P}(x)\end{array}\right)&\mapsto&-\det{\bf L}(x)={\cal P}(x)^{2}+{\cal
Q}(x){\cal R}(x)\;.\end{array}$ (3.4)

In view of the degree constraints on the entries of ${\bf L}(x)$, the
polynomial $-\det{\bf L}(x)$ has degree at most $2g+1$ and its constant term
is 1; moreover, every such polynomial is contained in the image of $\mu$. In
the $g=2$ case these curves are precisely the ones encountered in Section 2 in
the singularity analysis of (P.iv), see (1.8), (2.12) and (2.13), which in
part motivates the choice of constraints on the polynomials ${\cal P},{\cal
Q}$ and ${\cal R}$. (When $M_{g}$ is endowed with a Poisson structure, as in
Section 4, then $\mu$ can be viewed a momentum map.)

Throughout this section, $g>0$ is fixed. In each of the following subsections,
the results and phenomena being discussed will be specialized and illustrated
for the cases of $g=1$ and $g=2$, when $M_{g}$ has dimension 4 and 7,
respectively.

###  3.1 Stieltjes continued fractions

We start from a hyperelliptic curve $\Gamma_{f}$, defined by an odd
Weierstrass equation

$\Gamma_{f}:y^{2}=f(x)\;,\quad\hbox{with}\quad
f(x):=1+\sum_{j=1}^{2g+1}c_{j}x^{j}\in{\mathbb{C}}[x]\;.$ (3.5)

When $f$ has degree $2g+1$ and has no multiple roots, $\Gamma_{f}$ is non-
singular and its genus is $g$, which explains the notation used. Let $({\cal
P},{\cal Q},{\cal R})$ be any point in $\mu^{-1}(f)$, the fiber of $\mu$ over
$f$, so that $f(x)={\cal P}^{2}(x)+{\cal Q}(x){\cal R}(x)$, and the spectral
curve $\Gamma_{f}$ is the characteristic equation $\det({\bf
L}(x)-y\mathbf{1})=0$. We consider on $\Gamma_{f}$ the rational function,
given by

$F:=\frac{y+{\cal P}(x)}{{\cal Q}(x)}=\frac{{\cal R}(x)}{y-{\cal P}(x)}\;.$
(3.6)

In preparation for constructing the Stieltjes continued fraction of $F$, we
show in the following lemma how the triplet $({\cal P},{\cal Q},{\cal R})$
leads to another triplet $(\tilde{\cal P},\tilde{\cal Q},\tilde{\cal R})$ in
the same fiber of $\mu$, under the assumption that $({\cal P},{\cal Q},{\cal
R})$ is _regular_ , meaning that

$\frac{2{\cal P}(x)-{\cal Q}(x)+{\cal
R}(x)}{x}\big{|}_{x=0}=2p_{1}-q_{1}+r_{1}\neq 0\;.$ (3.7)

######  Lemma 3.1.

Given a regular triplet $({\cal P},{\cal Q},{\cal R})$ in $\mu^{-1}(f)$, there
exists a unique $w\in{\mathbb{C}}^{*}$ and a unique triplet $(\tilde{\cal
P},\tilde{\cal Q},\tilde{\cal R})$ in $\mu^{-1}(f)$ such that

$\frac{y+{\cal P}(x)}{{\cal Q}(x)}=1-\frac{w\,x}{\frac{y+\tilde{\cal
P}(x)}{\tilde{\cal Q}(x)}}\;.$ (3.8)

The two triplets are related by

$\tilde{\cal P}(x)={\cal Q}(x)-{\cal P}(x)\;,\quad\tilde{\cal
Q}(x)=\frac{2{\cal P}(x)-{\cal Q}(x)+{\cal R}(x)}{-wx}\;,\quad\tilde{\cal
R}(x)=-wx{\cal Q}(x)\;,$ (3.9)

and

$w=-\frac{2\,p_{1}-q_{1}+r_{1}}{2}=-\frac{\tilde{r}_{1}}{2}\;.$ (3.10)

###### Proof.

We will constructively show that we can achieve (3.8) with
$w\in{\mathbb{C}}^{*}$ and $(\tilde{\cal P},\tilde{\cal Q},\tilde{\cal
R})\in\mu^{-1}(f)$ uniquely determined. Clearing the denominators in (3.8) and
using $y^{2}=f(x)$ we get

$y({\cal P}(x)+\tilde{\cal P}(x)-{\cal Q}(x))+f(x)+{\cal P}(x)\tilde{\cal
P}(x)-\tilde{\cal P}(x){\cal Q}(x)+wx{\cal Q}(x)\tilde{\cal Q}(x)=0\;,$ (3.11)

an equality which holds in the field of fractions of
${\mathbb{C}}[x,y]/(y^{2}-f(x))$, so that the coefficients of $y$ and of
$y^{0}$ in (3.11) must be zero. The vanishing of the former coefficient gives
the first equation in (3.9) and guarantees $\tilde{\cal P}(0)=1$ and
$\deg\tilde{\cal P}\leqslant g$. The vanishing of the $y^{0}$ coefficient
gives

$-w\,\tilde{\cal Q}(x)=\frac{f(x)+({\cal P}(x)-{\cal Q}(x))\tilde{\cal
P}(x)}{x{\cal Q}(x)}=\frac{f(x)-({\cal P}(x)-{\cal Q}(x))^{2}}{x{\cal
Q}(x)}=\frac{2{\cal P}(x)-{\cal Q}(x)+{\cal R}(x)}{x}\;,$ (3.12)

where we have used in the last step that $f(x)={\cal P}^{2}(x)+{\cal
Q}(x){\cal R}(x)$. Notice that in view of the values of the constant terms in
(3.2), the last numerator in (3.12) vanishes for $x=0$ and is of degree at
most $g+1$. Also, as the triplet $({\cal P},{\cal Q},{\cal R})$ is assumed to
be regular, the polynomial $(2{\cal P}-{\cal Q}+{\cal R})/x$ does not vanish
at $x=0$, hence we can (uniquely) choose $w\in{\mathbb{C}}^{*}$ such that
$\tilde{\cal Q}(0)=2$. This gives the first equality in (3.10) and the second
equation in (3.9). The first equality in (3.12) also shows that
$f(x)-\tilde{\cal P}^{2}(x)$ is divisible by $\tilde{\cal Q}(x)$, with
quotient $-wx{\cal Q}(x)$; thus, if we take the third equality in (3.9) to
define $\tilde{\cal R}$, then $\deg\tilde{\cal R}=g+1$, $\tilde{\cal R}(0)=0$
and $f(x)=\tilde{\cal P}^{2}(x)+\tilde{\cal Q}(x)\tilde{\cal R}(x)$,
completing the proof that $(\tilde{\cal P},\tilde{\cal Q},\tilde{\cal R})$
belongs to $\mu^{-1}(f)$. Notice that the third equality in (3.9) implies the
alternative formula for $w$ in (3.10), since ${\cal Q}(0)=2$. ∎

Applying the lemma to all regular points of the fiber $\mu^{-1}(f)$ yields a
rational map of the fiber to itself, given by $({\cal P},{\cal Q},{\cal
R})\mapsto(\tilde{\cal P},\tilde{\cal Q},\tilde{\cal R})$ with $(\tilde{\cal
P},\tilde{\cal Q},\tilde{\cal R})$ given by (3.9). Since the latter can also
be solved rationally for $({\cal P},{\cal Q},{\cal R})$ in terms of
$(\tilde{\cal P},\tilde{\cal Q},\tilde{\cal R})$ by using the second
expression for $w$ in (3.10), this rational map is actually a birational
automorphism of the fiber. Iterating this map starting from a triplet $({\cal
P}_{0},{\cal Q}_{0},{\cal R}_{0})=({\cal P},{\cal Q},{\cal R})\in\mu^{-1}(f)$,
we get an infinite sequence $({\cal P}_{n},{\cal Q}_{n},{\cal
R}_{n})_{n\in{\mathbb{Z}}}$ of triplets as well as an infinite sequence
$(w_{n})_{n\in{\mathbb{Z}}}$ in ${\mathbb{C}}^{*}$, such that $({\cal
P}_{n+1},{\cal Q}_{n+1},{\cal R}_{n+1})$ and $w_{n+1}$ are related to $({\cal
P}_{n},{\cal Q}_{n},{\cal R}_{n})$ as dictated by the lemma:

${\cal P}_{n+1}(x)={\cal Q}_{n}(x)-{\cal P}_{n}(x)\;,\quad{\cal
Q}_{n+1}(x)=\frac{2{\cal P}_{n}(x)-{\cal Q}_{n}(x)+{\cal
R}_{n}(x)}{-w_{n+1}x}\;,\quad{\cal R}_{n+1}(x)=-w_{n+1}x{\cal Q}_{n}(x)\;.$
(3.13)

Writing, as in (3.2), ${\cal P}_{n}(x)=1+\sum_{i=1}^{g}p_{n,i}x^{i}$, and
similarly for ${\cal Q}_{n}(x)$ and ${\cal R}_{n}(x)$, the value of $w_{n+1}$
is given, according to (3.10), by

$w_{n+1}=-\frac{2p_{n,1}-q_{n,1}+r_{n,1}}{2}=-\frac{r_{n+1,1}}{2}\;,$ (3.14)

for all $n\in{\mathbb{Z}}$.

It is clear that $({\cal P}_{n},{\cal Q}_{n},{\cal R}_{n})$ is obtained from
$({\cal P}_{0},{\cal Q}_{0},{\cal R}_{0})$ by repeating the map $n$ times (or,
when $n<0$, repeating the inverse of the map $-n$ times). As pointed out in
the lemma, the starting triplet $({\cal P}_{0},{\cal Q}_{0},{\cal R}_{0})$
must be regular in order for $({\cal P}_{1},{\cal Q}_{1},{\cal R}_{1})$ and
$w_{1}\in{\mathbb{C}}^{*}$ to exist. But nothing guarantees that $({\cal
P}_{1},{\cal Q}_{1},{\cal R}_{1})$ will also be regular, and in general it
need not be so; assuming $({\cal P}_{1},{\cal Q}_{1},{\cal R}_{1})$ to be a
regular triplet puts an open linear condition on the coefficients of $({\cal
P}_{1},{\cal Q}_{1},{\cal R}_{1})$, namely that $2p_{1,1}-q_{1,1}+r_{1,1}\neq
0$, which amounts to an open polynomial condition on $({\cal P}_{0},{\cal
Q}_{0},{\cal R}_{0})$, and for the existence of every extra term of the
sequence such an extra condition is to be added to the triplet $({\cal
P}_{0},{\cal Q}_{0},{\cal R}_{0})$. However, since this amounts to a countable
number of open conditions on the latter, this means that when $({\cal
P}_{0},{\cal Q}_{0},{\cal R}_{0})$ is _generic_ , in the sense that it belongs
to a residual subset of the fiber $\mu^{-1}(f)$, the sequence of triplets of
polynomials $({\cal P}_{n},{\cal Q}_{n},{\cal R}_{n})\in\mu^{-1}(f)$ and the
sequence of constants $w_{n}\in{\mathbb{C}}^{*}$, both indexed by
$n\in{\mathbb{Z}}$, exist. For generic $({\cal P}_{0},{\cal Q}_{0},{\cal
R}_{0})$, iterating (3.8) gives

$F_{0}:=\frac{y+{\cal P}_{0}(x)}{{\cal
Q}_{0}(x)}=1-\frac{w_{1}x}{\frac{y+{\cal P}_{1}(x)}{{\cal
Q}_{1}(x)}}=\cdots=1-\cfrac{w_{1}x}{1-\cfrac{w_{2}x}{1-\cfrac{w_{3}x}{1-\cdots}}}\;,$
(3.15)

yielding the _Stieltjes continued fraction_ , also called _$S$ -fraction_, of
$F_{0}$. Similarly, each triple $({\cal P}_{n},{\cal Q}_{n},{\cal R}_{n})$,
$n\in{\mathbb{Z}}$, is associated with a rational function $F_{n}$, with a
corresponding S-fraction obtained by shifting each of the indices in (3.15),
which for $n>0$ appears on the $n$th line below the top.

######  Example 3.2.

Suppose that $g=1$. Then the entries of the triplets $({\cal P},{\cal Q},{\cal
R})$ and sequences $({\cal P}_{n},{\cal Q}_{n},{\cal
R}_{n})_{n\in{\mathbb{Z}}}$ in $M_{g}=M_{1}$ take the form

$\begin{array}[]{rcl}{\cal P}(x)&=&1+p_{1}x\;,\\\ {\cal Q}(x)&=&2+q_{1}x\;,\\\
{\cal
R}(x)&=&r_{1}x+r_{2}x^{2}\;,\end{array}\qquad\hbox{and}\qquad\begin{array}[]{rcl}{\cal
P}_{n}(x)&=&1+p_{n,1}x\;,\\\ {\cal Q}_{n}(x)&=&2+q_{n,1}x\;,\\\ {\cal
R}_{n}(x)&=&r_{n,1}x+r_{n,2}x^{2}\;.\end{array}$ (3.16)

The birational automorphism (3.9), constructed in Lemma 3.1, and its iterates
are given by

$\begin{array}[]{rcl}\tilde{p}_{1}&=&q_{1}-p_{1}\;,\\\
\tilde{q}_{1}&=&-r_{2}/w\;,\\\ \tilde{r}_{1}&=&-2w\;,\\\
\tilde{r}_{2}&=&-wq_{1}\;,\end{array}\qquad\hbox{and}\qquad\begin{array}[]{rcl}p_{n+1,1}&=&q_{n,1}-p_{n,1}\;,\\\
q_{n+1,1}&=&-r_{n,2}/w_{n+1}\;,\\\ r_{n+1,1}&=&-2w_{n+1}\;,\\\
r_{n+1,2}&=&-w_{n+1}q_{n,1}\;,\end{array}$ (3.17)

where $w=-\frac{2\,p_{1}-q_{1}+r_{1}}{2}$ and
$w_{n+1}=-\frac{2\,p_{n,1}-q_{n,1}+r_{n,1}}{2}$ for $n\in{\mathbb{Z}}$.

######  Example 3.3.

Suppose now that $g=2$. The entries of the triplets $({\cal P},{\cal Q},{\cal
R})$ and sequences $({\cal P}_{n},{\cal Q}_{n},{\cal
R}_{n})_{n\in{\mathbb{Z}}}$ in $M_{2}$ now take the form

$\begin{array}[]{rcl}{\cal P}(x)&=&1+p_{1}x+p_{2}x^{2}\;,\\\ {\cal
Q}(x)&=&2+q_{1}x+q_{2}x^{2}\;,\\\ {\cal
R}(x)&=&r_{1}x+r_{2}x^{2}+r_{3}x^{3}\;,\end{array}\qquad\hbox{and}\qquad\begin{array}[]{rcl}{\cal
P}_{n}(x)&=&1+p_{n,1}x+p_{n,2}x^{2}\;,\\\ {\cal
Q}_{n}(x)&=&2+q_{n,1}x+q_{n,2}x^{2}\;,\\\ {\cal
R}_{n}(x)&=&r_{n,1}x+r_{n,2}x^{2}+r_{n,3}x^{3}\;.\end{array}$ (3.18)

From the construction in Lemma 3.1, the birational automorphism (3.9) and its
iterates are given by (3.17), with the expression for $\tilde{q}_{1}$ modified
to

$\tilde{q}_{1}=-(2p_{2}-q_{2}+r_{2})/w,\quad\mathrm{and}\quad
q_{n+1,1}=-(2p_{n,2}-q_{n,2}+r_{n,2})/w_{n+1},$

further supplemented with the following formulae:

$\begin{array}[]{rcl}\tilde{p}_{2}&=&q_{2}-p_{2}\;,\\\
\tilde{q}_{2}&=&-r_{3}/w\;,\\\
\tilde{r}_{3}&=&-wq_{2}\;,\end{array}\qquad\hbox{and}\qquad\begin{array}[]{rcl}p_{n+1,2}&=&q_{n,2}-p_{n,2}\;,\\\
q_{n+1,2}&=&-r_{n,3}/w_{n+1}\;,\\\ r_{n+1,3}&=&-w_{n+1}q_{n,2}\;,\end{array}$
(3.19)

where, as in the genus 1 case, $w=-\frac{2\,p_{1}-q_{1}+r_{1}}{2}$ and
$w_{n+1}=-\frac{2\,p_{n,1}-q_{n,1}+r_{n,1}}{2}$ for $n\in{\mathbb{Z}}$.

###  3.2 Lax equation and invariants

In Section 3.1 we defined a birational automorphism of the fiber
$\mu^{-1}(f)\subset M_{g}$, where $f=f(x)$ is any polynomial of degree at most
$2g+1$, satisfying $f(0)=1$. This map, given by (3.9), is not just defined on
$\mu^{-1}(f)$, but is also as it stands a well-defined birational automorphism
of $M_{g}$. In view of its relation to the Volterra lattice (see Section 5),
we call it the _Volterra map_ , denoted ${\cal V}_{g}$; explicitly,

${\cal V}_{g}:\qquad({\cal P},{\cal Q},{\cal R})\mapsto(\tilde{\cal
P},\tilde{\cal Q},\tilde{\cal R}),$

where the entries of the latter are given by (3.9); also, we can write ${\cal
V}_{g}({\cal P}_{n},{\cal Q}_{n},{\cal R}_{n})=({\cal P}_{n+1},{\cal
Q}_{n+1},{\cal R}_{n+1})$ for all $n\in{\mathbb{Z}}$. For a fixed initial
triplet $({\cal P}_{0},{\cal Q}_{0},{\cal R}_{0})$, the entire sequence of
triplets $({\cal P}_{n},{\cal Q}_{n},{\cal R}_{n})_{n\in{\mathbb{Z}}}$ in
$M_{g}$ is called the orbit of ${\cal V}_{g}$ through $({\cal P}_{0},{\cal
Q}_{0},{\cal R}_{0})$. We also refer to a sequence of triplets that satisfies
the recursion relations (3.13) for all $n$ as a solution. The equations (3.9)
for the Volterra map, as well as the recursion relations (3.13) for its
iterates, are easily rewritten as discrete Lax equations; this fact has many
important consequences which will be worked out in what follows.

######  Proposition 3.4.

The Volterra map ${\cal V}_{g}$ can be written in the compact form

${\cal V}_{g}:\qquad{\bf L}(x){\bf M}(x)={\bf M}(x)\widetilde{\bf L}(x)\;,$
(3.20)

where ${\bf L}(x)$ is given by (3.3) and ${\bf
M}(x):=\left(\begin{array}[]{cc}1&-wx\\\ 1&0\end{array}\right),$ with $w$
given by (3.10). As a consequence, the $2g+1$ polynomial functions
$H_{1},\dots,H_{2g+1}$ on $M_{g}$, defined by

${\cal P}(x)^{2}+{\cal Q}(x){\cal R}(x)=1+\sum_{i=1}^{2g+1}H_{i}x^{i}$ (3.21)

are invariants of the Volterra map, i.e. $\tilde{H}_{i}=H_{i}$ for
$i=1,\dots,2g+1.$

###### Proof.

It is easily checked by direct computation that (3.9) and (3.20) are the same
set of equations. Since the latter says that $\widetilde{\bf L}(x)$ is
obtained from ${\bf L}(x)$ by conjugation with ${\bf M}(x)$, the spectrum of
${\bf L}(x)$ is preserved, hence also all coefficients of the determinant of
${\bf L}(x)$, i.e., the coefficients $H_{i}$ of $\mu({\bf L}(x))={\cal
P}(x)^{2}+{\cal Q}(x){\cal R}(x)$. ∎

Upon iterating the Volterra map, as discussed in Section 3.1, starting from a
generic triplet ${\bf L}_{0}(x)$ of $M_{g}$ we get a sequence of triplets

${\bf L}_{n}(x):=\left(\begin{array}[]{cr}{\cal P}_{n}(x)&{\cal R}_{n}(x)\\\
{\cal Q}_{n}(x)&-{\cal P}_{n}(x)\end{array}\right)$

of $M_{g}$. According to (3.20), a discrete Lax equation for this sequence is
given by

${\bf L}_{n}(x){\bf M}_{n}(x)={\bf M}_{n}(x){\bf L}_{n+1}(x)\;,$ (3.22)

where

${\bf M}_{n}(x):=\left(\begin{array}[]{cc}1&-w_{n+1}x\\\
1&0\end{array}\right)\;,\qquad\hbox{with}\qquad
w_{n+1}=-\frac{2p_{n,1}-q_{n,1}+r_{n,1}}{2}=-\frac{r_{n+1,1}}{2}\;.$ (3.23)

######  Example 3.5.

When $g=1$, respectively when $g=2$, the invariants $H_{i}$ can be computed
from (3.21):

$\begin{array}[]{rcl}H_{1}&=&2(p_{1}+r_{1})\;,\\\
H_{2}&=&p_{1}^{2}+q_{1}r_{1}+2r_{2}\;,\\\
H_{3}&=&q_{1}r_{2}\;,\end{array}\qquad\qquad\begin{array}[]{rcl}H_{1}&=&2(p_{1}+r_{1})\;,\\\
H_{2}&=&2p_{2}+p_{1}^{2}+q_{1}r_{1}+2r_{2}\;,\\\
H_{3}&=&2p_{1}p_{2}+2r_{3}+q_{1}r_{2}+q_{2}r_{1}\;,\\\
H_{4}&=&p_{2}^{2}+q_{1}r_{3}+q_{2}r_{2}\;,\\\
H_{5}&=&q_{2}r_{3}\;.\end{array}$ (3.24)

The formulae on the left, which correspond to $g=1$, can be obtained from the
first three formulae on the right by setting $p_{2}=q_{2}=r_{3}=0$ in them.

###  3.3 The Volterra map and its reductions

The invariants $H_{i}$ can be used to reduce the Volterra map to the
submanifolds obtained by fixing the values of some of these invariants. Here
we will use this to express the Volterra map in terms of the variables $w_{i}$
which we introduced when constructing the S-fraction (3.15).

We start from the linear coordinates
$p_{1},\dots,p_{g},q_{1},\dots,q_{g},r_{1},\dots,r_{g+1}$ of $M_{g}$, which we
identify with
$p_{0,1},\dots,p_{0,g},q_{0,1},\dots,q_{0,g},r_{0,1},\dots,r_{0,g+1}$. The
latter functions are used to define recursively $p_{n,1},\dots,p_{n,g},$
$q_{n,1},\dots,q_{n,g},r_{n,1},\dots,r_{n,g+1}$, as well as $w_{n}$, for all
$n\in{\mathbb{Z}}$. Recall that this is done using (3.13) and (3.14).

In a first step, we will use a birational map to replace our linear
coordinates for $M_{g}$ by $p_{0,1},\dots,p_{0,g}$ and some of its iterates
$p_{n,1},\dots p_{n,g}$. To do this, we fix the value of the invariant
$H_{1}=2(p_{1}+r_{1})$ to an arbitrary constant $c_{1}$. It means that we
consider the hyperplane $H_{1}=c_{1}$ of $M_{g}$, which denote by
$M_{g}^{c_{1}}$. On it, we can take
$p_{1},\dots,p_{g},q_{1},\dots,q_{g},r_{2},\dots,r_{g+1}$ as linear
coordinates (we left out $r_{1}$). The invariance of $H_{1}$ implies that
$2(p_{n,1}+r_{n,1})=c_{1}$ which, combined with $r_{n,1}=-2w_{n}$ (see
(3.14)), leads to

$w_{n}=\frac{1}{2}\left(p_{n,1}-\frac{c_{1}}{2}\right)\;$ (3.25)

for all $n\in{\mathbb{Z}}$. Using this and the first and last equations in
(3.13), we can express the above variables in terms of $p_{0,1},\dots,p_{0,g}$
and their iterates:

$q_{n,k}=p_{n+1,k}+p_{n,k}\;,\quad\hbox{and}\quad
r_{n,k+1}=-w_{n}q_{n-1,k}=\frac{1}{2}\left(\frac{c_{1}}{2}-p_{n,1}\right)(p_{n,k}+p_{n-1,k})\;,$
(3.26)

where $k=1,\dots,g$. Taking $n=0$ we get

$q_{0,k}=p_{1,k}+p_{0,k}\;,\quad\hbox{and}\quad
r_{0,k+1}=\frac{1}{2}\left(\frac{c_{1}}{2}-p_{0,1}\right)(p_{0,k}+p_{-1,k})\;,$
(3.27)

and we have expressed all coordinates of $M_{g}^{c_{1}}$ in terms of the $3g$
coefficients of the polynomials ${\cal P}_{-1},{\cal P}_{0}$ and ${\cal
P}_{1}$. It is clear that the (3.27) can be solved rationally for $p_{1,k}$
and $p_{-1,k}$ so that (3.27) defines a birational morphism from
$M_{g}^{c_{1}}$ to the space of triplets $({\cal P}_{-1},{\cal P}_{0},{\cal
P}_{1})$. For later use, we also express the iterates of the Volterra map as a
recursion relation on the polynomials ${\cal P}_{n}$. To do this, we apply
(3.13) several times to get

$\displaystyle{\cal P}_{n+2}$ $\displaystyle={\cal Q}_{n+1}(x)-{\cal
P}_{n+1}(x)=\frac{2{\cal P}_{n}(x)-{\cal Q}_{n}(x)+{\cal
R}_{n}(x)}{-w_{n+1}x}-{\cal P}_{n+1}(x)$ $\displaystyle=-{\cal
P}_{n+1}(x)+\frac{1}{w_{n+1}x}({\cal P}_{n+1}(x)-{\cal
P}_{n}(x))+\frac{w_{n}}{w_{n+1}}({\cal P}_{n}(x)+{\cal P}_{n-1}(x))\;.$ (3.28)

We will now go one step further and show how the above coordinates
$p_{0,k},p_{1,k}$ and $p_{-1,k}$ can be expressed birationally in terms of
$p_{0,1}$ and some of its iterates $p_{n,1}$. To do this we do a further
reduction, namely we also fix the value of each one of the invariants
$H_{2},\dots,H_{g}$ to an arbitrary constant $c_{2},\dots,c_{g}$ and consider
the (possibly singular) subvariety $\cap_{i=1}^{g}(H_{i}=c_{i})$ of $M_{g}$,
which we denote by $M_{g}^{c}$, so $c$ stands now for $(c_{1},\dots,c_{g})$.
Using (3.26), we get the following formula for the invariants $H_{i}$ in terms
of ${\cal P}_{n},{\cal P}_{n+1}$ and ${\cal P}_{n-1}$, valid for any
$n\in{\mathbb{Z}}$:

$1+\sum_{k=1}^{2g+1}H_{i}x^{i}={\cal P}_{n}^{2}(x)+{\cal Q}_{n}(x){\cal
R}_{n}(x)={\cal P}_{n}^{2}(x)-w_{n}x({\cal P}_{n}(x)+{\cal P}_{n+1}(x))({\cal
P}_{n}(x)+{\cal P}_{n-1}(x))\;,$

with $w_{n}$ given by (3.25). Upon comparing the coefficient of $x^{k}$ on
both sides, for $k=1$ we recover (3.25), while for $k=2,\dots,g$ we
recursively obtain $p_{n,k}$ in terms of $p_{0,1}$ and its iterates, via the
following formulae:

$\displaystyle c_{k}$
$\displaystyle=2p_{n,k}+\sum_{i=1}^{k-1}p_{n,i}p_{n,k-i}-2w_{n}(2p_{n,k-1}+p_{n-1,k-1}+p_{n+1,k-1})$
$\displaystyle\;\quad\quad-
w_{n}\sum_{i=1}^{k-2}(p_{n,i}+p_{n+1,i})(p_{n,k-1-i}+p_{n-1,k-1-i})\;.$ (3.29)

Indeed, aside from the linear term in $p_{n,k}$, the above equation contains
only the variables $p_{n,i}$ and $p_{n\pm 1,i}$ with $1\leqslant i<k$. For
$k=2$ one gets

$c_{2}=2p_{n,2}+p_{n,1}^{2}-2w_{n}(2p_{n,1}+p_{n-1,1}+p_{n+1,1})\;,$

from which it is clear that $p_{n,2}$ depends (polynomially) only on
$p_{n-1,1},p_{n,1}$ and $p_{n+1,1}$ (see (3.25) for the formula for $w_{n}$).
An easy recursion on $k$ using (3.3) shows that $p_{n,k}$ depends on
$p_{n-k+1,1},\dots,p_{n+k-1,1}$ only. Taking $n=-1$, $n=0$ and $n=-1$ we get
that the coefficients of ${\cal P}_{-1},{\cal P}_{0}$ and ${\cal P}_{1}$
depend only on $p_{-g,1},\dots,p_{g,1}$. Conversely, it is obvious from (3.3)
that the coefficients of ${\cal P}_{2}$, and hence of all ${\cal P}_{n}$ with
$n\geqslant 2$, are rational functions of the coefficients of ${\cal
P}_{-1},{\cal P}_{0}$ and ${\cal P}_{1}$. This applies in particular to
$p_{n,1}$ with $n\geqslant 2$. Using the inverse recursion, which yields a
formula similar to (3.3) expressing ${\cal P}_{n-2}$ in terms of ${\cal
P}_{n+1},{\cal P}_{n}$ and ${\cal P}_{n-1}$ one obtains similarly that all
$p_{n,1}$ with $n\in{\mathbb{Z}}$ are rational functions of the coefficients
of ${\cal P}_{-1},{\cal P}_{0}$ and ${\cal P}_{1}$. The upshot is that we have
a birational map between $M_{g}^{c}$ and ${\mathbb{C}}^{2g+1}$, equipped with
the coordinates $p_{-g,1},\dots,p_{g,1}$. In view of (3.25), which we now
write as

$p_{n,1}=2w_{n}+\frac{c_{1}}{2}\;,$ (3.30)

it amounts to a birational map between $M_{g}^{c}$ and
${\mathbb{C}}^{2g+1}_{w}$ where the latter denotes ${\mathbb{C}}^{2g+1}$,
equipped with the coordinates $w_{-g},\dots,w_{g}$.

Hence we can use the birational map between $M_{g}^{c}$ and
${\mathbb{C}}^{2g+1}_{w}$ to write the Volterra map on $M_{g}$, restricted to
$M_{g}^{c}$, as a birational automorphism of ${\mathbb{C}}^{2g+1}_{w}$. Since
we already gave in (3.3) the Volterra map and its iterates in terms of the
variables ${\cal P}_{i}$, we set $n=0$ therein, which gives the Volterra map
itself, and take the leading terms of both sides of (3.3):

$w_{1}(p_{2,g}+p_{1,g})=w_{0}(p_{0,g}+p_{-1,g})\;.$ (3.31)

In view of the dependence of $p_{-1,g},\dots,p_{2,g}$ on the variables
$w_{i}$, (3.31) gives an equation for $w_{g+1}$, which appears linearly in it,
and the birational automorphism
$(w_{-g},w_{1-g},\dots,w_{g})\mapsto(w_{1-g},\dots,w_{g},w_{g+1})$ is the
Volterra map on ${\mathbb{C}}_{w}^{2g+1}$. Explicit expressions for it will be
given in the examples below.

However, we can do a further reduction, restricting the Volterra map to the
subvariety $H_{k}=c_{k}$ for some $k$ with $g<k\leqslant 2g+1$. The relation

$H_{k}(w_{-g},\dots,w_{0},\dots,w_{g})=c_{k}$

defines $w_{g}$ as a rational function of $w_{-g},\dots,w_{g-1}$ because by
inspection $w_{g}$ appears linearly in it (the same applies to $w_{-g}$). As
we will see in the examples below, we can therefore take any of the invariants
$H_{k}$, with $g<k\leqslant 2g+1$ which will give a birational automorphism
which is an incarnation of the Volterra map ${\cal V}_{g}$ on
$M_{g}^{c}\cap(H_{k}=c_{k})$; precisely it is conjugate, via the above
birational map, to the Volterra map, restricted to
$M_{g}^{c}\cap(H_{k}=c_{k})$, where values of $c=(c_{1},\dots,c_{g})$ and
$c_{k}$ are arbitrary.

######  Example 3.6.

We first consider the case $g=1$. In this case, we only need to consider $k=1$
in (3.27), which combined with (3.30) yields the following birational map
between $M_{g}^{c}=M_{g}^{c_{1}}$ and ${\mathbb{C}}^{3}_{w}$:

$\displaystyle p_{1}$ $\displaystyle=p_{0,1}=2w_{0}+c_{1}/2\;,$ $\displaystyle
q_{1}$ $\displaystyle=q_{0,1}=p_{1,1}+p_{0,1}=2(w_{0}+w_{1})+c_{1}\;,$
$\displaystyle r_{1}$ $\displaystyle=r_{0,1}=c_{1}/2-p_{0,1}\;,$ (3.32)
$\displaystyle r_{2}$
$\displaystyle=r_{0,2}=-w_{0}(p_{0,1}+p_{-1,1})=-2w_{0}(w_{0}+w_{-1}+c_{1}/2)\;.$

In terms of the polynomials ${\cal P},{\cal Q}$ and ${\cal R}$ this can also
be written as

$\displaystyle{\cal P}(x)$ $\displaystyle=1+(2w_{0}+{c_{1}}/2)x\;,$
$\displaystyle{\cal Q}(x)$ $\displaystyle=2+2(w_{1}+w_{0}+{c_{1}}/2)x\;,$
(3.33) $\displaystyle{\cal R}(x)$
$\displaystyle=-2w_{0}x(1+(w_{0}+w_{-1}+{c_{1}}/2)x)\;.$

For $g=1$ the formula (3.31) takes the form
$w_{1}(p_{2,1}+p_{1,1})=w_{0}(p_{0,1}+p_{-1,1})$, and can be expressed
immediately in terms of the quantities $w_{-1},\dots,w_{2}$ since
$p_{n,1}=2w_{n}+c_{1}/2$, for all $n$:

$w_{1}(2w_{2}+2w_{1}+c_{1})=w_{0}(2w_{0}+2w_{-1}+c_{1})\;.$ (3.34)

It defines the Volterra map $(w_{-1},w_{0},w_{1})\mapsto(w_{0},w_{1},w_{2})$
on ${\mathbb{C}}^{3}_{w}$, being the same as equation (2) in [51], where
Svinin used continued fraction expansions to construct particular solutions;
equation (3.34) also appears in [24]. Substituting (3.6) in ${\cal
P}^{2}(x)+{\cal Q}(x){\cal R}(x)=1+H_{1}x+H_{2}x^{2}+H_{3}x^{3}$ we get the
following formulas for the invariants $H_{2}$ and $H_{3}$ of the Volterra map
in terms of the variables $w_{-1},w_{0}$ and $w_{1}$:

$\displaystyle H_{2}$
$\displaystyle=-4w_{0}(w_{1}+w_{0}+w_{-1}+{c_{1}}/2)+{c_{1}^{2}}/4\;,$ (3.35)
$\displaystyle H_{3}$
$\displaystyle=-4w_{0}(w_{1}+w_{0}+{c_{1}}/2)(w_{0}+w_{-1}+{c_{1}}/2)\;.$
(3.36)

We now fix $c_{3}$ and consider the Volterra map on the subvariety
$H_{3}=c_{3}$ of $M_{g}^{c}$. According to (3.36) we get

$4w_{0}(w_{1}+w_{0}+{c_{1}}/2)(w_{0}+w_{-1}+{c_{1}}/2)+c_{3}=0\;,$ (3.37)

which defines a $2D$ map $(w_{-1},w_{0})\mapsto(w_{0},w_{1})$, where $w_{1}$
is computed from (3.37). It has $H_{2}$ (from which $w_{1}$ is eliminated
using (3.37)) as invariant, given by

$H_{2}=\frac{c_{3}}{w_{0}+w_{-1}+c_{1}/2}-4w_{0}w_{-1}+\frac{c_{1}^{2}}{4}\;.$
(3.38)

We next fix $c_{2}$ and consider the Volterra map on the subvariety
$H_{2}=c_{2}$ of $M_{1}^{c}$. From (3.35) we now get

$4w_{0}(w_{1}+w_{0}+w_{-1}+{c_{1}}/2)-{c_{1}^{2}}/4+c_{2}=0\;.$ (3.39)

It defines a $2D$ map $(w_{-1},w_{0})\mapsto(w_{0},w_{1})$ with $H_{3}$ as
invariant, which (after using (3.39)) to eliminate $w_{1}$) takes the form

$H_{3}=\left(w_{0}+w_{-1}+\frac{c_{1}}{2}\right)\left(4w_{0}w_{-1}+c_{2}-\frac{c_{1}^{2}}{4}\right)\;.$
(3.40)

To finish the $g=1$ example we will present a slightly more involved
reduction, leading to a map which is closely related to Somos-5. To do this,
we first compare two different ways of writing of the genus 1 curve
$y^{2}=f(x)$, which lead to an alternative generating set of invariants of the
recursion. If we write

$y^{2}=1+c_{1}x+c_{2}x^{2}+c_{3}x^{3}=(1-c_{1}^{\prime}x)(1-c_{1}^{\prime}x+4c_{2}^{\prime}x^{2})-4c_{3}^{\prime}x^{3}\;,$
(3.41)

then the constants $c_{i}$ and $c_{i}^{\prime}$ are related by

$\begin{array}[]{rcl}c_{1}&=&-2c^{\prime}_{1}\;,\\\
c_{2}&=&4c_{2}^{\prime}+{c_{1}^{\prime}}^{2}\;,\\\
c_{3}&=&-4(c^{\prime}_{1}c^{\prime}_{2}+c^{\prime}_{3})\;,\end{array}\qquad\qquad\begin{array}[]{rcl}c^{\prime}_{1}&=&-c_{1}/2\;,\\\
c^{\prime}_{2}&=&\frac{1}{4}\left(c_{2}-\frac{c_{1}^{2}}{4}\right)\;,\\\
c^{\prime}_{3}&=&\frac{c_{1}}{8}\left(c_{2}-\frac{c_{1}^{2}}{4}\right)-\frac{c_{3}}{4}\;.\end{array}$
(3.42)

Next, if we write the recursion relations (3.37) and (3.39) in terms of the
constants $c_{i}^{\prime}$ using (3.42), we get respectively

$\displaystyle
w_{0}\left(w_{1}+w_{0}-{c^{\prime}_{1}}\right)\left(w_{0}+w_{-1}-{c^{\prime}_{1}}\right)=c^{\prime}_{1}c^{\prime}_{2}+c^{\prime}_{3}\;,$
(3.43) $\displaystyle
w_{0}\left(w_{1}+w_{0}+w_{-1}-{c^{\prime}_{1}}\right)+c^{\prime}_{2}=0\;.$
(3.44)

Notice that $c_{1}^{\prime}$ now appears linearly in (3.44), so we can easily
eliminate $c^{\prime}_{1}$ between (3.43) and (3.44), which yields the
following simple relation

$w_{1}w_{-1}=c_{2}^{\prime}+\frac{c_{3}^{\prime}}{w_{0}}\;$ (3.45)

on the generic level surface
$(H_{2}^{\prime}=c_{2}^{\prime})\cap((H_{3}^{\prime}=c_{3}^{\prime})$, which
is also birational with ${\mathbb{C}}^{2}$. It defines a $2D$ map
$(w_{-1},w_{0})\mapsto(w_{0},w_{1})$ which is a (multiplicative) QRT map with
$c_{2}^{\prime}$ and $c_{3}^{\prime}$ as parameters. To get an invariant for
this map, we eliminate $w_{1}$ between (3.43) and (3.44), to get

$c_{1}^{\prime}w_{0}w_{-1}=(w_{0}+w_{-1})w_{0}w_{-1}+c_{2}^{\prime}(w_{0}+w_{-1})+c_{3}^{\prime}.$
(3.46)

It leads upon division by $w_{0}w_{-1}$ to the following explicit formula for
the invariant:

$H_{1}^{\prime}=w_{0}+w_{-1}+c_{2}^{\prime}\left(\frac{1}{w_{0}}+\frac{1}{w_{-1}}\right)+\frac{c_{3}^{\prime}}{w_{0}w_{-1}}\;.$
(3.47)

Also, the tau function substitution

$w_{n}=\frac{\tau_{n}\tau_{n+3}}{\tau_{n+1}\tau_{n+2}}$

in (3.45), which we now write in the form of the recursion relation
$w_{n+1}w_{n-1}=c_{2}^{\prime}+{c_{3}^{\prime}}/{w_{n}}$, yields the general
form of the Somos-5 recursion relation, namely

$\tau_{n+5}\tau_{n}=c_{2}^{\prime}\tau_{n+4}\tau_{n+1}+c_{3}^{\prime}\tau_{n+3}\tau_{n+2}\;,\qquad
n\in{\mathbb{Z}}\;.$ (3.48)

In our previous work [24] we showed how to solve the initial value problem for
(3.48) explicitly in terms of the Weierstrass sigma function, but in Section
3.4 below we will show how it can also be solved in terms of Hankel
determinants, using the S-fraction (3.15).

Note that, apart from (3.45), the maps (3.37) and (3.39) are also examples of
symmetric QRT maps [44], and the orbits of all three maps can be identified by
restricting to particular level sets of their invariants, which is a common
feature of families of these maps [29]. To see how the orbits of these
different 2D maps coincide, it is necessary to identify the parameters and
values of the invariants (3.38), (3.40) and (3.47) in an appropriate way, from
which it can be seen that $H_{2}$, $H_{3}$ and $H_{1}^{\prime}$ define an
identical biquadratic curve in the $(w_{-1},w_{0})$ plane, namely (3.46),
whose coefficients can be rewritten in terms of $c_{1},c_{2},c_{3}$ using
(3.42). Moreover, this curve is birationally equivalent to the Weierstrass
cubic (3.41).

######  Example 3.7.

Using (3.31) when $g=2$, on $M_{2}^{c}$ (which is birational to
${\mathbb{C}}^{5}$) the formula for the Volterra map takes the form

$w_{1}(p_{2,2}+p_{1,2})=w_{0}(p_{0,2}+p_{-1,2})\;,$ (3.49)

so we need to express $p_{n,2}$ for $n=-1,\dots,2$ in terms of the variables
$w_{i}$. To do this, we use (3.3), keeping in mind that
$p_{n,1}=2w_{n}+c_{1}/2$ for all $n$:

$\displaystyle 2p_{n,2}$
$\displaystyle=-p_{n,1}^{2}+2w_{n}(p_{n+1,1}+2p_{n,1}+p_{n-1,1})+c_{2}$
$\displaystyle=-\left(2w_{n}+\frac{c_{1}}{2}\right)^{2}+4w_{n}(w_{n+1}+2w_{n}+w_{n-1}+c_{1})+c_{2}$
$\displaystyle=4w_{n}\left(w_{n+1}+w_{n}+w_{n-1}+\frac{c_{1}}{2}\right)+c_{2}-\frac{c_{1}^{2}}{4}\;.$
(3.50)

Substituted in (3.49) and slightly reordering the terms, we get the following
symmetric relation:

$\displaystyle\quad
w_{1}\left(2w_{2}(w_{3}+w_{2})+2w_{1}(w_{1}+w_{0})+4w_{2}w_{1}+(w_{2}+w_{1})c_{1}+c_{2}-\frac{c_{1}^{2}}{4}\right)$
$\displaystyle=\ $ $\displaystyle
w_{0}\left(2w_{0}(w_{1}+w_{0})+2w_{-1}(w_{-1}+w_{-2})+4w_{0}w_{-1}+(w_{0}+w_{-1})c_{1}+c_{2}-\frac{c_{1}^{2}}{4}\right)\;.$
(3.51)

This defines a $5D$ map
$(w_{-2},w_{-1},w_{0},w_{1},w_{2})\mapsto(w_{-1},w_{0},w_{1},w_{2},w_{3})$,
where $w_{3}$ is computed from (3.7). Using the first equation in (3.9), the
above formulae for $p_{n,1}$ and $p_{n,2}$ lead at once to the following
expressions for the coefficients of ${\cal Q}_{n}$:

$\displaystyle q_{n,1}$ $\displaystyle=2w_{n+1}+2w_{n}+c_{1}\;,$
$\displaystyle q_{n,2}$
$\displaystyle=2(w_{n}w_{n-1}+w_{n+1}w_{n+2})+2(w_{n+1}+w_{n})^{2}+(w_{n+1}+w_{n})c_{1}+c_{2}-\frac{c_{1}^{2}}{4}\;.$
(3.52)

The formulae for $r_{n,k}$ then follow from $r_{n,1}=-2w_{n}$ and
$r_{n,k}=-w_{n}q_{n-1,k-1}$ for $k>1$, by applying the third equation in
(3.9). With these formulae we can express the invariants $H_{3},\dots,H_{5}$
in terms of the variables $w_{i}$. We write this out for $H_{3}$, in order to
find a recursion relation of order 4. According to (3.24), and using the above
expressions for $r_{n,k}$,

$\displaystyle H_{3}$
$\displaystyle=2p_{1}p_{2}+2r_{3}+q_{1}r_{2}+q_{2}r_{1}=p_{0,1}p_{0,2}+2r_{0,3}+q_{0,1}r_{0,2}+q_{0,2}r_{0,1}$
$\displaystyle=2p_{0,1}p_{0,2}-w_{0}(q_{0,1}q_{-1,1}+2q_{-1,2}+2q_{0,2})\;,$

which can be written completely in terms of the variables $w_{i}$ using (3.7),
(3.7) and the fundamental formula $p_{n,1}=2w_{n}+c_{1}/2$. After some
simplification, we get on the hypersurface $H_{3}=c_{3}$, which is birational
to ${\mathbb{C}}^{4}$,

$\displaystyle
w_{2}w_{1}w_{0}+w_{0}w_{-1}w_{-2}+2w_{0}^{2}(w_{1}+w_{-1})+w_{0}^{3}+w_{0}(w_{1}^{2}+w_{1}w_{-1}+w_{-1}^{2})$
$\displaystyle\quad+\frac{c_{1}}{2}w_{0}(w_{1}+w_{0}+w_{-1})+\frac{1}{2}\left(c_{2}-\frac{c_{1}^{2}}{4}\right)w_{0}+\frac{c_{3}}{4}-\frac{c_{1}c_{2}}{8}+\frac{c_{1}^{3}}{32}=0\;.$

This is exactly the equation (1.2) defining the $4D$ map (P.iv), after setting
$n=-2$ and

$\nu=\frac{c_{1}}{2}\;,\quad
a=\frac{c_{3}}{4}-\frac{c_{1}c_{2}}{8}+\frac{c_{1}^{3}}{32}\;,\quad
b=\frac{1}{2}\left(c_{2}-\frac{c_{1}^{2}}{4}\right)\;.$ (3.53)

The invariants $H_{4}$ and $H_{5}$ yield the invariants for (P.iv), given in
(1.3) and (1.4).

###  3.4 Hankel determinant solutions

The function $F_{0}$ in (3.15) that defines the S-fraction admits a series
expansion in $x$ around $(0,1)\in\Gamma_{f}$ that we shall use to give
explicit solutions to the recurrence relation defined by the Volterra map. For
a generic point $({\cal P},{\cal Q},{\cal R})$ in $\mu^{-1}(f)$, with $f$ as
in (3.5), we introduce new variables $s_{1},s_{2},\dots$ by writing

$1-\cfrac{w_{1}x}{1-\cfrac{w_{2}x}{1-\cfrac{w_{3}x}{1-\cdots}}}=1-\sum_{j=1}^{\infty}s_{j}x^{j}=1-S(x)\;,$
(3.54)

where the latter equality defines the power series $S(x)$, which can be
regarded as a generating function for the moments $s_{j}$. The moments can be
defined from the integral

$s_{j}=\frac{1}{2\pi\mathrm{i}}\oint\frac{(1-F_{0})}{x^{j+1}}\,\mathrm{d}x,$

for a sufficiently small contour around the point $(0,1)$ on $\Gamma_{f}$, and
this leads to a linear functional (defined on polynomials in $x^{-1}$), and
the connection with the classical theory of orthogonal polynomials [46], but
we shall not pursue this further here. In view of (3.15), $F_{0}=1-S(x)$ in
the sense that the latter is a Taylor series expansion of the rational
function $F_{0}$ at $(0,1)$. It was shown by Stieltjes [48] that the variables
$w_{i}$ can be expressed as Hankel determinants of the variables $s_{i}$.
Precisely, he showed that

$w_{n}=\frac{\Delta_{n-3}\Delta_{n}}{\Delta_{n-2}\Delta_{n-1}}\qquad\mathrm{for}\,\,n\geqslant
1\;,$ (3.55)

where $\Delta_{2k-1}=\det(s_{i+j-1})_{i,j=1,\ldots,k}$ and
$\Delta_{2k}=\det(s_{i+j})_{i,j=1,\ldots,k}$, for $k\geqslant 1$, that is,

$\Delta_{2k-1}=\left|\begin{array}[]{cccc}s_{1}&s_{2}&\cdots&s_{k}\\\
s_{2}&&\iddots&\vdots\\\ \vdots&\iddots&&\vdots\\\
s_{k}&\cdots&\cdots&s_{2k-1}\end{array}\right|\quad\hbox{and}\quad\Delta_{2k}=\left|\begin{array}[]{cccc}s_{2}&s_{3}&\cdots&s_{k+1}\\\
s_{3}&&\iddots&\vdots\\\ \vdots&\iddots&&\vdots\\\
s_{k+1}&\cdots&\cdots&s_{2k}\end{array}\right|\;.$ (3.56)

Also, by definition, $\Delta_{-2}=\Delta_{-1}=\Delta_{0}=1$. For example,
$w_{1}=s_{1}$, $w_{2}=s_{2}/s_{1}$,
$w_{3}=(s_{1}s_{3}-s_{2}^{2})/(s_{1}s_{2})$, and so on. It is clear from
(3.54) that, conversely, $s_{i}$ can be expressed as a polynomial in
$w_{1},\dots,w_{i}$, for example $s_{1}=w_{1}$, $s_{2}=w_{1}w_{2}$,
$s_{3}=w_{1}w_{2}(w_{2}+w_{3})$, and so on. (While these expressions can be
found by expanding geometric series, a systematic method using continuants is
presented in Section 3.5.)

######  Theorem 3.8.

The terms $w_{n}$, $n\geqslant 1$, of the recurrence sequence defined by the
Volterra map on $\mu^{-1}(f)$ can be written as

$w_{n}=\frac{\Delta_{n-3}\Delta_{n}}{\Delta_{n-2}\Delta_{n-1}}\;,$

where the entries of the Hankel matrices satisfy the recursion relation

$s_{j}=\sum_{k=1}^{g}\big{(}p_{k}-q_{k}\big{)}s_{j-k}+\sum_{i=1}^{j-1}s_{i}s_{j-i}+\frac{1}{2}\sum_{k=1}^{g}q_{k}\sum_{i=1}^{j-k-1}s_{i}s_{j-k-i},\quad
j\geqslant g+2\;.$ (3.57)

The initial values $s_{1},s_{2},\ldots,s_{g+1}$ and the coefficients for the
recursion (3.57) are provided by a generic triple $({\cal P},{\cal Q},{\cal
R})\in\mu^{-1}(f)$.

###### Proof.

In order to prove the recursion formula (3.57), we will derive a quadratic
formula for $S(x)=1-F_{0}$, introduced in (3.54). We use (3.15) to write

$y=-{\cal P}(x)+F_{0}{\cal Q}(x)=-{\cal P}(x)+(1-S(x)){\cal Q}(x)\;,$ (3.58)

which we substitute in

$y^{2}=f(x)={\cal P}(x)^{2}+{\cal Q}(x){\cal R}(x)$ (3.59)

to get the following quadratic equation for $S(x)$:

$\big{(}{\cal Q}(x)-{\cal P}(x)\big{)}\,S(x)-\frac{1}{2}{\cal
Q}(x)\,S^{2}(x)+{\cal P}(x)-\frac{1}{2}{\cal Q}(x)+\frac{1}{2}{\cal
R}(x)=0\;.$ (3.60)

Substituting the power series for $S(x)$ into the quadratic, as well as the
polynomials ${\cal P},{\cal Q},{\cal R}$, the coefficients of $x^{j}$ for
$1\leqslant j\leqslant g+1$ allow the $g+1$ initial values
$s_{1},\ldots,s_{g+1}$ to be determined from these polynomials. Then, upon
taking the coefficient of $x^{j}$ for $j\geqslant g+2$, the recursion relation
(3.57) is obtained directly. Observe that the number of initial values plus
independent coefficients appearing linearly in the recursion is
$g+1+2g=3g+1=\,$dim$\,M_{g}$. ∎

As we will see in the examples, it is often more practical not to fix the
curve $y^{2}=f(x)$, i.e. not to fix the values of all invariants, but only fix
some of them and take $w_{0},\dots,w_{g}$ as extra initial conditions.

######  Example 3.9.

We specialize the above results to $g=1$, fixing arbitrary constants $c_{1}$
and $c_{2}$ and taking the Volterra map on the surface
$H_{1}=c_{1},\;H_{2}=c_{2}$, as in one of the reductions considered in Example
3.6. For the recursion (3.57) we can take $p_{1},q_{1}$ as initial conditions,
since given $p_{1}$ and $q_{1}$, specifying the values of $H_{1}$ and $H_{2}$
is equivalent to specifying the values of $r_{1}$ and $r_{2}$ (see the
explicit formulas for $H_{1}$ and $H_{2}$ in the left column of (3.24)). It
follows from (3.6) that

$p_{1}-q_{1}=-2w_{1}-\frac{c_{1}}{2}\;,\quad\hbox{and}\quad
q_{1}=2\left(w_{1}+w_{0}+\frac{c_{1}}{2}\right)=-2w_{2}+\frac{1}{2w_{1}}\left(\frac{c_{1}^{2}}{4}-c_{2}\right)\;,$

where we obtained the last equality by using the recursion relation (3.39),
shifted by 1, to replace $w_{0}$ by $w_{2}$. Substituted in (3.57), we get the
following recursive formula for $s_{j}$ ($j\geqslant 3$) in terms of
$w_{1},w_{2}$, which we can take as initial conditions, instead of $p_{1}$ and
$q_{1}$:

$s_{j}=\left(-2w_{1}-\frac{c_{1}}{2}\right)s_{j-1}+\sum_{i=1}^{j-1}s_{i}s_{j-i}+\left(\frac{1}{4w_{1}}\left(\frac{c_{1}^{2}}{4}-c_{2}\right)-w_{2}\right)\sum_{i=1}^{j-2}s_{i}s_{j-1-i}\;.$
(3.61)

Then specifying the two initial values $w_{1},w_{2}$ fixes the initial
conditions $s_{1},s_{2}$ for the above, as $s_{1}=w_{1}$ and
$s_{2}=w_{1}w_{2}$. Notice that $s_{j}$ is a polynomial of degree $j$ in
$w_{1},w_{2}$, with $w_{1}|s_{j}$ for all $j$.

As a concrete example, consider the curve $y^{2}=1-10x+29x^{2}-24x^{3}$, with
initial conditions $s_{1}=1$, $s_{2}=2$ (or, equivalently, $w_{1}=1$,
$w_{2}=2$). Since $c_{1}=-10$ and $c_{2}=29$, the recursion (3.61) becomes

$s_{j}=3s_{j-1}+\sum_{i=1}^{j-1}s_{i}s_{j-i}-3\sum_{i=1}^{j-2}s_{i}s_{j-i-1},\quad
j\geqslant 3,$

which generates the sequence

$(s_{j})_{j\geqslant 1}:\quad 1,2,7,27,109,456,1969,8746,39825,185266,\ldots,$

producing $\Delta_{1}=1$, $\Delta_{2}=2$,

$\Delta_{3}=\left|\begin{array}[]{cc}1&2\\\
2&7\end{array}\right|=3,\,\Delta_{4}=\left|\begin{array}[]{cc}2&7\\\
7&27\end{array}\right|=5,\,\Delta_{5}=\left|\begin{array}[]{ccc}1&2&7\\\
2&7&27\\\
7&27&109\end{array}\right|=11,\,\Delta_{6}=\left|\begin{array}[]{ccc}2&7&27\\\
7&27&109\\\ 27&109&456\end{array}\right|=37,\ldots,$

which extends symmetrically to all $n\in{\mathbb{Z}}$ to produce the original
Somos-5 sequence [39],

$\ldots,3,2,1,1,1,1,1,2,3,5,11,37,83,274,1217,6161,22833,\ldots,$

generated by the bilinear recurrence

$\Delta_{n+5}\Delta_{n}=\Delta_{n+4}\Delta_{n+1}+\Delta_{n+3}\Delta_{n+2}.$
(3.62)

It is a particular case of (3.48) with $c_{2}^{\prime}=c_{3}^{\prime}=1$,
where the latter are computed from $c_{1}=-10,$ $c_{2}=29$, $c_{3}=-24$, using
(3.42).

######  Remark 3.10.

Note that Hankel determinant formulae for Somos-5 were previously obtained in
the work of Chang, Hu and Xin, using a bilinear Bäcklund transformation for
Somos-4. We will return to the connection with Somos-4 in Section 5, but for
now we just point out that the Hankel determinant expressions found in [8] are
more complicated than the above, because two different moment sequences are
required for the terms with even/odd indices. In particular, for the original
Somos-5 sequence, there are two sequences of moments, namely
$(\bar{s}_{j})_{j\geqslant
0}:\,1,-1,4,-8,25,-65,197,-571,1753,-5351,16746,-52626,\ldots$, and
$(\hat{s}_{j})_{j\geqslant
0}:\,2,-1,3,-1,12,2,61,39,352,374,2210,3162,\ldots$, which are defined by

$\bar{s}_{0}=1,\,\bar{s}_{1}=-1,\,\bar{s}_{j+1}=-\bar{s}_{j}+2\bar{s}_{j-1}+\sum_{i=0}^{j-1}\bar{s}_{i}\bar{s}_{j-i-1},\quad\mathrm{and}\quad\hat{s}_{0}=2,\,\hat{s}_{1}=-1,\,\hat{s}_{j+1}=\hat{s}_{j}+\sum_{i=0}^{j-1}\hat{s}_{i}\hat{s}_{j-i-1},$

respectively, where the recursions hold for $j\geq 1$, and (with the indexing
convention of Theorem 1.2 in [8]) the terms of the Somos-5 sequence [39] are
then given by
$\mathrm{S}_{0}=1,\mathrm{S}_{1}=1,\mathrm{S}_{2}=\bar{s}_{0}=1,\mathrm{S}_{3}=\hat{s}_{0}=2$,
and

$\mathrm{S}_{4}=\left|\begin{array}[]{cc}1&-1\\\
-1&4\end{array}\right|=3,\,\mathrm{S}_{5}=\left|\begin{array}[]{cc}2&-1\\\
-1&3\end{array}\right|=5,\,\mathrm{S}_{6}=\left|\begin{array}[]{ccc}1&-1&4\\\
-1&4&-8\\\
4&-8&25\end{array}\right|=11,\,\mathrm{S}_{7}=\left|\begin{array}[]{ccc}2&-1&3\\\
-1&3&-1\\\ 3&-1&12\end{array}\right|=37,\ldots,$

which are not related to the determinants in Example 3.9 in a straightforward
manner.

######  Example 3.11.

We now specialize the above results to $g=2$, thereby continuing Example 3.7.
From (3.53) it is clear that fixing the values $c_{1},c_{2},c_{3}$ is
equivalent to specifying the parameters $a,b$ and $\nu$, which we fix, since
these are the coefficients in the recurrence relation (P.iv), and we can take
$w_{0},w_{1},w_{2},w_{3}$ (or $w_{1},w_{2},w_{3},w_{4}$) as initial data for
the latter. We now have

$s_{1}=w_{1},\quad s_{2}=w_{1}w_{2},\quad s_{3}=w_{1}w_{2}(w_{2}+w_{3}),$

providing the 3 initial values for the recursion (3.57), which takes the form

$s_{j}=\hat{\alpha}\,s_{j-1}+\hat{\beta}\,s_{j-2}+\sum_{i=1}^{j-1}s_{i}s_{j-i}+\hat{\gamma}\,\sum_{i=1}^{j-2}s_{i}s_{j-i-1}+\hat{\delta}\,\sum_{i=1}^{j-3}s_{i}s_{j-i-2}\;,\quad
j\geqslant 4\;.$ (3.63)

While $s_{1},s_{2},s_{3}$ are determined by $w_{1},w_{2},w_{3}$ only, $w_{0}$
and $a,b,\nu$ are required to find the coefficients
$\hat{\alpha},\dots,\hat{\delta}$, which are computed using $p_{1}=2w_{0}+\nu$
(recall that $\nu=c_{1}/2$) and (3.7), (3.7) for $n=1$, to give

$\displaystyle\hat{\alpha}$ $\displaystyle=p_{1}-q_{1}=-2w_{1}-\nu\;,$
$\displaystyle\hat{\beta}$
$\displaystyle=p_{2}-q_{2}=-2w_{1}(w_{2}+w_{1}+w_{0}+\nu)-b\;,$
$\displaystyle\hat{\gamma}$
$\displaystyle=\frac{1}{2}q_{1}=w_{1}+w_{0}+\nu\;,$ (3.64)
$\displaystyle\hat{\delta}$
$\displaystyle=\frac{1}{2}q_{2}=w_{-1}w_{0}+w_{1}w_{2}+(w_{0}+w_{1})^{2}+\nu(w_{0}+w_{1})+b$
$\displaystyle=-\Big{(}w_{2}w_{3}+w_{1}w_{2}+w_{2}^{2}+w_{0}w_{2}+\nu
w_{2}+\frac{a}{w_{1}}\Big{)}\;,$

where, in the last equality, we have used the recurrence relation (1.2) to
replace $w_{-1}$ by $w_{3}$. If desired, one can apply (1.2) once again to
replace $w_{0}$ by $w_{4}$ in the above expressions, but we have not done
this.

As a particular numerical example, we take the rational orbit (2.1) of (P.iv)
considered in Section 2. Upon fixing $w_{0}=w_{1}=w_{2}=w_{3}=1$ and $\nu=3$,
$a=5$, $b=7$, we see that for $j\geq 4$ the recursion (3.63) becomes

$s_{j}=-5\,s_{j-1}-19\,s_{j-2}+\sum_{i=1}^{j-1}s_{i}s_{j-i}+5\,\sum_{i=1}^{j-2}s_{i}s_{j-i-1}-12\,\sum_{i=1}^{j-3}s_{i}s_{j-i-2}\;,$

and the three initial values $s_{1}=s_{2}=1$, $s_{3}=2$ lead to the following
moment sequence and Hankel determinants:

$(s_{j})_{j\geqslant
1}:\,\,\,1,1,2,-26,45,11,-116,553,1151,-26727,108897,-169157,-310959,3004412,-4722005,\ldots,$
$\displaystyle\Delta_{1}=$ $\displaystyle
1,\,\Delta_{2}=1,\,\Delta_{3}=\left|\begin{array}[]{cc}1&1\\\
1&2\end{array}\right|=1,\,\Delta_{4}=\left|\begin{array}[]{cc}1&2\\\
2&-26\end{array}\right|=-30,\,\Delta_{5}=\left|\begin{array}[]{ccc}1&1&2\\\
1&2&-26\\\ 2&-26&45\end{array}\right|=-743,$ $\displaystyle\Delta_{6}=$
$\displaystyle\left|\begin{array}[]{ccc}1&2&-26\\\ 2&-26&45\\\
-26&45&11\end{array}\right|=10541,\qquad\Delta_{7}=\left|\begin{array}[]{cccc}1&1&2&-26\\\
1&2&-26&45\\\ 2&-26&45&11\\\ -26&45&11&-116\end{array}\right|=127318,\,\dots.$

This reproduces the sequence of tau functions in Example 2.2, if we identify
$\Delta_{n-3}=\tau_{n}$ in (2.7).

The preceding explicit form of the recursion for the entries of the Hankel
determinants when $g=2$ yields a simple proof of the Laurent property for
(2.4).

  * Proof of Proposition 2.3 (reprise):

For $n\geqslant 1$ we have

$w_{n}=\frac{\tau_{n}\tau_{n+3}}{\tau_{n+1}\tau_{n+2}}=\frac{\Delta_{n-3}\Delta_{n}}{\Delta_{n-2}\Delta_{n-1}}\;,$

where $\tau_{n}$ satisfies (2.4). Hence the tau functions are given by Hankel
determinants, up to a shift of index and a gauge transformation of the form
(2.6), with a different scaling for even/odd $n$. Comparing with the values
$\Delta_{-2}=\Delta_{-1}=\Delta_{0}=1$, we see that the relation between the
two sequences is

$\tau_{2k+1}=\tau_{1}\,\left(\frac{\tau_{3}}{\tau_{1}}\right)^{k}\,\Delta_{2k-2},\qquad\tau_{2k+2}=\frac{\tau_{1}\tau_{2}}{\tau_{3}}\,\left(\frac{\tau_{3}}{\tau_{1}}\right)^{k+1}\,\Delta_{2k-1}$
(3.65)

for $k\geqslant 0$. Recall that $\cal R$ denotes the ring formed of Laurent
polynomials in $\tau_{2},\tau_{3},\tau_{4}$ and polynomials in
$\tau_{0},\tau_{1},\tau_{5},\tau_{6}$ with coefficients in
${\mathbb{Z}}[a,b,\nu]$. Upon rewriting the formulae (3.11) for the
coefficients in (3.63) in terms of the 7 initial tau functions, we see that
$\hat{{\alpha}}$, $\hat{{\beta}}\in{\cal R}$, but (due to the presence of
terms involving $w_{0}$ and $1/w_{1}$), $\hat{{\gamma}}$ and $\hat{\delta}$
both have $\tau_{1}$ appearing in the denominator, so instead
$\hat{{\gamma}}$, $\hat{\delta}\in\tau_{1}^{-1}{\cal R}$. However, the three
initial values are

$s_{1}=\frac{\tau_{1}\tau_{4}}{\tau_{2}\tau_{3}},\quad
s_{2}=\frac{\tau_{1}\tau_{5}}{\tau_{3}^{2}},\quad
s_{3}=\tau_{1}\left(\frac{\tau_{2}\tau_{5}^{2}}{\tau_{3}^{3}\tau_{4}}+\frac{\tau_{6}}{\tau_{3}\tau_{4}}\right),$

so $s_{j}\in\tau_{1}{\cal R}$ for $j=1,2,3$. Then by induction, since
$\hat{{\gamma}}$ and $\hat{\delta}$ appear in front of terms of degree 2 in
$s_{i}$ in the recursion (3.63), it follows that $s_{j}\in\tau_{1}{\cal R}$
for all $j\geqslant 1$. Then since $\Delta_{2k-1}$ and $\Delta_{2k}$ are
$k\times k$ determinants, a factor of $\tau_{1}$ can be taken out of each row
(or column), so they are each given by an overall factor of $\tau_{1}^{k}$
times an element of ${\cal R}$. Thus the powers of $\tau_{1}$ exactly cancel
in (3.65), and hence $\tau_{n}\in{\cal R}$ for all $n\geqslant 1$. ∎

######  Remark 3.12.

Hankel determinant formula for negative indices: As was previously noted, the
Laurent property for negative indices $n$ follows automatically from the
birationality and reversing symmetry of (2.4), but it can also be shown
directly from an appropriate extension of (3.65) to $k<0$. In fact, for any
$g$ there is a version of the Hankel determinant formula (3.55) which is valid
for $n\leqslant 0$. Indeed, the Volterra map arises from the S-fraction
expansion (3.15) of the function $F_{0}$, based on the power series $S(x)$,
with $F_{0}=1-S$, as in (3.54), but more precisely this is the expansion
around the point $(0,1)$ on the curve $\Gamma_{f}$ given by (3.5), with $x$
being a local parameter. The inverse Volterra map arises from another
S-fraction, associated with a power series $S^{*}(x)$, corresponding to the
expansion of the same function $F_{0}$ around the point $(0,-1)\in\Gamma_{f}$,
that is

$F_{0}=\cfrac{w_{0}x}{1-\cfrac{w_{-1}x}{1-\cfrac{w_{-2}x}{1-\cdots}}}=\sum_{j=1}^{\infty}s_{j}^{*}x^{j}=:S^{*}(x)\;.$
(3.66)

Then the extension of (3.55) to non-positive values of the index is

$w_{n}=\frac{\Delta^{*}_{-n-2}\Delta^{*}_{-n+1}}{\Delta^{*}_{-n}\Delta^{*}_{-n-1}}\qquad\mathrm{for}\,\,n\leqslant
0\;,$ (3.67)

where $\Delta^{*}_{2k-1}=\det(s^{*}_{i+j-1})_{i,j=1,\ldots,k}$ and
$\Delta^{*}_{2k}=\det(s^{*}_{i+j})_{i,j=1,\ldots,k}$ for $k\geqslant 1$, with
$\Delta^{*}_{-2}=\Delta^{*}_{-1}=\Delta^{*}_{0}=1$. Mutatis mutandis, this is
proved in the same way as Theorem 3.8, and the moments $s_{j}^{*}$ satisfy
another recursion of the form (3.57). The two sets of Hankel determinants
combine to produce a single sequence of tau functions
$(\tau_{n})_{n\in{\mathbb{Z}}}$, consistently defined by taking
$\tau_{n}=\Delta_{n-3}$ for $n\geqslant 1$, and $\tau_{n}=\Delta^{*}_{-n+1}$
for $n\leqslant 3$.

###  3.5 The birational map $M_{g}\to{\mathbb{C}}^{3g+1}_{w}$

Elaborating further on the S-fraction of $F=\frac{y+{\cal P}(x)}{{\cal
Q}(x)}$, we construct a birational map between $M_{g}$ and
${\mathbb{C}}^{3g+1}_{w}$, where the latter stands for ${\mathbb{C}}^{3g+1}$
equipped with $w_{1},\dots,w_{3g+1}$ as affine coordinates. We may call it the
_unreduced_ birational map, in view of the birational map
$M_{g}^{c}\to{\mathbb{C}}^{2g+1}_{w}$ which we obtained by reduction (fixing
$c=(c_{1},\ldots,c_{g})$). This unreduced map turns out to be less convenient
for deriving the Volterra map in the $w_{i}$ coordinates, but it is
nevertheless useful for obtaining abstract results in these coordinates.

We start from the equation (3.60), which we can view as a linear relation for
${\cal P}(x),{\cal Q}(x)$ and ${\cal R}(x)$. It amounts to an infinite linear
system of equations for the coefficients
$p_{1},\dots,p_{g},q_{1},\dots,q_{g},r_{1},\dots,r_{g+1}$ of these
polynomials, given in terms of the coefficients $s_{j}$ of the power series
$S(x)$ defined in (3.54). We show that its solution can be expressed
rationally in terms of $s_{1},\dots,s_{3g+1}$, and hence in terms of
$w_{1},\dots,w_{3g+1}$, yielding the birational map. To do this, we
investigate the first $3g+1$ equations only, namely the ones corresponding to
the coefficient of $x^{j}$ in (3.60). For convenience, we set

$\rho_{k}:=\sum_{i+j=k}s_{i}s_{j},$

for $k\geqslant 2$, so that
$S^{2}=\sum_{i+j=k}s_{i}s_{j}x^{k}=\sum_{k=2}^{\infty}\rho_{k}x^{k}$, and then
consider the terms of (3.60) at each order in $x$. At leading order, the
constant term cancels, and the coefficients of $x,x^{2},\dots,x^{g+1}$ can be
used to obtain $r_{1},\dots,r_{g+1}$ in terms of
$p_{1},\dots,p_{g},q_{1},\dots,q_{g},$ and $s_{j}$ for $1\leqslant j\leqslant
g+1$, so it is sufficient to show that we can solve the coefficients of
$x^{j}$ with ${g+2}\leqslant j\leqslant{3g+1}$ for
$p_{1},\dots,p_{g},q_{1},\dots,q_{g}$ in terms of $s_{1},\dots,s_{3g+1}$. For
each such $j$, the equation at order $x^{j}$ in (3.60) is given by (3.57),
which we can rewrite as

$\sum_{i=1}^{g}s_{j-i}(p_{i}-q_{i})+\frac{1}{2}\sum_{i=1}^{g}\rho_{j-i}q_{i}=s_{j}-\rho_{j}\;,\qquad\mathrm{for}\quad
g+2\leqslant j\leqslant 3g+1\;.$

Viewed as a linear system in the $2g$ variables $p_{j}-q_{j}$ and $q_{j}/2$,
with $i=1,\dots,g$, the associated matrix takes the $2\times 2$ block form,
each block being a Toeplitz matrix,

$\begin{pmatrix}T_{g}(s,g)&T_{g}(\rho,g)\\\ T_{g}(s,2g)&T_{g}(\rho,2g)\\\
\end{pmatrix}\quad\hbox{where}\quad
T_{g}(\sigma,k):=\begin{pmatrix}\sigma_{k+1}&\sigma_{k}&\dots&\sigma_{k-g+1}\\\
\sigma_{k+2}&\sigma_{k+1}&\ddots\\\ \vdots&\ddots&\ddots&\sigma_{k}\\\
\sigma_{k+g}&&\sigma_{k+2}&\sigma_{k+1}\end{pmatrix}$ (3.68)

Our claim that $p_{1},\dots,p_{g},q_{1},\dots,q_{g}$ can be solved rationally
in terms of $s_{1},\dots,s_{3g+1}$ then follows from the fact that its
determinant is non-zero, which is shown in the following lemma:

######  Lemma 3.13.

The block $2\times 2$ matrix in (3.68) is non-degenerate.

###### Proof.

Since the entries of the matrix are polynomials in $s_{1},\dots,s_{3g+1}$, so
is their determinant. To show that the determinant is non-zero for generic
values of the $s_{i}$, it is therefore sufficient to show it for one
particular set of values of the $s_{i}$. We pick

$s_{i}:=\left\\{\begin{array}[]{cc}i&1\leqslant i\leqslant g\;,\\\
1/4&1\leqslant i=g+1\;,\\\ 0&g+1\leqslant i\leqslant
3g+1\;.\end{array}\right.$

Then $T_{g}(s,g)$ is upper triangular, with all diagonal entries $1/4$, while
$T_{g}(s,2g)$ is the zero matrix. So we only need to show that

$V_{g}:=T_{g}(\rho,2g)=\begin{pmatrix}1&2&3&\dots&\dots&g\\\
1/4&1&2&3&\dots&g-1\\\ 0&1/4&1&2&\dots&g-2\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\\ \vdots&&0&1/4&1&2\\\
0&\dots&\dots&0&1/4&1\end{pmatrix}$

is non-singular. We show by complete induction that $\det V_{g}=2^{2-2g}g$ for
all $g\geqslant 1$. The formula being true for $g=1$, we assume it to be true
for all values smaller than $g$ and compute by successive development with
respect to the first column, to find that $\det V_{g}$ equals

$\displaystyle\det V_{g-1}-\frac{1}{4}\left|\begin{matrix}2&3&4&\dots&g\\\
1/4&1&2&\dots&g-2\\\ 0&\ddots&\ddots&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\ddots&2\\\ 0&\dots&0&1/4&1\end{matrix}\right|$
$\displaystyle=\det V_{g-1}-\frac{1}{4}\times 2\det
V_{g-2}+\left(\frac{1}{4}\right)^{2}\det\left|\begin{matrix}3&4&5&\dots&g\\\
1/4&1&2&\dots&g-3\\\ 0&\ddots&\ddots&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\ddots&2\\\ 0&\dots&0&1/4&1\end{matrix}\right|\;,$

and hence (defining $\det V_{0}:=1$ for convenience)

$\det V_{g}=\sum_{i=1}^{g}\left(-\frac{1}{4}\right)^{i-1}i\det
V_{g-i}\;=\left(\frac{1}{4}\right)^{g-2}\sum_{i=1}^{g-1}(-1)^{i-1}i(g-i)+(-1)^{g-1}2^{2-2g}g\;,$

where we have used the induction hypothesis in the last step. It remains to be
shown that

$\sum_{i=1}^{g-1}(-1)^{i-1}i(g-i)=\left\\{\begin{array}[]{cl}g/2&\hbox{when
$g$ is even,}\\\ 0&\hbox{when $g$ is odd.}\end{array}\right.$

The even case follows from an easy induction, while the odd case is completely
trivial by symmetry. ∎

######  Example 3.14.

The simplest example is the case of $g=1$. Then $3g+1=4$. The birational map
between $s_{1},\dots,s_{4}$ and $w_{1},\dots,w_{4}$ is given by

$s_{1}=w_{1}\;,\quad s_{2}=w_{1}w_{2}\;,\quad
s_{3}=w_{1}w_{2}(w_{2}+w_{3})\;,\quad
s_{4}=w_{1}w_{2}\left((w_{2}+w_{3})^{2}+w_{3}w_{4}\right)\;.$ (3.69)

The linear system for $p_{1},q_{1},r_{1},r_{2}$ then corresponds to the
coefficients of $x^{j}$ in (3.60) with $j=1,\dots,4$. They are explicitly
given by

$\displaystyle q_{1}-2p_{1}-2s_{1}-r_{1}=0\;,$ $\displaystyle
2p_{1}s_{1}-2q_{1}s_{1}-2s_{2}+2\rho_{2}-r_{2}=0\;,$ $\displaystyle
2\rho_{3}-2s_{3}+(\rho_{2}-2s_{2})q_{1}+2s_{2}p_{1}=0\;,$ (3.70)
$\displaystyle 2\rho_{4}-2s_{4}+(\rho_{3}-2s_{3})q_{1}+2s_{3}p_{1}=0\;.$

As in the general case, we first solve the last $2g=2$ equations for
$p_{1},q_{1}$, which gives, upon writing each $\rho_{k}$ in terms of the
$s_{j}$,

$\displaystyle p_{1}$
$\displaystyle=2\frac{-3s_{1}^{2}s_{2}^{2}-2s_{2}^{3}+2(s_{1}s_{2}+s_{1}^{3}-s_{3})s_{3}+(2s_{2}-s_{1}^{2})s_{4}}{s_{1}(2s_{2}^{2}-s_{1}s_{3})}=-w_{1}-w_{2}+w_{3}+\frac{w_{3}(w_{1}-w_{4})(w_{1}-2w_{2})}{w_{1}(w_{2}-w_{3})}\;,$
$\displaystyle q_{1}$
$\displaystyle=2\frac{s_{2}s_{4}-s_{3}^{2}-s_{2}^{3}}{s_{1}(2s_{2}^{2}-s_{1}s_{3})}=-2w_{2}\left(1+\frac{w_{3}(w_{1}-w_{4})}{w_{1}(w_{2}-w_{3})}\right)\;,$

where we have used (3.69) to write the formulae in terms of
$w_{1},\dots,w_{4}$. The first two equations can now be solved for $r_{1}$ and
$r_{2}$, leading to the following expressions:

$r_{1}=-2w_{3}\left(1+\frac{(w_{1}-w_{2})(w_{1}-w_{4})}{w_{1}(w_{2}-w_{3})}\right)\;,\qquad
r_{2}=2w_{1}w_{3}\left(1+\frac{w_{1}-w_{4}}{w_{2}-w_{3}}\right)\;.$

Note also that in this case the unreduced version of the Volterra map is the
birational map of ${\mathbb{C}}^{4}_{w}$ defined by

$\displaystyle(w_{1},w_{2},w_{3},w_{4})\mapsto(w_{2},w_{3},w_{4},w_{5})\;,$
$\displaystyle\frac{w_{5}w_{4}+w_{4}^{2}-w_{3}^{2}-w_{3}w_{2}}{w_{4}-w_{3}}+\frac{w_{1}w_{2}+w_{2}^{2}-w_{3}^{2}-w_{3}w_{4}}{w_{3}-w_{2}}=0\;,$
(3.71)

which (after replacing each $w_{i}\to w_{i-2}$) is the relation obtained by
eliminating $c_{1}$ from (3.34).

For any $g$, we can describe an explicit algorithmic procedure for obtaining
the exact expressions for the birational map between $s_{1},\ldots,s_{3g+1}$
and $w_{1},\ldots,w_{3g+1}$. One way round, this is given by the Hankel
determinant formula (3.55), but to describe the inverse map more explicitly we
must recall how this relates to the approximation problem originally
considered by Stieltjes [48], as well as other classical results on
convergents of continued fractions in terms of continuants (for a concise
review of the latter, see [16]).

The convergents of the S-fraction (3.15) are the sequence of rational
functions of $x$ obtained by truncating the continued fraction at some finite
line $n$, which approximate the series for $F_{0}=1-S$ exactly up to and
including the coefficient of $x^{n}$, that is

${\cal
F}_{n}(x):=1-\cfrac{w_{1}x}{1-\cfrac{w_{2}x}{1-\cfrac{\cdots}{1-w_{n}x}}}=1-S(x)+\mathcal{O}(x^{n+1}),$
(3.72)

Thus, for the first few convergents we have

${\cal F}_{0}=1,\quad{\cal F}_{1}=1-w_{1}x,\quad{\cal
F}_{2}=\frac{1-(w_{1}+w_{2})x}{1-w_{2}x},\quad{\cal
F}_{3}=\frac{1-(w_{1}+w_{2}+w_{3})x+w_{1}w_{3}x^{2}}{1-(w_{2}+w_{3})x},$

and so on. By the usual correspondence between convergents and $2\times 2$
matrices, we see that the monodromy product over the conjugation matrices
(3.23) appearing in the discrete Lax equation is given by

${\bm{\Phi}}_{n+1}:={\bf M}_{1}(x){\bf M}_{2}(x)\cdots{\bf
M}_{n+1}(x)=\left(\begin{array}[]{cc}{\cal
K}_{n+1}(w_{1},\ldots,w_{n};x)&-w_{n}x\,{\cal
K}_{n}(w_{1},\ldots,w_{n-1};x)\\\ {\cal
K}_{n}(w_{2},\ldots,w_{n};x)&-w_{n}x\,{\cal
K}_{n-1}(w_{2},\ldots,w_{n-1};x)\end{array}\right),$ (3.73)

with the $n$th convergent being the ratio of the entries in the first column,
that is

${\cal F}_{n}(x)=\frac{{\cal K}_{n+1}(w_{1},\ldots,w_{n};x)}{{\cal
K}_{n}(w_{2},\ldots,w_{n};x)},$

where the polynomial ${\cal K}_{n}$ is a continuant of size $n$, that is

${\cal
K}_{n}(w_{1},\ldots,w_{n-1};x):=\left|\begin{array}[]{rrrrr}{1}&{-1}&{0}&{\dots}&{0}\\\
{-w_{1}x}&{1}&{-1}&{\ddots}&{\vdots}\\\ {0}&{-w_{2}x}&{1}&{\ddots}&{0}\\\
{\vdots}&&{\ddots}&{\ddots}&{-1}\\\
{0}&{\dots}&{0}&{-w_{n-1}x}&{1}\end{array}\right|.$

From (3.73), the continuants are generated recursively from the linear
relation ${\bm{\Phi}}_{n+1}={\bm{\Phi}}_{n}{\bf M}_{n+1}$, starting from
${\bm{\Phi}}_{0}={\bm{1}}$ (the identity matrix). Hence, by replacing the
series for $F_{0}=1-S$ in (3.72) by its truncation at the $n$th term, then
multiplying by ${\cal K}_{n}(w_{2},\ldots,w_{n};x)$ on both sides, we find the
relation

$\Big{(}1-\sum_{j=1}^{n}s_{j}x^{j}\Big{)}{\cal
K}_{n}(w_{2},\ldots,w_{n};x)={\cal K}_{n+1}(w_{1},\ldots,w_{n};x)+{\cal
O}(x^{n+1}),$

and comparing the coefficients of $x^{j}$ for $j=1,\ldots,n$ allows
$s_{1},s_{2},\ldots,s_{n}$ to be calculated recursively as polynomial
expressions in $w_{1},\ldots,w_{n}$. For instance, when $n=4$ the expressions
(3.69) are obtained from the numerator and denominator of

${\cal
F}_{4}=\frac{1-(w_{1}+w_{2}+w_{3}+w_{4})x+(w_{1}w_{3}+w_{1}w_{4}+w_{2}w_{4})x^{2}}{1-(w_{2}+w_{3}+w_{4})x+w_{2}w_{4}x^{2}}.$

Observe that these expressions are universal, in the sense that they depend
only on the structure of the S-fraction, so that each $s_{n}$ is always given
by the same polynomial in $w_{j}$ for $1\leqslant j\leqslant n$, independent
of $g$.

For $g\geqslant 2$, the corresponding formulae for the coefficients of ${\cal
P}(x),{\cal Q}(x),{\cal R}(x)$ in terms of $w_{1},\ldots,w_{3g+1}$ become very
complicated, and we do not have a compact way to write them. However, we will
not need the explicit form of these formulae in what follows, even when
dealing with the example of genus 2.

##  4 Discrete integrability

We show in this section that the map (P.iv) is a discrete a.c.i. system, in a
sense which will be defined below. To do this, we will first show that the
affine space $M_{g}$ which was constructed in the previous section has the
same integrability properties as the odd and even Mumford systems (see [38,
53]): the invariants $H_{i}$ of the Volterra map are in involution with
respect to a large family of compatible Poisson structures and their generic
level sets are affine parts of Jacobi varieties. We then show that the
Volterra map is a Poisson map with respect to these Poisson structures and
that it is a translation on the latter complex tori. In the Appendix (Section
7) we further show the precise relation between the Mumford-like system,
introduced here, and the even Mumford system.

###  4.1 Compatible Poisson structures for the Mumford-like system

Recall that $g$ is a fixed positive integer, and $M_{g}$ denotes the affine
space defined by

$M_{g}:=\left\\{({\cal P}(x),{\cal Q}(x),{\cal
R}(x))\in{\mathbb{C}}[x]^{3}~{}\Bigl{|}~{}\begin{array}[]{ll}\deg{\cal
P}(x)\leqslant g\;,&{\cal P}(0)=1\\\ \deg{\cal Q}(x)\leqslant g\;,&{\cal
Q}(0)=2\\\ \deg{\cal R}(x)\leqslant g+1\;,&{\cal
R}(0)=0\end{array}\right\\}\;,$ (4.1)

with the $3g+1$ non-constant coefficients of ${\cal P},\,{\cal Q}$ and ${\cal
R}$ being linear coordinates on $M_{g}$.

We first introduce a $g+1$-dimensional family of compatible Poisson structures
on $M_{g}$. The family is parametrized by the $g+1$-dimensional vector space
of polynomials $\phi\in{\mathbb{C}}[x]$ of degree at most $g+1$, vanishing at
$0$, so $\phi(0)=0$. For such a polynomial $\phi$, the corresponding Poisson
structure is most naturally written by viewing ${\cal P},\,{\cal Q}$ and
${\cal R}$ as generating functions, and is defined by

$\displaystyle\left\\{{\cal P}(x),{\cal P}(y)\right\\}^{\phi}$
$\displaystyle=\left\\{{\cal Q}(x),{\cal Q}(y)\right\\}^{\phi}=0\;,\
\left\\{{\cal R}(x),{\cal R}(y)\right\\}^{\phi}=\phi(y){\cal
R}(x)-\phi(x){\cal R}(y)\;,$ $\displaystyle\left\\{{\cal P}(x),{\cal
Q}(y)\right\\}^{\phi}$ $\displaystyle=\frac{\phi(x)y{\cal Q}(y)-\phi(y)x{\cal
Q}(x)}{x-y}\;,\ \left\\{{\cal P}(x),{\cal
R}(y)\right\\}^{\phi}=-y\frac{\phi(x){\cal R}(y)-\phi(y){\cal R}(x)}{x-y}\;,$
(4.2) $\displaystyle\left\\{{\cal Q}(x),{\cal R}(y)\right\\}^{\phi}$
$\displaystyle=2y\frac{\phi(x){\cal P}(y)-\phi(y){\cal
P}(x)}{x-y}-\phi(y){\cal Q}(x)\;.$

Of course, one needs to verify that the above definition is coherent, in the
sense that the right-hand side of each of these formulae is indeed a
polynomial in $x$ and $y$, and also that the right-hand side does not contain
any monomials $x^{i}y^{j}$ which are absent in the left-hand side. For
example, $\left\\{{\cal P}(x),{\cal Q}(y)\right\\}^{\phi}$ is a polynomial in
$x,y$ with only non-zero coefficients of $x^{i}y^{j}$ when $1\leqslant
i,j\leqslant g$, while $\phi(x)y{\cal Q}(y)-\phi(y)x{\cal Q}(x)$ is clearly
divisible by $x-y$ and the quotient has only non-zero coefficients of
$x^{i}y^{j}$ in the same range. The same argument applies to the other
formulae. Moreover, we need to verify that
$\left\\{\cdot\,,\cdot\right\\}^{\phi}$ is a Poisson bracket, i.e., that it
satisfies the Jacobi identity. This follows easily from the formulae (4.1).
For example,

$\displaystyle\left\\{\cancel{\left\\{{\cal Q}(x),{\cal
Q}(y)\right\\}^{\phi}},{\cal R}(z)\right\\}^{\phi}+\left\\{\left\\{{\cal
Q}(y),{\cal R}(z)\right\\}^{\phi},{\cal
Q}(x)\right\\}^{\phi}+\left\\{\left\\{{\cal R}(z),{\cal
Q}(x)\right\\}^{\phi},{\cal Q}(y)\right\\}^{\phi}$
$\displaystyle=\left\\{2z\frac{\phi(y){\cal P}(z)-\phi(z){\cal
P}(y)}{y-z}-\bcancel{\phi(z){\cal Q}(y)},{\cal Q}(x)\right\\}^{\phi}-\langle
x\leftrightarrow y\rangle$
$\displaystyle=\frac{2z\phi(y)}{y-z}\frac{\phi(z)x{\cal
Q}(x)-\xcancel{\phi(x)z{\cal
Q}(z)}}{z-x}-\frac{2z\phi(z)}{y-z}\frac{\phi(y)x{\cal Q}(x)-\phi(x)y{\cal
Q}(y)}{y-x}-\langle x\leftrightarrow y\rangle$
$\displaystyle=\left(\frac{1}{(y-z)(z-x)}+\frac{1}{(x-y)(y-z)}+\frac{1}{(z-x)(x-y)}\right)2xz\phi(y)\phi(z){\cal
Q}(x)-\langle x\leftrightarrow y\rangle=0\;.$

Notice that since $\left\\{\cdot\,,\cdot\right\\}^{\phi}$ is a Poisson bracket
for any $\phi\in{\mathbb{C}}[x]$ with $\deg\phi\leqslant g+1$ and $\phi(0)=0$,
the Poisson brackets $\left\\{\cdot\,,\cdot\right\\}^{\phi}$ are compatible,
simply because
$\left\\{\cdot\,,\cdot\right\\}^{\phi}+\left\\{\cdot\,,\cdot\right\\}^{\phi^{\prime}}=\left\\{\cdot\,,\cdot\right\\}^{\phi+\phi^{\prime}}$
for any such polynomials $\phi,\phi^{\prime}$.

We show in the next proposition that the Volterra map ${\cal V}_{g}$ is a
Poisson map with respect to each one of these Poisson brackets. Recall from
(3.9) and (3.10) that ${\cal V}_{g}$ is given by

$\tilde{\cal P}(x)={\cal Q}(x)-{\cal P}(x)\;,\qquad\tilde{\cal
Q}(x)=\frac{2{\cal P}(x)-{\cal Q}(x)+{\cal R}(x)}{-wx}\;,\qquad\tilde{\cal
R}(x)=-wx{\cal Q}(x)\;,$ (4.3)

where $w=-\frac{2\,p_{1}-q_{1}+r_{1}}{2}$.

######  Proposition 4.1.

For any $\phi=\sum_{i=1}^{g+1}\phi_{i}x^{i}$, the Volterra map ${\cal
V}_{g}:M_{g}\to M_{g}$ is a (birational) Poisson map with respect to
$\left\\{\cdot\,,\cdot\right\\}^{\phi}$.

###### Proof.

We need to check that ${\cal V}_{g}$ preserves the Poisson bracket
$\left\\{\cdot\,,\cdot\right\\}^{\phi}$. In formulae, this means that
$\left\\{\tilde{\cal S}(x),\tilde{\cal
T}(y)\right\\}^{\phi}=\widetilde{\left\\{{\cal S}(x),{\cal
T}(y)\right\\}^{\phi}}$, where ${\cal S}$ and ${\cal T}$ stand for any of the
polynomials ${\cal P},{\cal Q},{\cal R}$. To do this, it helps to first use
(4.1) to derive the following formulae for the Poisson brackets of $w$ with
the polynomials ${\cal P},{\cal Q},{\cal R}$:

$\left\\{w,{\cal P}(y)\right\\}^{\phi}=\left\\{w,{\cal
Q}(y)\right\\}^{\phi}=\frac{\phi_{1}}{2}{\cal
Q}(y)-\frac{\phi(y)}{y}\;,\quad\left\\{w,{\cal
R}(y)\right\\}^{\phi}=\frac{\phi(y)}{y}-w\phi(y)-\frac{\phi_{1}}{2}(2{\cal
P}(y)+{\cal R}(y))\;.$ (4.4)

To derive these formulae from (4.1), it suffices to use that $-2w$ is the
linear term of $2{\cal P}(x)-{\cal Q}(x)+{\cal R}(x)$. For example,

$\displaystyle\left\\{-2w,{\cal R}(y)\right\\}^{\phi}$
$\displaystyle=\lim_{x\to 0}\frac{1}{x}\left\\{2{\cal P}(x)-{\cal Q}(x)+{\cal
R}(x),{\cal R}(y)\right\\}^{\phi}$ $\displaystyle=\lim_{x\to
0}\frac{1}{x}\left[-2y\frac{\phi(x){\cal R}(y)-\phi(y){\cal
R}(x)}{x-y}-2y\frac{\phi(x){\cal P}(y)-\phi(y){\cal P}(x)}{x-y}+\phi(y)({\cal
Q}(x)+{\cal R}(x))-\phi(x){\cal R}(y)\right]$ $\displaystyle=\phi_{1}{\cal
R}(y)-\phi(y)r_{1}+2\phi_{1}{\cal P}(y)+\phi(y)\lim_{x\to 0}\frac{2y{\cal
P}(x)+(x-y){\cal Q}(x)}{x(x-y)}$ $\displaystyle=\phi_{1}{\cal
R}(y)-\phi(y)r_{1}+2\phi_{1}{\cal
P}(y)-2\frac{\phi(y)}{y}-(2p_{1}-q_{1})\phi(y)$
$\displaystyle={\phi_{1}}(2{\cal P}(y)+{\cal
R}(y))-2\frac{\phi(y)}{y}+2w\phi(y)\;,$

which proves the last formula in (4.4). It is now easy to prove that the
Volterra map is a Poisson map. For example, it is clear from (4.4) that
$\left\\{{\cal P}(x)-{\cal Q}(x),w\right\\}^{\phi}=0$, and so

$\displaystyle\left\\{\tilde{\cal P}(x),\tilde{\cal R}(y)\right\\}^{\phi}$
$\displaystyle=\left\\{{\cal Q}(x)-{\cal P}(x),-wy{\cal
Q}(y)\right\\}^{\phi}=wy\left\\{{\cal P}(x),{\cal
Q}(y)\right\\}^{\phi}=wy\frac{\phi(x)y{\cal Q}(y)-\phi(y)x{\cal Q}(x)}{x-y}$
$\displaystyle=-y\frac{\phi(x)\tilde{\cal R}(y)-\phi(y)\tilde{\cal
R}(x)}{x-y}=\widetilde{\left\\{{\cal P}(x),{\cal R}(y)\right\\}^{\phi}}\;.$

The other verifications are done in the same way. This shows that the Volterra
map is a Poisson map. ∎

We show in the next proposition that the invariants of the Volterra map, as
introduced in the previous section, are in involution with respect to any of
the Poisson brackets $\left\\{\cdot\,,\cdot\right\\}^{\phi}$, where we recall
that $\deg\phi\leqslant g+1$ and $\phi(0)=0$.

######  Proposition 4.2.

The $2g+1$ invariants $H_{1},\dots,H_{2g+1}$ of the Volterra map, defined by

${\cal P}(x)^{2}+{\cal Q}(x){\cal R}(x)=1+\sum_{i=1}^{2g+1}H_{i}x^{i}\;.$
(4.5)

are in involution with respect to $\left\\{\cdot\,,\cdot\right\\}^{\phi}$.

###### Proof.

We first compute from (4.1)

$\displaystyle\left\\{{\cal P}(x),{\cal P}(y)^{2}+{\cal Q}(y){\cal
R}(y)\right\\}^{\phi}$ $\displaystyle={\cal Q}(y)\left\\{{\cal P}(x),{\cal
R}(y)\right\\}^{\phi}+{\cal R}(y)\left\\{{\cal P}(x),{\cal
Q}(y)\right\\}^{\phi}$ $\displaystyle=-y{\cal Q}(y)\frac{\cancel{\phi(x){\cal
R}(y)}-\phi(y){\cal R}(x)}{x-y}+{\cal R}(y)\frac{\cancel{y\phi(x){\cal
Q}(y)}-x\phi(y){\cal Q}(x)}{x-y}$ $\displaystyle=\phi(y)\frac{y{\cal
Q}(y){\cal R}(x)-x{\cal Q}(x){\cal R}(y)}{x-y}\;,$ (4.6)

and similarly

$\displaystyle\left\\{{\cal Q}(x),{\cal P}(y)^{2}+{\cal Q}(y){\cal
R}(y)\right\\}^{\phi}$ $\displaystyle=2\phi(y)\frac{x{\cal P}(y){\cal
Q}(x)-y{\cal P}(x){\cal Q}(y)}{x-y}-\phi(y){\cal Q}(x){\cal Q}(y)\;,$ (4.7)
$\displaystyle\left\\{{\cal R}(x),{\cal P}(y)^{2}+{\cal Q}(y){\cal
R}(y)\right\\}^{\phi}$ $\displaystyle=2x\phi(y)\frac{{\cal P}(x){\cal
R}(y)-{\cal P}(y){\cal R}(x)}{x-y}+\phi(y){\cal Q}(y){\cal R}(x)\;.$ (4.8)

These formulae imply

$\displaystyle\left\\{{\cal P}^{2}(x)+{\cal Q}(x){\cal R}(x),{\cal
P}^{2}(y)+{\cal Q}(y){\cal R}(y)\right\\}^{\phi}$ $\displaystyle=2{\cal
P}(x)\left\\{{\cal P}(x),{\cal P}^{2}(y)+{\cal Q}(y){\cal
R}(y)\right\\}^{\phi}+{\cal Q}(x)\left\\{{\cal R}(x),{\cal P}^{2}(y)+{\cal
Q}(y){\cal R}(y)\right\\}^{\phi}+{\cal R}(x)\left\\{{\cal Q}(x),{\cal
P}^{2}(y)+{\cal Q}(y){\cal R}(y)\right\\}^{\phi}$ $\displaystyle=2{\cal
P}(x)\phi(y)\frac{\cancel{y{\cal Q}(y){\cal R}(x)}-\bcancel{x{\cal Q}(x){\cal
R}(y)}}{x-y}+{\cal Q}(x)2x\phi(y)\frac{\bcancel{{\cal P}(x){\cal
R}(y)}-\xcancel{{\cal P}(y){\cal R}(x)}}{x-y}+{\cal Q}(x)\phi(y){\cal
Q}(y){\cal R}(x)$ $\displaystyle\qquad+{\cal
R}(x)2\phi(y)\frac{\xcancel{x{\cal P}(y){\cal Q}(x)}-\cancel{y{\cal P}(x){\cal
Q}(y)}}{x-y}-{\cal R}(x)\phi(y){\cal Q}(x){\cal Q}(y)=0\;,$

so that the invariants $H_{i}$ are in involution,
$\left\\{H_{i},H_{j}\right\\}^{\phi}=0$ for $1\leqslant i,j\leqslant 2g+1$. ∎

As before, we often fix the Hamiltonians $H_{1},\dots,H_{g}$ to generic values
so that we actually work on a (non-singular) subvariety of $M_{g}^{c}$, which
is birational with ${\mathbb{C}}^{2g+1}$. We show in the following proposition
that $M_{g}^{c}$ is a bi-Hamiltonian manifold, i.e., that it is equipped with
a pencil of compatible Poisson structures.

######  Proposition 4.3.

When $\phi=\phi_{g}x^{g}+\phi_{g+1}x^{g+1}$ the Hamiltonians
$H_{1},\dots,H_{g}$ are Casimirs of the Poisson structure
$\left\\{\cdot\,,\cdot\right\\}^{\phi}$. Moreover,

$C_{\phi}:=\sum_{i=0}^{g}(-1)^{i}\phi_{g}^{g-i}\phi_{g+1}^{i}H_{2g+1-i}$ (4.9)

is also a Casimir function of $\left\\{\cdot\,,\cdot\right\\}^{\phi}$.

###### Proof.

Consider for $i=1,\dots,g$ the Hamiltonian vector field $\mathrm{X}_{H_{i}}$,
which is given as the coefficient of $y^{i}$ in $\left\\{\cdot\,,{\cal
P}(y)^{2}+{\cal Q}(y){\cal R}(y)\right\\}^{\phi}$, which we computed in (4.1)
– (4.8). These Hamiltonians are Casimir functions for
$\left\\{\cdot\,,\cdot\right\\}^{\phi}$ if and only if their Hamiltonian
vector fields are zero, which is in turn equivalent to the fact that the right
hand sides in (4.1) – (4.8) are divisible by $y^{g+1}$. Notice that the latter
right hand sides, without the factor $\phi(y)$, are divisible by $y$ since
${\cal P}(0)=1,$ ${\cal Q}(0)=2$ and ${\cal R}(0)=0$, without being divisible
by $y^{2}$. Therefore, we need $\phi(y)$ to be divisible by $y^{g}$; since
$\deg\phi\leqslant g+1$, it follows that the Poisson structures which make
$M_{g}^{c}$ into a bi-Hamiltonian manifold are the restrictions to $M_{g}^{c}$
of the Poisson pencil $\left\\{\cdot\,,\cdot\right\\}^{\phi}$ with
$\phi(x)=\phi_{g}x^{g}+\phi_{g+1}x^{g+1}$. It is easily shown by direct
computation that $C_{\phi}$, given by (4.9) is a Casimir of the Poisson
pencil. For example,

$\displaystyle\left\\{{\cal
P}(x),\sum_{i=0}^{g}(-1)^{i}\phi_{g}^{g-i}\phi_{g+1}^{i}H_{2g+1-i}\right\\}^{\phi}$
$\displaystyle=\mathop{\textrm{Res}}\limits_{y=0}\frac{1}{y^{2g+2}}\left\\{{\cal
P}(x),H_{2g+1-i}y^{2g+1}\right\\}^{\phi}\sum_{i=0}^{g}(-1)^{i}\phi_{g}^{g-i}\phi_{g+1}^{i}$
$\displaystyle=\mathop{\textrm{Res}}\limits_{y=0}\frac{1}{y^{2g+2}}\left\\{{\cal
P}(x),{\cal P}(y)^{2}+{\cal Q}(y){\cal
R}(y)\right\\}^{\phi}\sum_{i=0}^{g}(-1)^{i}\phi_{g}^{g-i}\phi_{g+1}^{i}y^{i}$
$\displaystyle=\mathop{\textrm{Res}}\limits_{y=0}\frac{1}{y^{2g+2}}\left[\frac{\phi(x)y{\cal
Q}(y)-\phi(y)x{\cal Q}(x)}{x-y}{\cal R}(y)-y{\cal Q}(y)\frac{\phi(x){\cal
R}(y)-\phi(y){\cal
R}(x)}{x-y}\right]\sum_{i=0}^{g}(-1)^{i}\phi_{g}^{g-i}\phi_{g+1}^{i}y^{i}$
$\displaystyle=\mathop{\textrm{Res}}\limits_{y=0}\frac{1}{y^{2g+2}}\;\frac{y{\cal
Q}(y){\cal R}(x)-x{\cal Q}(x){\cal
R}(y)}{x-y}\sum_{i=0}^{g}(-1)^{i}\phi_{g}^{g-i}\phi_{g+1}^{i}y^{i}\phi(y)$
$\displaystyle=\mathop{\textrm{Res}}\limits_{y=0}\frac{1}{y^{2g+2}}\;\frac{y{\cal
Q}(y){\cal R}(x)-x{\cal Q}(x){\cal
R}(y)}{x-y}\left(\phi_{g}^{g+1}y^{g}+(-1)^{g}\phi_{g+1}^{g+1}y^{2g+1}\right)$
$\displaystyle=\phi_{g}^{g+1}\mathop{\textrm{Res}}\limits_{y=0}\frac{1}{y^{g+2}}\;\frac{y{\cal
Q}(y){\cal R}(x)-x{\cal Q}(x){\cal R}(y)}{x-y}=0\;,$

where we have used in the last two equalities respectively that ${\cal R}(y)$
is divisible by $y$ and that the polynomial $\frac{y{\cal Q}(y){\cal
R}(x)-x{\cal Q}(x){\cal R}(y)}{x-y}$ has degree at most $g$ in $y$. ∎

######  Example 4.4.

Continuing Example 3.6, we specialize the above results to $g=1$ and make them
more explicit. Recall that $M_{1}$ is the $4$-dimensional vector space of
polynomials $({\cal P},{\cal Q},{\cal R})$ with

${\cal P}(x)=1+p_{1}x\;,\quad{\cal Q}(x)=2+q_{1}x\;,\quad{\cal
R}(x)=r_{1}x+r_{2}x^{2}\;,$ (4.10)

and the invariants $H_{1},H_{2},H_{3}$ are given in (3.24). The Poisson
structures $\left\\{\cdot\,,\cdot\right\\}^{\phi}$ on $M_{1}$ are parametrized
by $\phi(x)=\phi_{1}x+\phi_{2}x^{2}$ and they all have $H_{1}$ as a Casimir
function. The Poisson matrices of the basic Poisson structures
$\left\\{\cdot\,,\cdot\right\\}^{x}$ and
$\left\\{\cdot\,,\cdot\right\\}^{x^{2}}$ (with the coordinates taken in the
following order: $p_{1},\,q_{1},\,r_{1},\,r_{2}$) are easily determined from
(4.1) and are given by

$\left\\{\cdot\,,\cdot\right\\}^{x}=\begin{pmatrix}0&-q_{1}&0&r_{2}\\\
q_{1}&0&-q_{1}&0\\\ 0&q_{1}&0&-r_{2}\\\
-r_{2}&0&r_{2}&0\end{pmatrix}\;,\qquad\left\\{\cdot\,,\cdot\right\\}^{x^{2}}=\begin{pmatrix}0&2&0&-r_{1}\\\
-2&0&2&2q_{1}-p_{1}\\\ 0&-2&0&r_{1}\\\
r_{1}&p_{1}-2q_{1}&-r_{1}&0\end{pmatrix}\;.$ (4.11)

For a generic $c_{1}\in{\mathbb{C}}$, the subvariety $M_{1}^{c}$ is defined by
$H_{1}=c_{1}$. The pencil of Poisson structures
$\left\\{\cdot\,,\cdot\right\\}^{\phi}$ can be restricted to $M_{1}^{c}$ and
the Casimir function $C_{\phi}$ on
$(M_{1}^{c},\left\\{\cdot\,,\cdot\right\\}^{\phi})$, given by (4.9), takes the
simple form $C_{\phi}=\phi_{2}H_{2}-\phi_{1}H_{3}$. In particular, $H_{2}$ and
$H_{3}$ are Casimir functions of $\left\\{\cdot\,,\cdot\right\\}^{x^{2}}$ and
$\left\\{\cdot\,,\cdot\right\\}^{x}$, respectively, and
$\left\\{\cdot\,,\cdot\right\\}^{x}$ can be restricted to $H_{3}=c_{3}$ while
$\left\\{\cdot\,,\cdot\right\\}^{x^{2}}$ can be restricted to $H_{2}=c_{2}$
(for generic $c_{2},c_{3}$). Since, by Proposition 4.1, the Volterra map is a
Poisson map with respect to any such $\phi$ (of degree at most $g+1$ with
$\phi(0)=0$), it follows that

  1. $\bullet$

(3.34) defines a Poisson map with respect to the full Poisson pencil
$\left\\{\cdot\,,\cdot\right\\}^{\phi}$;

  2. $\bullet$

(3.37) defines a Poisson map with respect to
$\left\\{\cdot\,,\cdot\right\\}^{x}$;

  3. $\bullet$

(3.39) defines a Poisson map with respect to
$\left\\{\cdot\,,\cdot\right\\}^{x^{2}}$.

It follows from the Poisson matrices (4.11) and from (3.6) that the Poisson
structures $\left\\{\cdot\,,\cdot\right\\}^{x}$ and
$\left\\{\cdot\,,\cdot\right\\}^{x^{2}}$ are respectively given by
$\left\\{w_{n-1},w_{n}\right\\}^{x}=-(w_{n-1}+w_{n}+c_{1}/2)/2$, and
$\left\\{w_{n-1},w_{n}\right\\}^{x^{2}}=1/2$.

In Example 3.6 we also considered the recursion relation on the surface
$H_{2}^{\prime}=c_{2}^{\prime},\,H_{3}^{\prime}=c_{3}^{\prime}$. None of the
Poisson structures $\left\\{\cdot\,,\cdot\right\\}^{\phi}$ considered above
can be restricted to these surfaces, but a Nambu-Poisson structure with
$H_{2}^{\prime}$ and $H_{3}^{\prime}$ as Casimirs can be so restricted. It
leads to the quadratic Poisson bracket

$\left\\{w_{n-1},w_{n}\right\\}=w_{n-1}w_{n}$ (4.12)

with respect to which (3.45) is a Poisson map. The above bracket can also be
derived via reduction of a presymplectic structure for the tau functions, by
regarding (3.48) as a mutation in a cluster algebra [18].

To see how the quadratic bracket arises here, recall from [52] that, for some
fixed choice of $m$-form $\Omega$, a Nambu-Poisson bracket of order $m$ is
defined by

$\\{f_{1},f_{2},\ldots,f_{m}\\}\,\Omega=\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}\wedge\cdots\wedge\mathrm{d}f_{m}.$

In the case at hand (taking $m=4$), observe that the unreduced version (3.14)
of the Volterra map $\varphi$ on ${\mathbb{C}}^{4}_{w}$ with coordinates
$(w_{-1},w_{0},w_{1},w_{2})$ preserves the rational volume form

$\Omega=\frac{w_{0}w_{1}}{w_{0}-w_{1}}\,\mathrm{d}w_{-1}\wedge\mathrm{d}w_{0}\wedge\mathrm{d}w_{1}\wedge\mathrm{d}w_{2},\qquad\varphi^{*}(\Omega)=\Omega.$

Then the corresponding Nambu-Poisson bracket defines a Poisson bracket on
${\mathbb{C}}^{4}_{w}$ according to

$\\{f_{1},f_{2}\\}:=\\{f_{1},f_{2},H_{2}^{\prime},H_{3}^{\prime}\\},$

which by construction has Casimirs $H_{2}^{\prime}$ and $H_{3}^{\prime}$, and
restricts to (4.12) on the surface $H_{2}^{\prime}=c_{2}^{\prime}$,
$H_{3}^{\prime}=c_{3}^{\prime}$. The same Nambu-Poisson bracket also produces
any member of the pencil $\left\\{\cdot\,,\cdot\right\\}^{\phi}$ for $g=1$: in
particular, the Poisson structures $\left\\{\cdot\,,\cdot\right\\}^{x}$ and
$\left\\{\cdot\,,\cdot\right\\}^{x^{2}}$ arise in this way, by taking (up to
scaling) $\\{f_{1},f_{2},H_{1},H_{3}\\}$ and $\\{f_{1},f_{2},H_{1},H_{2}\\}$,
respectively. However, this construction does not extend to $g>1$ in a
straightforward manner.

######  Example 4.5.

Continuing Example 3.7, taking $\phi=x^{3}$ when $g=2$ we find that the
Poisson brackets on $M_{2}$ (with coordinates
$p_{1},p_{2},q_{1},q_{2},r_{1},r_{2},r_{3}$) take the form

$\displaystyle\\{r_{1},r_{3}\\}^{x^{3}}=r_{1},$
$\displaystyle\\{r_{2},r_{3}\\}^{x^{3}}=r_{2},\qquad\qquad$
$\displaystyle\\{p_{1},q_{2}\\}^{x^{3}}=2=\\{p_{2},q_{1}\\}^{x^{3}},$
$\displaystyle\\{p_{2},q_{2}\\}^{x^{3}}=q_{1},$
$\displaystyle\\{p_{2},r_{3}\\}^{x^{3}}=-r_{2},\quad\qquad$
$\displaystyle\\{p_{1},r_{3}\\}^{x^{3}}=-r_{1}=\\{p_{2},r_{2}\\}^{x^{3}},$
$\displaystyle\\{q_{1},r_{2}\\}^{x^{3}}=2=\\{q_{2},r_{1}\\}^{x^{3}},$
$\displaystyle\\{q_{1},r_{3}\\}^{x^{3}}=2p_{1}-q_{1},\,\quad$
$\displaystyle\\{q_{2},r_{2}\\}^{x^{3}}=2p_{1},\quad\\{q_{2},r_{3}\\}^{x^{3}}=2p_{2}-q_{2},$

where only the non-zero brackets are specified here. By restricting this
Poisson structure to $M_{g}^{c}\cap(H_{3}=c_{3})$, we find that (up to an
overall factor of $1/2$) it coincides with Poisson bracket (1.5) for the
(P.iv) map that was derived from a discrete Lagrangian in [21], where the
parameters $\nu,a,b$ are fixed according to (3.53) in terms of the values
$c_{1},c_{2},c_{3}$ of the 3 Casimirs of
$\left\\{\cdot\,,\cdot\right\\}^{x^{3}}$.

###  4.2 The generic fibers of the momentum map

We will now give an algebro-geometric description of the generic fibers of the
momentum map $\mu:M_{g}\to{\mathbb{C}}[x]$, which we recall is given by
$\mu({\cal P},{\cal Q},{\cal R})={\cal P}^{2}+{\cal Q}{\cal R}$. Precisely, we
will describe the fiber over any polynomial $f\in{\mathbb{C}}[x]$ of degree
$2g+1$, without multiple roots, and satisfying $f(0)=1$. Notice that such
polynomials are exactly those $f$ in the image of $\mu$ for which $y^{2}=f(x)$
defines a non-singular affine curve $\Gamma_{f}$ of genus $g$; we use
$\bar{\Gamma}_{f}$ to denote the completion of the latter, which can be
thought of as a compact Riemman surface.

######  Proposition 4.6.

Let $f$ be polynomial of degree $2g+1$, without multiple roots, such that
$f(0)=1$. Denote by $\Gamma_{f}$ the non-singular affine curve of genus $g$,
defined by $y^{2}=f(x)$, with $\bar{\Gamma}_{f}$ being its completion. Then
$\mu^{-1}(f)$ is isomorphic to an affine part of the Jacobian variety of
$\bar{\Gamma}_{f}$ minus three translates of the theta divisor,

$\mu^{-1}(f)\cong\operatorname{Jac}(\bar{\Gamma}_{f})\setminus(\Theta\cup\Theta_{+}\cup\Theta_{-})\;.$
(4.13)

The $(-1)$-involution on $\operatorname{Jac}(\bar{\Gamma}_{f})$ leaves
$\Theta$ invariant and permutes $\Theta_{+}$ and $\Theta_{-}$.

###### Proof.

Let $({\cal P},{\cal Q},{\cal R})\in\mu^{-1}(f)$, so that

${\cal P}^{2}(x)+{\cal Q}(x){\cal R}(x)=f(x)\;,\quad\deg{\cal P},\deg{\cal
Q},\deg{\cal R}-1\leqslant g\;,\quad({\cal P}(0),{\cal Q}(0),{\cal
R}(0))=(1,2,0)\;.$ (4.14)

Since $f$ has degree $2g+1$, $\deg{\cal Q}=g$ and $\deg{\cal R}=g+1$; it also
implies that $\bar{\Gamma}_{f}$ is obtained from $\Gamma_{f}$ by adding a
single point, which we denote by $\infty$. The hyperelliptic involution on
$\bar{\Gamma}_{f}$ is denoted by $\imath$; it fixes $\infty$ and sends
$(x,y)\in\Gamma_{f}$ to $(x,-y)$. To $({\cal P},{\cal Q},{\cal R})$ we
associate a divisor $\sum_{i=1}^{g}(x_{i},y_{i})-g\infty$ on
$\bar{\Gamma}_{f}$ as follows: $x_{1},x_{2},\dots,x_{g}$ are the roots of
${\cal Q}(x)$ and $y_{i}:={\cal P}(x_{i})$ for $i=1,\dots,g$. It is indeed a
divisor on $\bar{\Gamma}_{f}$ since for $j=1,\dots,g$,

$y_{j}^{2}-f(x_{j})={\cal P}^{2}(x_{j})-({\cal P}^{2}(x_{j})+{\cal
Q}(x_{j}){\cal R}(x_{j}))=0\;.$

Of course, $(x_{i},y_{i})\neq\infty$ for all $i$. Notice that when ${\cal
Q}(x)$ has multiple roots, say $x_{1}=\dots=x_{k}$, then
$y_{1}=y_{2}=\dots=y_{k}$. We show by contradiction that if $x_{i}$ is a root
of ${\cal Q}(x)$ and $y_{i}=0$ (so that $x_{i}$ is also a root of ${\cal
P}(x)$), then $x_{i}$ is a simple root of ${\cal Q}(x)$. Indeed, if $x_{i}$ is
a multiple root of ${\cal Q}(x)$ and is also a root of ${\cal P}(x)$, then
$x_{i}$ is a multiple root of $f(x)={\cal P}(x)^{2}+{\cal Q}(x){\cal R}(x)$,
so that $f$ is not square-free, contrary to the assumptions. The upshot is
that the obtained divisors are of the form $\sum_{i=1}^{g}P_{i}-g\infty$,
where $P_{i}\in\Gamma_{f}$ for $i=1,\dots,g$ and $P_{i}\neq\imath(P_{j})$ when
$i\neq j$. It is well-known that two such divisors are linearly equivalent if
and only if they are the same; also, that none of these divisors are
equivalent to a divisor of the form $\sum_{i=1}^{g-1}Q_{i}-(g-1)\infty$, with
$Q_{i}\in\bar{\Gamma}_{f}$ for $i=1,\dots,g-1$. Since
$\operatorname{Jac}(\bar{\Gamma}_{f})$ is the group of degree zero divisors on
$\bar{\Gamma}_{f}$, modulo linear equivalence, this shows that the map
$\mu^{-1}(f)\to\operatorname{Jac}(\bar{\Gamma}_{f})$, which associates to
$({\cal P},{\cal Q},{\cal R})$ the divisor class
$\left[\sum_{i=1}^{g}P_{i}-g\infty\right]$, is injective. This map is of
course not surjective. In order to determine the image, let

$\displaystyle\Theta$
$\displaystyle:=\left\\{\left[\sum_{i=1}^{g-1}P_{i}-(g-1)\infty\right]\mid\forall
i\ P_{i}\in\bar{\Gamma}_{f}\right\\}\;,$ $\displaystyle\Theta_{+}$
$\displaystyle:=\left\\{\left[(0,1)+\sum_{i=1}^{g-1}P_{i}-g\infty\right]\mid\forall
i\ P_{i}\in\bar{\Gamma}_{f}\right\\}=\Theta+[(0,1)-\infty]\;,$
$\displaystyle\Theta_{-}$
$\displaystyle:=\left\\{\left[(0,-1)+\sum_{i=1}^{g-1}P_{i}-g\infty\right]\mid\forall
i\ P_{i}\in\bar{\Gamma}_{f}\right\\}=\Theta+[(0,-1)-\infty]\;.$

The first one is the theta divisor and the other two are translates of it.
Notice that $\imath(\Theta)=\Theta$ and $\imath(\Theta_{+})=\Theta_{-}$, since
$\imath(\infty)=\infty$. As we already said, the image contains no point of
the form $\left[\sum_{i=1}^{g-1}P_{i}-(g-1)\infty\right]$, i.e., is disjoint
from $\Theta$. Since $P_{i}=(x_{i},y_{i})$ where $x_{i}$ is a root of ${\cal
Q}(x)$ and since ${\cal Q}(0)=2$, it is clear that $x_{i}\neq 0$, so that
every $P_{i}=(x_{i},y_{i})$ is different from $(0,1)$ and from $(0,-1)$; hence
the image is also disjoint from $\Theta_{+}$ and $\Theta_{-}$. Take now any
point in
$\operatorname{Jac}(\bar{\Gamma}_{f})\setminus(\Theta\cup\Theta_{+}\cup\Theta_{-})$.
It can as above be written uniquely as
$\left[\sum_{i=1}^{g}P_{i}-g\infty\right]$ with
$P_{i}\notin\\{\infty,(1,0),(-1,0)\\}$ and $P_{i}\neq\imath(P_{j})$ for all
$i\neq j$. When all $P_{i}=(x_{i},y_{i})$ are different, there is a unique
polynomial ${\cal Q}(x)$ whose roots are the $x_{i}$ and with ${\cal Q}(0)=2$,
and there is a unique polynomial ${\cal P}(x)$ of degree $g$, with ${\cal
P}(x_{i})=y_{i}$ for $i=1,\dots,g$ and ${\cal P}(0)=1$: setting
$(x_{0},y_{0})=(0,1)$, they are given by

${\cal Q}(x)=2\prod_{i=1}^{g}\left(1-\frac{x}{x_{i}}\right)\;,\qquad{\cal
P}(x)=\sum_{i=0}^{g}y_{i}\prod_{j\neq i}\frac{x-x_{j}}{x_{i}-x_{j}}\;.$ (4.15)

This also works in the limiting case when some of the $P_{i}$ are the same
upon adding in the definition of ${\cal P}(x)$ a tangency condition (see [38,
page 3.18]), which assures that $f(x)-{\cal P}^{2}(x)$ is divisible by ${\cal
Q}(x)$. The quotient is a polynomial ${\cal R}(x)$ of degree $g$ satisfying
$f(x)={\cal P}(x)^{2}+{\cal Q}(x){\cal R}(x)$; by uniqueness, $\mu({\cal
P},{\cal Q},{\cal R})=\left[\sum_{i=1}^{g}P_{i}-g\infty\right]$, as required.
∎

######  Example 4.7.

We specialize Proposition 4.6 to the case of $g=1$. Notice that in this case
we should say _elliptic_ rather than _hyperelliptic_ , and in this case the
(hyper-) elliptic involution is not unique. Another peculiarity about $g=1$ is
the well-known fact that a complete non-singular genus one curve (i.e., any
compact elliptic Riemann surface) is isomorphic to its Jacobian, a fact that
we will be able to illustrate here. Let $f(x)=1+c_{1}x+c_{2}x^{2}+c_{3}x^{3}$
be an arbitrary polynomial of degree 3 with no multiple roots. We investigate
$\mu^{-1}(f)$, which is the affine curve defined by the following equations,
which are found by expressing that ${\cal P}^{2}(x)+{\cal Q}(x){\cal
R}(x)=f(x)$:

$\displaystyle 2(p_{1}+r_{1})=c_{1}\;,$ $\displaystyle
p_{1}^{2}+q_{1}r_{1}+2r_{2}=c_{2}\;,$ $\displaystyle q_{1}r_{2}=c_{3}\;.$

The curve $\mu^{-1}(f)$ can be equivalently written as a plane algebraic curve
by first eliminating $r_{1}$ from the first two equations and then $r_{2}$
from the two remaining equations; writing $p_{1}$ and $q_{1}$ simply as $p$
and $q$, the final equation takes the simple form

$\mu^{-1}(f)\ :\
\frac{pq}{2}(p-q)+c_{1}\frac{q^{2}}{4}-c_{2}\frac{q}{2}+c_{3}=0\;.$ (4.16)

It is easy to see that, thanks to the conditions on $f$, this curve is non-
singular, just like $\Gamma_{f}$. In fact, if we denote the left-hand side of
(4.16) by $F$ then a singular point $(q_{0},p_{0})$ of $\mu^{-1}(f)$ must
satisfy

$\frac{\partial F}{\partial
p}(q_{0},p_{0})=\frac{q_{0}}{2}(q_{0}-2p_{0})=0\;,\quad\frac{\partial
F}{\partial
q}(q_{0},p_{0})=\frac{1}{2}(p_{0}^{2}-2p_{0}q_{0}+c_{1}q_{0}-c_{2})=0\;.$
(4.17)

Since $q_{0}\neq 0$ (as $F(q_{0},p_{0})=0$ and $c_{3}\neq 0$), we get
$q_{0}=2p_{0}$; substituted in $F(q_{0},p_{0})=0$ and in the second equation
of (4.17) we get

$p_{0}^{3}-c_{1}p_{0}^{2}+c_{2}p_{0}-c_{3}=0\;,\quad
3p_{0}^{2}-2c_{1}p_{0}+c_{2}=0\;,$

which can be written as $f(-p_{0})=f^{\prime}(-p_{0})=0$. Since $f$ has no
multiple roots these equations have no common solution, which shows that
$\mu^{-1}(f)$ is non-singular.

To see that $\mu^{-1}(f)$ and $\Gamma_{f}$ are birationally isomorphic, it
suffices to consider the following rational map:

$q=-\frac{2}{x}\;,\ p=\frac{y-1}{x}\;,\quad\hbox{ with inverse }\quad
x=-\frac{2}{q}\;,\ y=\frac{q-2p}{q}\;.$ (4.18)

Notice that, despite the appearance of $q$ in the denominator, the inverse map