# Calligraphs and sphere realizations

Matteo Gallet⋄ Georg Grasegger∘ Niels Lubbes∗ Josef Schicho

###### Abstract

We introduce a recursive procedure for computing the number of realizations of
a minimally rigid graph on the sphere up to rotations. We accomplish this by
combining two ingredients. The first is a framework that allows us to think of
such realizations as of elements of a moduli space of stable rational curves
with marked points. The second is the idea of splitting a minimally rigid
graph into two subgraphs, called calligraphs, that admit one degree of freedom
and that share only a single edge and a further vertex. This idea has been
recently employed for realizations of graphs in the plane up to isometries.
The key result is that we can associate to a calligraph a triple of natural
numbers with a special property: whenever a minimally rigid graph is split
into two calligraphs, the number of realizations of the former equals the
product of the two triples of the latter, where this product is specified by a
fixed quadratic form. These triples and quadratic form codify the fact that we
express realizations as intersections of two curves on the blowup of a sphere
along two pairs of complex conjugate points.

††footnotetext:  ⋄ The project leading to this paper started when Matteo
Gallet was supported by the Austrian Science Fund (FWF): P33003. At the
moment, he is member of the Gruppo Nazionale Strutture Algebriche, Geometriche
e le loro Applicazioni (GNSAGA) of the Istituto Nazionale di Alta Matematica
(INdAM).
∘ Supported by the Austrian Science Fund (FWF): P31888.
∗ Supported by the Austrian Science Fund (FWF): P33003.

##  1 Introduction

Given a graph without loops or multiedges, and a choice of length for each of
its edges, one can ask in how many ways one can realize the graph in the plane
so that the distance between two vertices connected by an edge coincides with
the given edge length. Clearly, once a realization compatible with the edge
length assignment is found, applying any isometry of the plane yields another
realization compatible with the same edge length assignment. Hence, it is
reasonable to consider realizations up to isometries. In this context, one
family of graphs emerges as particularly interesting, namely those graphs that
admit only finitely many realizations, up to isometries, compatible with
algebraically independent edge lengths. These graphs are called _minimally
rigid_. If we allow complex coordinates for the points of the realizations,
then the number of realizations, up to isometries, of a minimally rigid graph
$G$ is the same, regardless of the choice of algebraically independent edges;
it is denoted by $\mathbf{c}_{2}(G)$. Similarly as in the plane, one can
consider the problem of computing the number of realizations on the sphere up
to direct isometries, i. e., rotations. In this case, the number is denoted by
$\mathbf{c}_{S^{2}}(G)$. It turns out that graphs that are minimally rigid for
the plane are also minimally rigid for the sphere; see [Pog73, SW07, Izm09,
EJN+19].

Computing the number of realizations, up to isometries, of a minimally rigid
graph, or obtaining bounds for this number, is a question that has attracted
recent interest, as witnessed by several works; see, for example, [BS04, ST10,
ETV10, CGG+18b, JO19, GKT20, BET21, BELT21, BET23, SHK+21].

Our result In this paper, we combine the technique developed in [GEHL22b] with
the framework from [GGS20] to provide new ways for counting complex
realizations of minimally rigid graphs on the sphere up to rotations.

Previous work The technique developed in [GEHL22b], which can be applied to
realizations in the plane, splits a minimally rigid graph into two
_calligraphs_ that share an edge and a further vertex. A calligraph is a graph
that, differently from a rigid one, admits a one-dimensional set of
realizations up to isometries when algebraically independent edge lengths are
chosen; see Figure 1.

Figure 1: Examples of calligraphs.

The realizations of the minimally rigid graph are then given by the
intersections of the two curves, called _coupler curves_ , obtained by tracing
the common vertex of the two calligraphs. One could then think that it would
be enough to understand the degree of the two coupler curves, and to use
Bézout’s theorem to compute their number of intersections. However, this turns
out not to provide the correct number, which means that we are in presence of
intersections “at infinity”. The key observation is that, even if we change
the edge lengths of the minimally rigid graph, the coupler curves of the two
calligraphs may always intersect at two points at infinity. Even more: these
curves may pass through these two points with the same slope, which in
classical algebraic geometry language can be expressed by saying that they
further pass through four more points, which are “infinitely near” to the two
initial ones. Summing up, whatever minimally rigid graph we start from, the
coupler curves of two calligraphs that form a split may intersect at six fixed
points, regardless of the choice of edge lengths.

A standard technique in algebraic geometry is to change the space where the
curves lie in such a way that these unavoidable intersections are taken into
account by the geometry of the space itself. Employing this approach in the
specific situation of calligraphs amounts to moving from the plane to a
particular surface (i. e., the blowup of $\mathbb{P}^{2}$ at the fixed
intersections) and to use its intersection theory. In the plane, as far as
intersections are concerned, the only relevant information about a curve is
its degree: this is a way to re-state Bézout’s theorem. In other words, the
intersection theory of the plane attaches to every curve a single integer and
to the intersection of two curves the product of these numbers. In general,
the geometry of a surface specifies how to attach a tuple of integers to each
curve lying on it and a quadratic form to the whole surface: the quadratic
form specifies how to “multiply” two tuples in order to get the number of
intersections of the two corresponding curves.

In the case of calligraphs, the authors of [GEHL22b] attach a triple of
integers, called the _class_ , to each calligraph. The problem of computing
the number $\mathbf{c}_{2}(G)$ of realizations in the plane up to isometries
for a minimally rigid graph $G$ then reduces to computing the class $(a,b,c)$
of any given calligraph $H$: in fact, once we split the graph $G$ into two
calligraphs, it is enough to multiply the corresponding classes according to a
specific quadratic form. The solution to the problem of computing the class is
to construct, out of the calligraph $H$, three minimally rigid graphs $H_{1}$,
$H_{2}$, and $H_{3}$. Each of these three graphs splits into $H$ and one of
three “basic calligraphs” with at most four vertices, for which the classes
are known; see Figure 2. Computing the number of realizations of $H_{1}$,
$H_{2}$, and $H_{3}$ allows us to recover $(a,b,c)$. This enables us to set up
a recursive scheme for the computation of $\mathbf{c}_{2}(G)$.

$L$$R$$C_{v}$ Figure 2: The three “basic calligraphs”, whose classes are known
to be $(1,1,0)$ for $L$, $(1,0,1)$ for $R$, and $(2,0,0)$ for $C_{v}$.

We see in this paper that, maybe surprisingly, the number
$\mathbf{c}_{S^{2}}(G)$ of realizations on the sphere up to rotations can be
computed in the same way. However, the proof technique of this result is
fundamentally different from the one of [GEHL22b] and, additionally, it avoids
several technical issues that are present in the plane case. This is due to
the fact that, in the case of the sphere, we have a well-behaved ambient space
for the sets of realizations of graphs up to sphere rotations, which are
called _configurations_. In fact, the paper [GGS20] explains how we can think
of configurations of a graph on the sphere as elements of the so-called
_moduli space $\overline{\mathscr{M}}_{0,2n}$ of rational stable curves with
marked points_, in which two marked points correspond to one vertex of the
graph and $n$ is the number of vertices. Here, a _rational stable curve_ is a
connected union of copies of the projective line that intersect each other
transversely in at most one point. The moduli space
$\overline{\mathscr{M}}_{0,2n}$ is a geometric object (more precisely, a
smooth projective variety) whose elements correspond to equivalence classes of
rational stable curves on which $2n$ points (the markings) are specified, up
to the action of the automorphisms of the projective line. The paper [GGS20]
provides an algorithm to compute the number of configurations of a minimally
rigid graph by exploiting the structure of the Chow ring of the moduli space
$\overline{\mathscr{M}}_{0,2n}$. The Chow ring of an algebraic variety is a
ring whose elements, roughly speaking, are the subvarieties of the given
variety and in which the product of two elements corresponds to the
intersection of subvarieties. In other words, the Chow ring encodes the
intersection theory of an algebraic variety. In our case, for each graph $G$
we can consider the subvariety $\mathscr{K}_{G}^{\Lambda}$ of configurations
of $G$ with an assignment $\Lambda$ of edge lengths. This determines an
element $K_{G}$ in the Chow ring of $\overline{\mathscr{M}}_{0,2n}$, which
does not depend on $\Lambda$. When $G$ is minimally rigid and $\Lambda$ is
given by algebraically independent values, the subvariety
$\mathscr{K}_{G}^{\Lambda}$ is a collection of points, one for each
configuration of $G$. The cardinality of $\mathscr{K}_{G}^{\Lambda}$ then
equals $\mathbf{c}_{S^{2}}(G)$. In turn, this cardinality is equal to the so-
called _degree_ of $K_{G}$; we can then perform the computation of
$\mathbf{c}_{S^{2}}(G)$ by working only in the Chow ring, since in addition we
have that $K_{G}$ can be written as a product of particular elements
corresponding to codimension one subvarieties (the so-called _divisors_), one
for each of the edges of the graph. This allows us to set up a recursive
procedure for the computation of the degree of $K_{G}$.

The technique The goal of this paper is to enhance the algorithm just
described to compute the number $\mathbf{c}_{S^{2}}(G)$ by the same procedure
of splitting a minimally rigid graph into calligraphs explained in the case of
the plane.

A powerful feature of moduli spaces of stable curves with marked points is
that they come equipped with _forgetful morphisms_ of the form
$\overline{\mathscr{M}}_{0,2n}\longrightarrow\overline{\mathscr{M}}_{0,2m}$
whenever $m\leq n$, which forget a certain number of marked points of a
rational stable curve. Via these morphisms, we can relate the configuration
space of a minimally rigid graph $G$ to those of two calligraphs $G_{1}$ and
$G_{2}$ constituting a split of $G$ and to the one of the graph $G_{12}$
constituted of the common edge and vertex of $G_{1}$ and $G_{2}$. All these
morphisms are both proper and flat, which means that they are most suited for
operating between Chow rings. Using them, we can express $K_{G}$ as a product
of three contributions, coming from $G_{1}$, $G_{2}$, and $G_{12}$ (these
contributions are pullbacks, under suitable forgetful morphisms, of classes
determined by the corresponding graphs in the Chow rings of the appropriate
moduli spaces). By elementary computations in intersection theory, we show
that we can “pushforward” all these computations to the moduli space
corresponding to $G_{12}$. There, what we get is a product of two quantities
depending on $G_{1}$ and $G_{2}$. This product reduces to computing the
intersection of two curves lying on the configuration space of $G_{12}$. It
turns out that the latter is isomorphic to the blow up of
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ at four points. Therefore, in terms of
intersection theory, each of the two curves is identified by six integers,
which actually become three because these curves have a real structure. Hence,
we are in the same setting as in the case of realizations in the plane
described before and so we can apply the same recursive strategy, attaching to
each calligraph a triple of integer numbers, which we call the _$S^{2}$
-class_.

Structure of the paper The paper is structured as follows. In Section 2, we
recall the notions of minimally rigid graph, of calligraph, and of split of a
minimally rigid graph as the union of two calligraphs that share a common edge
and a common vertex. At the end, we state the main result of this paper,
Theorem 2.6. In Section 3, we briefly summarize how to encode the realizations
of a graph on the sphere up to rotations as a subvariety of a moduli space of
rational stable curves with marked points. Section 4 and Section 5 are the
technical core of the paper. Section 4 proves how it is possible to reduce the
computation of the degree of the configuration space of a minimally rigid
graph to the degree of the intersection of three surfaces in a three-
dimensional moduli space. Section 5 provides an explicit description of one of
these surfaces, which turns out to be isomorphic to the blowup of
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ at four points. Section 6 combines the
results from the previous two sections to define the $S^{2}$-class of a
calligraph and to show that the product of the two $S^{2}$-classes in a
calligraphic split of a minimally rigid graph gives the number of realizations
of the latter on the sphere up to rotations, thus proving Theorem 2.6. Section
7 describes the recursive procedure that we obtain and reports some proof-of-
concept comparison between the timings of the original version of the
algorithm from [GGS20] against the “enhanced” version using the recursion of
this paper.

Acknowledgments. The authors would like to thank Paolo Aluffi, Valentina
Beorchia, Renzo Cavalieri, Boulos El Hilany, Neithalath Mohan Kumar, Danilo
Lewański, Andrea Ricolfi, and Elaine Wong for useful discussions. They would
like to thank especially Paolo Aluffi for pointing out to us the precise
references that led to the proof of Lemma 4.5.

##  2 Minimally rigid graphs and calligraphs

We start by introducing the concepts of _minimally rigid graph_ and of
_calligraph_. The first one is a well-established and studied notion, while
the second one has recently been introduced in [GEHL22b]. Once their edge
lengths have been fixed, minimally rigid graphs can be drawn in finitely many
non-congruent ways in the plane, while calligraphs allow a flex. We define the
notion of realization of a graph in the complex plane (and later on the
complex sphere) and of their congruence. This allows us to speak about the
number of realizations of a minimally rigid graph up to isometries. After
defining what we mean by a _split_ of a minimally rigid graph into two
calligraphs, we state the main theorem of the paper (Theorem 2.6), which is
the theoretical bedrock of the recursive procedure we propose in Section 7.
Notice that, in this paper, all considered graphs are _simple_ , namely
without loops or multiedges.

A _(complex) realization_ of a graph $G=(V,E)$ is a function $\rho\colon
V\longrightarrow\mathbb{C}^{2}$. Given a choice of _edge lengths_
$\lambda=(\lambda_{e})_{e\in E}$ with $\lambda_{e}\in\mathbb{R}_{>0}$ for all
$e\in E$, we say that a realization $\rho$ _induces_ $\lambda$ if

$\bigl{\langle}\rho(u)-\rho(v),\rho(u)-\rho(v)\bigr{\rangle}=\lambda_{e}$

for all edges $e=\\{u,v\\}$ in $E$, where
$\left\langle\cdot,\cdot\right\rangle$ is the quadratic form associated to the
identity matrix. Notice that, although realizations of vertices can have
complex coordinates, we always ask edge lengths to be non-zero _real_ numbers.
This is of crucial importance for the development of our results, because from
this assumption it follows that all the objects that we are going to define
are endowed with a real structure. Two realizations are considered _congruent_
up to (complex) isometries if there exists a matrix
$A\in\mathrm{SO}_{2}(\mathbb{C})$ and a vector $b\in\mathbb{C}^{2}$ such that
the map $v\mapsto Av+b$ sends one realization to the other.

######  Definition 2.1.

A graph $G=(V,E)$ is called _minimally rigid_ if it admits only finitely many,
but not zero, (complex) realizations in the plane for a general choice of edge
lengths, up to (complex) isometries. This number of realizations is denoted by
$\mathbf{c}_{2}(G)$.

A theorem of Pollaczek-Geiringer [Pol27] and Laman [Lam70] characterizes
minimally rigid graphs in combinatorial terms.

######  Theorem 2.2 (Pollaczek-Geiringer, Laman).

A graph $G=(V,E)$ is _minimally rigid_ if and only if

  * $\triangleright$

$|E|=2|V|-3$ and

  * $\triangleright$

$|E^{\prime}|\leq 2|V^{\prime}|-3$ for every subgraph
$G^{\prime}=(V^{\prime},E^{\prime})$ with at least two vertices.

We can adapt all notions we introduced so far in order to define realizations
of graphs on the (complex) sphere

$S^{2}_{\mathbb{C}}:=\bigl{\\{}(x,y,z)\in\mathbb{C}^{3}\,\mid\,x^{2}+y^{2}+z^{2}=1\bigr{\\}}\,.$

Hence, we can speak of minimally rigid graphs on the sphere and if $G$ is one
of them, one denotes by $\mathbf{c}_{S^{2}}(G)$ the number of its (complex)
realizations on $S^{2}_{\mathbb{C}}$ up to (complex) rotations. One then finds
out that a graph is minimally rigid on the sphere if and only if it is so in
the plane; see [Pog73, SW07, Izm09, EJN+19].

The paper [GGS20] provides a recursive procedure for the computation of
$\mathbf{c}_{S^{2}}(G)$ and the goal of this paper is to set up a technique
that provides sometimes very sensible speed ups, mimicking the one of
[GEHL22b]. To do so, we introduce another type of graphs, called
_calligraphs_. These graphs, roughly speaking, admit a one-dimensional set of
realizations up to isometries and are the key players in the recursive
procedure.

######  Definition 2.3.

A _calligraph_ is a graph $H=(V,E)$ together with a choice of an edge
$\bar{e}=\\{\bar{u}_{1},\bar{u}_{2}\\}$ and of a vertex $\bar{v}$ such that

  * $\triangleright$

$|E|=2|V|-4$ and

  * $\triangleright$

$(V,E\cup\\{\\{\bar{v},\bar{u}_{1}\\}\\})$ is rigid or
$(V,E\cup\\{\\{\bar{v},\bar{u}_{2}\\}\\})$ is rigid.

######  Notation.

Throughout this paper, if $(H,\bar{e},\bar{v})$ is a calligraph, we always
suppose that $\bar{e}=\\{1,2\\}$ and $\bar{v}=0$. Hence, from now on we always
refer to a calligraph simply by its underlying graph.

The key point of the recursive technique that we establish is to split a
minimally rigid graph into a union of two calligraphs with a common edge and a
further common vertex.

######  Definition 2.4.

A _split_ of a graph $G=(V,E)$ is a pair $(G_{1},G_{2})$ of subgraphs of $G$
such that if we write $G_{i}=(V_{i},E_{i})$, then

  * $\triangleright$

$V=V_{1}\cup V_{2}$ and $E=E_{1}\cup E_{2}$;

  * $\triangleright$

$V_{1}\cap V_{2}=\\{0,1,2\\}$ and $E_{1}\cap E_{2}=\\{\\{1,2\\}\\}$.

If both $G_{1}$ and $G_{2}$ are calligraphs, then $(G_{1},G_{2})$ is called a
_calligraphic split_.

As already mentioned in Section 1, three among all calligraphs are going to
play a special role in our recursion.

######  Definition 2.5.

We define three “basic calligraphs”, namely

  * $\triangleright$

$L:=\bigl{(}\\{0,1,2\\},\\{\\{1,2\\},\\{0,1\\}\\}\bigr{)}$,

  * $\triangleright$

$R:=\bigl{(}\\{0,1,2\\},\\{\\{1,2\\},\\{0,2\\}\\}\bigr{)}$, and

  * $\triangleright$

for $v\in\mathbb{N}$ with $v>2$, we set
$C_{v}:=\bigl{(}\\{0,1,2,v\\},\\{\\{1,2\\},\\{0,v\\},\\{1,v\\},\\{2,v\\}\\}\bigr{)}$.

These are the three graphs from Figure 1.

We conclude this section by stating the main theorem of the paper. This result
shows that we can attach to each calligraph a triple of integer numbers so
that, once we have a calligraphic split of a minimally rigid graph $G$, the
number $\mathbf{c}_{S^{2}}(G)$ can be computed from the two triples of the
split.

######  Theorem 2.6.

There exist a function

$[\,\\_\,]\colon\\{\text{calligraphs}\\}\longrightarrow\mathbb{Z}^{3}$

and a quadratic form

$\begin{array}[]{rccc}\\_\cdot\\_\colon&\mathbb{Z}^{3}\times\mathbb{Z}^{3}&\longrightarrow&\mathbb{Z}\\\
&(a_{1},b_{1},c_{1}),(a_{2},b_{2},c_{3})&\mapsto&2(a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2})\end{array}$

such that, if $G$ is a minimally rigid graph and $(G_{1},G_{2}$) is a
calligraphic split of $G$, then the number of complex realizations of $G$ on
the sphere up to complex rotations, i. e., the quantity
$\mathbf{c}_{S^{2}}(G)$, is $[G_{1}]\cdot[G_{2}]$. The triple $[H]$ is called
the _$S^{2}$ -class_ of $H$.

Moreover, the $S^{2}$-classes of the three “basic calligraphs” from Definition
2.5 are

$[L]=(1,1,0)\,,\quad[R]=(1,0,1)\,,\quad[C_{v}]=(2,0,0)\,.$

######  Remark 2.7.

As explained in Section 6, one could also choose $(6,4,4)$ for the
$S^{2}$-class of $C_{v}$ and the rest of Theorem 2.6 would stay the same.

Theorem 2.6 is the spherical analogue of [GEHL22b, Theorem I]. However, the
proof strategy that we employ here is completely different from the one of
[GEHL22b]. We believe it would be interesting to have a “moduli-based” proof
of this result also in the case of the plane, but this remains an open
problem.

##  3 Graph sphere configurations in moduli spaces

In this section, we briefly recall the construction of [GGS20] that assigns to
each graph $G$ with $n$ vertices together with a choice of edge lengths a
subvariety of the moduli space of rational stable curves with $2n$ marked
points. This subvariety is called the _configuration space_ of the graph and
it is defined in such a way that an open subset of it is in bijection with the
congruence classes of realizations of $G$ on the sphere inducing the given
edge lengths up to (complex) isometries. The idea is that:

  * $\triangleright$

the projective closure of the complex sphere $S^{2}_{\mathbb{C}}$ intersects
the plane at infinity in a smooth conic, called the _absolute conic_ ;

  * $\triangleright$

through each point on the complex sphere $S^{2}_{\mathbb{C}}$ pass two lines,
each of which intersects the absolute conic in a point.

Hence, to a realization of $n$ points on the sphere, we can associate $2n$
points on the absolute conic. Then, one notices that two realizations of $n$
points are (complex) rotation-congruent if and only if the corresponding two
$2n$ tuples of points on the absolute conic differ by a projective
automorphism. Thus, realizations of $n$ vertices up to (complex) rotations can
be encoded as elements of the moduli space of rational stable curves with $2n$
markings, which is usually denoted by $\overline{\mathscr{M}}_{0,2n}$.
Moreover, the cosine of the angle between two points on the sphere can be
expressed as the cross-ratio between the four corresponding points on the
absolute conic. For the precise details, we refer to [GGS20] and the
references therein.

Notice that the real projective automorphisms of $S^{2}_{\mathbb{C}}$ that
preserve the absolute conic correspond to exactly rotations of
$S^{2}_{\mathbb{R}}$. Moreover, the real structure of $S^{2}_{\mathbb{C}}$
given by componentwise complex conjugation swaps the two lines passing through
each point of $S^{2}_{\mathbb{R}}$. This induces a real structure on the
moduli space of rational stable curves with $2n$ markings that swaps two
$n$-tuples of markings.

######  Definition 3.1.

For a set $V$, we write $\overline{\mathscr{M}}_{0,2V}$ to denote the moduli
space $\overline{\mathscr{M}}_{0,2|V|}$ of rational stable curves with $2|V|$
marked points, where the marked points are indicated by $\\{P_{v}\\}_{v\in
V}\cup\\{Q_{v}\\}_{v\in V}$. The moduli space $\overline{\mathscr{M}}_{0,2V}$
is a smooth projective variety.

With this notation, the real structure on $\overline{\mathscr{M}}_{0,2V}$
described before swaps the $P$-marked points with the $Q$-marked points.

To encode the computation of the cross-ratio in the framework of moduli
spaces, we introduce a particular type of morphisms.

######  Definition 3.2.

Let $G=(V,E)$ be a graph and let $e\in E$ with $e=\\{v,w\\}$. We define
$\pi_{e}$ to be the _edge forgetful morphism_

$\pi_{e}\colon\overline{\mathscr{M}}_{0,2V}\longrightarrow\overline{\mathscr{M}}_{0,2e}$

that forgets all the marked points except for $P_{v}$, $Q_{v}$, $P_{w}$, and
$Q_{w}$.

By unraveling the definitions, one sees that the edge forgetful morphisms
compute the cross-ratio of the four points that are not forgotten, when we
identify $\overline{\mathscr{M}}_{0,2e}$ with $\mathbb{P}^{1}$. Hence, in our
perspective, fixing the value of an edge forgetful morphism amounts to fixing
the spherical length of an edge. Therefore, if we are given an assignment of
edge lengths $\lambda=(\lambda_{e})_{e\in E}$ for a graph $G=(V,E)$, then we
can think of it as an element $\Lambda\in\prod_{e\in
E}\overline{\mathscr{M}}_{0,2e}$ and we can look after all elements in the
moduli space $\overline{\mathscr{M}}_{0,2V}$ that are mapped to $\Lambda$ by
the edge forgetful morphisms. These elements constitute the configurations of
the graph, namely a compactification of the set of realizations up to
rotations.

######  Definition 3.3.

Let $G=(V,E)$ be a graph and let $\Lambda\in\prod_{e\in
E}\overline{\mathscr{M}}_{0,2e}$. The _configuration space_
$\mathscr{K}_{G}^{\Lambda}$ of $G$ with labeling $\Lambda$ is defined as

$\mathscr{K}_{G}^{\Lambda}:=\Phi_{G}^{-1}(\Lambda)\,,$

where

$\Phi_{G}\colon\overline{\mathscr{M}}_{0,2V}\longrightarrow\prod_{e\in
E}\overline{\mathscr{M}}_{0,2e}$

is the product $\prod_{e\in E}\pi_{e}$. If the element $\Lambda$ comes from a
real assignment of edge lengths, then the variety $\mathscr{K}_{G}^{\Lambda}$
is real with respect to the real structure that swaps the $P$-marked points
and the $Q$-marked points. We denote by $K_{G}^{\Lambda}$ the rational
equivalence class of $\mathscr{K}_{G}^{\Lambda}$ in the Chow ring of
$\overline{\mathscr{M}}_{0,2V}$. Since this class does not depend on
$\Lambda$, from now on we simply write $K_{G}$.

For reasons that are made clear in Section 4, once we are given a split of a
graph, it is convenient to remove the constraint given by the edge common to
the two calligraphs and define two new configuration spaces.

######  Definition 3.4.

Let $G=(V,E)$ and let $(G_{1},G_{2})$ be a split of $G$. Let
$\Lambda\in\prod_{e\in E}\overline{\mathscr{M}}_{0,2e}$ be an edge labeling
for $G$. For $i\in\\{1,2\\}$, we set

$\widetilde{E}_{i}:=E_{i}\setminus\\{\\{1,2\\}\\}\quad\text{and}\quad\widetilde{\Lambda}_{i}:=(\Lambda_{e})_{e\in\widetilde{E}_{i}}\,.$

Moreover, we define

$\widetilde{\Phi}_{G_{i}}\colon\overline{\mathscr{M}}_{0,2V}\longrightarrow\prod_{e\in\widetilde{E}_{i}}\overline{\mathscr{M}}_{0,2e}$

as the product $\prod_{e\in\widetilde{E}_{i}}\pi_{e}$. We define the _reduced
configuration space_
$\widetilde{\mathscr{K}}_{G_{i}}^{\widetilde{\Lambda}_{i}}$ of $G_{i}$ to be

$\widetilde{\mathscr{K}}_{G_{i}}^{\widetilde{\Lambda}_{i}}:=\widetilde{\Phi}_{G_{i}}^{-1}(\widetilde{\Lambda}_{i})\,.$

As in Definition 3.3, if $\widetilde{\Lambda}_{i}$ comes from a real edge
length assignment, then
$\widetilde{\mathscr{K}}_{G_{i}}^{\widetilde{\Lambda}_{i}}$ is a real
subvariety of $\overline{\mathscr{M}}_{0,2V}$. We denote by
$\widetilde{K}_{G_{i}}$ the rational equivalence class of
$\widetilde{\mathscr{K}}_{G_{i}}^{\widetilde{\Lambda}_{i}}$ in the Chow ring,
which does not depend on $\widetilde{\Lambda}_{i}$ either.

When a graph is minimally rigid, its configuration space for a general choice
of edge lengths is a finite set, whose cardinality equals the number of
realizations of the graph on the sphere up to rotations. Therefore, this
number is equal to the degree of the class corresponding to the configuration
space of the graph, as stated in Proposition 3.5.

######  Proposition 3.5 ([GGS20]).

Let $G=(V,E)$ be a minimally rigid graph. Then

$\mathbf{c}_{S^{2}}(G)=\int_{\overline{\mathscr{M}}_{0,2V}}K_{G}\,.$

Here and in the following, $\displaystyle\int_{X}D$ indicates the _degree_ of
a class $D$ in the Chow ring of a complete variety $X$ over $\mathbb{C}$,
which is the number $s_{\ast}(D)$, where $s\colon
X\longrightarrow\mathrm{Spec}(\mathbb{C})$ is the structural morphism of $X$
and one identifies the Chow group of classes of dimension zero in
$\mathrm{Spec}(\mathbb{C})$ with $\mathbb{Z}$; see [Ful98, Definition 1.4].
When $D$ is the class of a finite reduced scheme, its degree is the number of
closed points of that scheme.

##  4 Reduction to three classes in $\overline{\mathscr{M}}_{0,6}$

The goal of this section is to show that it is possible to express the degree
of $K_{G}$ as an intersection of three contributions in the Chow ring of
$\overline{\mathscr{M}}_{0,6}$ (see Proposition 4.2). From now on, in this
section we fix the following notation:

  * $\triangleright$

$G=(V,E)$ is a graph;

  * $\triangleright$

$(G_{1},G_{2})$ is a split of $G$ and $G_{i}=(V_{i},E_{i})$ for
$i\in\\{1,2\\}$;

  * $\triangleright$

$G_{12}$ is the graph with vertices $V_{12}:=\\{0,1,2\\}$ and edges
$E_{12}:=\\{\\{1,2\\}\\}$.

From this setting, we get the following commutative diagram, where all the
maps are forgetful morphisms (hence, in particular, all maps are flat and
proper):

${\overline{\mathscr{M}}_{0,2V}}$${\overline{\mathscr{M}}_{0,2V_{1}}}$${\overline{\mathscr{M}}_{0,2V_{2}}}$${\overline{\mathscr{M}}_{0,2V_{12}}}$$\scriptstyle{\sigma_{1}}$$\scriptstyle{\sigma_{2}}$$\scriptstyle{\vartheta}$$\scriptstyle{\eta_{1}}$$\scriptstyle{\eta_{2}}$
(1)

First of all, we write $K_{G}$ as the product of three contributions in
$\overline{\mathscr{M}}_{0,2V}$, two of which are the pullbacks of the classes
of the reduced configuration spaces from Definition 3.4.

######  Lemma 4.1.

$K_{G}=\vartheta^{\ast}(K_{G_{12}})\cdot\sigma_{1}^{\ast}(\widetilde{K}_{G_{1}})\cdot\sigma_{2}^{\ast}(\widetilde{K}_{G_{2}})$
.

###### Proof.

The statement follows from inspecting the description of $K_{G}$ provided in
[GGS20], which we now recall.

We can write $K_{G}$ as a product of the classes of particular divisors in
$\overline{\mathscr{M}}_{0,2V}$, denoted by $D_{I,J}$, where $(I,J)$ is a
partition of $2V$:

$K_{G}=\prod_{\begin{subarray}{c}e\in E\\\
e=\\{v,w\\}\end{subarray}}\sum_{\begin{subarray}{c}(I,J)\text{ s.t.}\\\
P_{v},Q_{v}\in I\\\ P_{w},Q_{w}\in J\end{subarray}}D_{I,J}\,.$

The precise description of the classes $D_{I,J}$ is not needed in this paper;
for the interested reader, we refer to [GGS20] and the references therein.

With this description at hand, we decompose $K_{G}$ into three contributions:

$K_{G}=\prod_{\begin{subarray}{c}e\in E\\\
e=\\{v,w\\}\end{subarray}}\sum_{\begin{subarray}{c}(I,J)\text{ s.t.}\\\
P_{v},Q_{v}\in I\\\ P_{w},Q_{w}\in J\end{subarray}}D_{I,J}=\\\
\underbrace{\prod_{\begin{subarray}{c}e\in E_{12}\\\
e=\\{v,w\\}\end{subarray}}\sum_{\begin{subarray}{c}(I,J)\text{ s.t.}\\\
P_{v},Q_{v}\in I\\\ P_{w},Q_{w}\in
J\end{subarray}}D_{I,J}}_{\vartheta^{\ast}(K_{G_{12}})}\,\cdot\,\underbrace{\prod_{\begin{subarray}{c}e\in\widetilde{E}_{1}\\\
e=\\{v,w\\}\end{subarray}}\sum_{\begin{subarray}{c}(I,J)\text{ s.t.}\\\
P_{v},Q_{v}\in I\\\ P_{w},Q_{w}\in
J\end{subarray}}D_{I,J}}_{\sigma_{1}^{\ast}(\widetilde{K}_{G_{1}})}\,\cdot\,\underbrace{\prod_{\begin{subarray}{c}e\in\widetilde{E}_{2}\\\
e=\\{v,w\\}\end{subarray}}\sum_{\begin{subarray}{c}(I,J)\text{ s.t.}\\\
P_{v},Q_{v}\in I\\\ P_{w},Q_{w}\in
J\end{subarray}}D_{I,J}}_{\sigma_{2}^{\ast}(\widetilde{K}_{G_{2}})}$

where the $\widetilde{E}_{i}$’s are as in Definition 3.4. ∎

The main result that we prove in this section is the following.

######  Proposition 4.2.

$\displaystyle\int_{\overline{\mathscr{M}}_{0,2V}}K_{G}=\int_{\overline{\mathscr{M}}_{0,2V_{12}}}K_{G_{12}}\cdot(\eta_{1})_{\ast}(\widetilde{K}_{G_{1}})\cdot(\eta_{2})_{\ast}(\widetilde{K}_{G_{2}})$
.

Proposition 4.2 results immediately from the following, stronger, result.

######  Proposition 4.3.

$\vartheta_{\ast}(K_{G})=K_{G_{12}}\cdot(\eta_{1})_{\ast}(\widetilde{K}_{G_{1}})\cdot(\eta_{2})_{\ast}(\widetilde{K}_{G_{2}})$
.

The rest of the section is devoted to proving Proposition 4.3.

First of all, using the so-called _projection formula_ (see [Ful98,
Proposition 8.3(c)]) and Lemma 4.1, we get:

$\vartheta_{\ast}(K_{G})=\vartheta_{\ast}\bigl{(}\vartheta^{\ast}(K_{G_{12}})\cdot\sigma_{1}^{\ast}(\widetilde{K}_{G_{1}})\cdot\sigma_{2}^{\ast}(\widetilde{K}_{G_{2}})\bigr{)}=K_{G_{12}}\cdot\vartheta_{\ast}\bigl{(}\sigma_{1}^{\ast}(\widetilde{K}_{G_{1}})\cdot\sigma_{2}^{\ast}(\widetilde{K}_{G_{2}})\bigr{)}\,.$

Hence, to prove Proposition 4.3, it is enough to show that

$\vartheta_{\ast}\bigl{(}\sigma_{1}^{\ast}(\widetilde{K}_{G_{1}})\cdot\sigma_{2}^{\ast}(\widetilde{K}_{G_{2}})\bigr{)}=(\eta_{1})_{\ast}(\widetilde{K}_{G_{1}})\cdot(\eta_{2})_{\ast}(\widetilde{K}_{G_{2}})\,.$
(2)

Equation 2 is implied by the following claim.

Claim.
$(\sigma_{2})_{\ast}(\sigma_{1})^{\ast}=(\eta_{2})^{\ast}(\eta_{1})_{\ast}$ .

In fact, if the claim holds we have that

$\displaystyle\vartheta_{\ast}\bigl{(}\sigma_{1}^{\ast}(\widetilde{K}_{G_{1}})\cdot\sigma_{2}^{\ast}(\widetilde{K}_{G_{2}})\bigr{)}$
$\displaystyle=(\eta_{2}\,\sigma_{2})_{\ast}\bigl{(}\sigma_{1}^{\ast}(\widetilde{K}_{G_{1}})\cdot\sigma_{2}^{\ast}(\widetilde{K}_{G_{2}})\bigr{)}$
$\displaystyle=(\eta_{2})_{\ast}\bigl{(}(\sigma_{2})_{\ast}\sigma_{1}^{\ast}(\widetilde{K}_{G_{1}})\cdot\widetilde{K}_{G_{2}}\bigr{)}\quad\text{(proj.\
formula)}$
$\displaystyle=(\eta_{2})_{\ast}\bigl{(}\eta_{2}^{\ast}(\eta_{1})_{\ast}(\widetilde{K}_{G_{1}})\cdot\widetilde{K}_{G_{2}}\bigr{)}\quad\text{(claim)}$
$\displaystyle=(\eta_{1})_{\ast}(\widetilde{K}_{G_{1}})\cdot(\eta_{2})_{\ast}(\widetilde{K}_{G_{2}})\quad\text{(proj.\
formula)}\,.$

Thus, we are left with proving the claim. Notice that if the diagram in
Equation 1 was a fiber product, then the claim would immediately follow from
the commutativity of flat pullbacks with proper pushforwards (see [Ful98,
Proposition 1.7]) since all maps are both flat and proper. However, this is
not the case. Still, we can prove that the diagram is “not too far” from a
fiber square, in the following sense. Consider

${\overline{\mathscr{M}}_{0,2V}}$${\overline{\mathscr{M}}_{0,2V_{1}}\times_{\overline{\mathscr{M}}_{0,2V_{12}}}\overline{\mathscr{M}}_{0,2V_{2}}}$${\overline{\mathscr{M}}_{0,2V_{1}}}$${\overline{\mathscr{M}}_{0,2V_{2}}}$${\overline{\mathscr{M}}_{0,2V_{12}}}$$\scriptstyle{\sigma_{1}}$$\scriptstyle{\sigma_{2}}$$\scriptstyle{\gamma}$$\scriptstyle{\alpha_{1}}$$\scriptstyle{\alpha_{2}}$$\scriptstyle{\eta_{1}}$$\scriptstyle{\eta_{2}}$
(3)

where $\gamma$ is the morphism determined by the universal property of the
fiber product.

######  Lemma 4.4.

The map $\gamma$ is proper and birational.

###### Proof.

The morphism $\gamma$ is proper since both $\overline{\mathscr{M}}_{0,2V}$ and
$\overline{\mathscr{M}}_{0,2V_{1}}\times_{\overline{\mathscr{M}}_{0,2V_{12}}}\overline{\mathscr{M}}_{0,2V_{2}}$
are projective varieties. We now show that it is birational. For this, recall
that, from the general construction of $\overline{\mathscr{M}}_{0,2V}$, there
exists an open subset $\mathscr{M}_{2V}$ of $\overline{\mathscr{M}}_{0,2V}$
whose elements are irreducible stable curves with marked points, namely,
tuples of $2|V|$ distinct points in $\mathbb{P}^{1}$. We restrict $\gamma$ to
$\mathscr{M}_{2V}$ and prove that it is an isomorphism. The map $\gamma$ sends
a tuple $(P_{v})_{v\in V}\cup(Q_{v})_{v\in V}$ to the pair of tuples
$(P_{v})_{v\in V_{1}}\cup(Q_{v})_{v\in V_{1}}$ and $(P_{v})_{v\in
V_{2}}\cup(Q_{v})_{v\in V_{2}}$, which share the tuples $(P_{v})_{v\in
V_{12}}\cup(Q_{v})_{v\in V_{12}}$, hence $\gamma$ is an isomorphism between
$\mathscr{M}_{2V}$ and its image. ∎

Now that we know that Equation 1 “differs” from a fiber square by a proper
birational morphism, we can use the following result to “transfer” the
commutativity of maps from the fiber square to the outer square in Equation 3.

We do not claim any originality about Lemma 4.5, see the Acknowledgments. Any
errors in adapting the explained proof should be ascribed only to us.

######  Lemma 4.5.

Let $f\colon X\longrightarrow Y$ and $g\colon Y\longrightarrow Z$ be morphisms
of varieties, let $h:=g\circ f$. Suppose that:

  * $\triangleright$

$f$ is proper and birational;

  * $\triangleright$

$g$ and $h$ are flat;

  * $\triangleright$

$Z$ is non-singular.

Then $f_{\ast}\,h^{\ast}=g^{\ast}$.

###### Proof.

This result follows from the combination of [Ful98, Proposition 8.1.1(c)] and
[Ful98, Proposition 8.1.2(a)]. In fact, in [Ful98, Proposition 8.1.1(c)] we
take:

  * $\triangleright$

$f$, $g$, and $h$ as in our statement;

  * $\triangleright$

$X^{\prime}=X$, $Y^{\prime}=Y$, and $Z^{\prime}=Z$, furthermore
$p_{X}=\mathrm{id}_{X}$, $p_{Y}=\mathrm{id}_{Y}$, and $p_{Z}=\mathrm{id}_{Z}$;

  * $\triangleright$

$x=[X]$.

With this choice, the following formula holds:

$f_{\ast}([X]\cdot_{h}z)=f_{\ast}([X])\cdot_{g}z$

for any class $z$ in $Z$, as an equality in the Chow group of $Y$. Since $f$
is birational, we have that $f_{\ast}([X])=[Y]$. Both $g$ and $h$ are flat, so
[Ful98, Proposition 8.1.2(a)] gives us that

$[X]\cdot_{h}z=h^{\ast}(z)\quad\text{and}\quad[Y]\cdot_{g}z=g^{\ast}(z)\,.$

Therefore, altogether the previous equalities tell us that for any class $z$
in $Z$, we have

$f_{\ast}\,h^{\ast}(z)=g^{\ast}(z)\,,$

which is the desired result. ∎

We have all the tools to prove the claim and conclude this section. Notice
that both the maps $\alpha_{1}$ and $\alpha_{2}$ are proper and flat since
$\eta_{1}$ and $\eta_{2}$ are so. Hence, Lemma 4.4 allows us to apply Lemma
4.5 with $f=\gamma$, $g=\alpha_{1}$, and $h=\sigma_{1}$ to get

$\gamma_{\ast}\sigma_{1}^{\ast}=\alpha_{1}^{\ast}\,.$

Since flat pullbacks commute with proper pushforwards in a fiber square, we
have
$(\alpha_{2})_{\ast}(\alpha_{1})^{\ast}=(\eta_{2})^{\ast}(\eta_{1})_{\ast}$.
Then

$\displaystyle(\eta_{2})^{\ast}(\eta_{1})_{\ast}$
$\displaystyle=(\alpha_{2})_{\ast}(\alpha_{1})^{\ast}$
$\displaystyle=(\alpha_{2})_{\ast}\gamma_{\ast}(\sigma_{1})^{\ast}$
$\displaystyle=(\sigma_{2})_{\ast}(\sigma_{1})^{\ast}\,.$

This concludes the proof of the claim and hence of Proposition 4.3. Therefore,
we reduced the computation of the class of the configuration space of a graph
$G$ to the intersection of three classes in the $3$-fold
$\overline{\mathscr{M}}_{0,6}$, one of which does not depend on $G$. Notice
that so far we have neither supposed that $G$ is minimally rigid, nor that
$G_{1}$ and $G_{2}$ are calligraphs.

##  5 Description of the configurations of the fixed edge

Recall from Section 4 that the graph $G_{12}$ has vertices
$V_{12}=\\{0,1,2\\}$ and edges $E_{12}=\\{\\{1,2\\}\\}$. Therefore, the map
$\Phi_{G_{12}}$ as in Definition 3.3 has
$\overline{\mathscr{M}}_{0,2V_{12}}\cong\overline{\mathscr{M}}_{0,6}$ as
domain and
$\overline{\mathscr{M}}_{0,2\\{1,2\\}}\cong\overline{\mathscr{M}}_{0,4}\cong\mathbb{P}^{1}$
as codomain. Moreover, recall again from Definition 3.3 that
$\mathscr{K}_{G_{12}}^{\Lambda_{12}}$ is the fiber of $\Phi_{G_{12}}$ over an
element $\Lambda_{12}\in\overline{\mathscr{M}}_{0,2\\{1,2\\}}$. The goal of
this section is to prove that, for a general $\Lambda_{12}$, the variety
$\mathscr{K}_{G_{12}}^{\Lambda_{12}}$ is isomorphic to the blowup of
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ at four points (Proposition 5.1). This
may be well-known to experts, but we could not locate this result in the
literature; moreover, we believe that the proof that we provide may be
accessible with some acquaintance about moduli spaces and intersection theory.

Let us start by remarking that the moduli space
$\overline{\mathscr{M}}_{0,2V_{12}}$ of stable rational curves with six marked
points, where $\mathscr{K}_{G_{12}}^{\Lambda_{12}}$ lies, is isomorphic to the
blowup of the Segre cubic $S^{3}$ threefold along its $10$ nodes (see [BB22,
Section 3]). The Segre cubic threefold $S_{3}$ is the unique (up to projective
transformations) cubic hypersurface in $\mathbb{P}^{4}$ with $10$ nodes (see
for example [Dol16]). It admits an interpretation as a moduli space for
$6$-tuples of points in $\mathbb{P}^{1}$ with at most three coinciding points
that we are soon going to recall.

Recalling from Definition 3.3 the description of the map $\Phi_{G_{12}}$, we
have a commutative diagram

${\overline{\mathscr{M}}_{0,2V_{12}}}$${S_{3}}$${\overline{\mathscr{M}}_{0,2\\{1,2\\}}}$blowup$\scriptstyle{\Phi_{G_{12}}}$$\scriptstyle{\Psi_{G_{12}}}$
(4)

that defines the rational map $\Psi_{G_{12}}$. We know that
$\mathscr{K}_{G_{12}}^{\Lambda_{12}}$ is a fiber of $\Phi_{G_{12}}$ over a
general point. We compute the (closure of) a fiber of $\Psi_{G_{12}}$ over a
general point of $\overline{\mathscr{M}}_{0,2\\{1,2\\}}$ and in this way we
obtain the desired description of $\mathscr{K}_{G_{12}}^{\Lambda_{12}}$.

######  Proposition 5.1.

For a general $\Lambda_{12}\in\overline{\mathscr{M}}_{0,2\\{1,2\\}}$, the
variety $\mathscr{K}_{G_{12}}^{\Lambda_{12}}$ is isomorphic to the blowup of
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ at four points.

First of all, we recall the construction of the Segre cubic $S_{3}$ as a
moduli space of $6$-tuples of points in $\mathbb{P}^{1}$ from [HMSV09]. The
construction proceeds as follows:

  * $\triangleright$

We consider all possible graphs with vertex set
$\\{P_{0},P_{1},P_{2},Q_{0},Q_{1},Q_{2}\\}$ where all the vertices have
valency one and such that, once we draw the vertices as vertices of a regular
hexagon and the edges as straight segments, no two edges intersect. We obtain
five graphs denoted by $X_{0}$, $X_{1}$, $X_{2}$, $Y_{1}$, and $Y_{2}$,
depicted in Figure 3. We select an orientation for each of the edges of these
graphs. In our case, we orient an edge from its vertex above to its vertex
below.

$P_{2}$$P_{0}$$P_{1}$$Q_{2}$$Q_{0}$$Q_{1}$$X_{0}$$P_{2}$$P_{0}$$P_{1}$$Q_{2}$$Q_{0}$$Q_{1}$$X_{1}$$P_{2}$$P_{0}$$P_{1}$$Q_{2}$$Q_{0}$$Q_{1}$$X_{2}$$P_{2}$$P_{0}$$P_{1}$$Q_{2}$$Q_{0}$$Q_{1}$$Y_{1}$$P_{2}$$P_{0}$$P_{1}$$Q_{2}$$Q_{0}$$Q_{1}$$Y_{2}$
Figure 3: The five auxiliary graphs that define $S_{3}$.

  * $\triangleright$

Each of the vertices of the hexagon corresponds to a copy of $\mathbb{P}^{1}$.
We associate a polynomial to each of the five graphs from Figure 3 as follows.
Suppose that $e=\\{v,w\\}$ is an edge of one of those graphs and suppose that
$e$ is oriented from $v$ to $w$. Let $(s_{v}:t_{v})$ be the coordinates on the
$\mathbb{P}^{1}$ corresponding to $v$ and let $(s_{w}:t_{w})$ be the
coordinates on the $\mathbb{P}^{1}$ corresponding to $w$. We set

$\ell_{e}:=s_{v}t_{w}-s_{w}t_{v}\,.$

Then for each graph $N=(E_{N},V_{N})$ in Figure 3, we define

$p_{N}:=\prod_{e\in E_{N}}\ell_{e}\,.$

  * $\triangleright$

We define the rational map
$\Gamma\colon(\mathbb{P}^{1})^{6}\dashrightarrow\mathbb{P}^{4}$ as
$\Gamma:=(p_{X_{0}}:p_{X_{1}}:p_{X_{2}}:p_{Y_{1}}:p_{Y_{2}})$. On $6$-tuples
of distinct points, the map $\Gamma$ behaves as a quotient map, namely sends
all $\mathbb{P}\mathrm{GL}_{2}$-equivalent $6$-tuples to the same point.

  * $\triangleright$

We consider the subset $\mathcal{U}\subset(\mathbb{P}^{1})^{6}$ defined by

$\mathcal{U}:=\bigl{\\{}(a_{1},\dotsc,a_{6})\in(\mathbb{P}^{1})^{6}\,\mid\,\text{
at most three }a_{i}\text{'s coincide}\bigr{\\}}\,.$

The image of $\mathcal{U}$ under $\Gamma$ is the Segre cubic $S_{3}$. We write
$x_{0}$, $x_{1}$, $x_{2}$, $y_{1}$, and $y_{2}$ for the coordinates of
$\mathbb{P}^{4}$ where $S_{3}$ lives. In these coordinates, the threefold
$S_{3}$ has the equation

$x_{0}x_{1}x_{2}-y_{1}y_{2}(x_{0}+x_{1}+x_{2}-y_{1}-y_{2})=0\,.$

The real structure in $(\mathbb{P}^{1})^{6}$ that swaps the coordinates
labeled by $P_{i}$ with those labeled by $Q_{i}$ for $i\in\\{0,1,2\\}$ induces
a real structure on $S_{3}\subset\mathbb{P}^{4}$ for which the $10$ nodes come
into five conjugated pairs. This real structure is compatible, via the diagram
in Equation 4, with the real structure on $\overline{\mathscr{M}}_{0,2V_{12}}$
described in Section 3. Moreover, it determines an automorphism of the
coordinate ring of $\mathbb{P}^{4}$ such that the variables $x_{i}$ are
conjugated to $-x_{i}$ for $i\in\\{0,1,2\\}$, while $y_{1}$ is conjugated to
$-y_{2}$. The reason for this is that, when in Figure 3 we swap the vertices
labelled with $P_{i}$ by those labelled with $Q_{i}$, what happens is that
each directed graph $X_{i}$ is mapped to the same graph with all the edge
orientations reversed, while the directed graph $Y_{1}$ is mapped to $Y_{2}$
with all the edge orientations reversed. Due to the rule associating a
directed graph to its polynomial explained above, changing the orientation of
three edges changes the sign of the polynomial. This concludes the explanation
of the automorphism provided by the real structure.

Notice that, from the construction of $S_{3}$ that we reported above, it
follows that we can define a rational map
$S_{3}\dashrightarrow\overline{\mathscr{M}}_{0,2\\{1,2\\}}$ by specifying a
rational map $\mathcal{U}\dashrightarrow\mathbb{P}^{1}$ that is
$\mathbb{P}\mathrm{GL}_{2}$-invariant. In particular, by inspecting Equation
4, the map $\Psi_{12}\colon
S_{3}\dashrightarrow\overline{\mathscr{M}}_{0,2\\{1,2\\}}$ is the one induced
by the rational map $\mathcal{U}\dashrightarrow\mathbb{P}^{1}$ that takes the
cross-ratio of the points $P_{1}$, $Q_{1}$, $P_{2}$, and $Q_{2}$. In fact,
this is precisely what $\Psi_{12}$ does when we apply it to irreducible
rational stable curves for which at most three of the marked points $P_{1}$,
$Q_{1}$, $P_{2}$, and $Q_{2}$ coincide. Let $(s_{1}^{P}:t_{1}^{P})$,
$(s_{1}^{Q}:t_{1}^{Q})$, $(s_{2}^{P}:t_{2}^{P})$, and $(s_{2}^{Q}:t_{2}^{Q})$
be their coordinates, respectively. Their cross-ratio is:

$\bigl{(}(s_{2}^{P}t_{1}^{Q}-s_{1}^{Q}t_{2}^{P})(s_{2}^{Q}t_{1}^{P}-s_{1}^{P}t_{2}^{Q}):(s_{2}^{P}t_{1}^{P}-s_{1}^{P}t_{2}^{P})(s_{2}^{Q}t_{1}^{Q}-s_{1}^{Q}t_{2}^{Q})\bigr{)}\,.$
(5)

Our goal is now to infer from Equation 5 an explicit expression of
$\Psi_{G_{12}}$ as a map from $S_{3}$ to $\mathbb{P}^{1}$. If we
multiply***This may enlarge the subset where the rational map is not defined,
but we can recover the missing points by expressing the rational map in a
different way. both the homogeneous coordinates in Equation 5 by
$(s_{0}^{P}t_{0}^{Q}-s_{0}^{Q}t_{0}^{P})$, the first component of
$\Psi_{G_{12}}$ equals the polynomial $p_{X_{0}}$ from the construction of
$S_{3}$. This means that, when we pass to the map $\Psi_{G_{12}}$, its first
coordinate is $x_{0}$. The second coordinate of the cross ratio from Equation
5 becomes

$(s_{0}^{P}t_{0}^{Q}-s_{0}^{Q}t_{0}^{P})(s_{2}^{P}t_{1}^{P}-s_{1}^{P}t_{2}^{P})(s_{2}^{Q}t_{1}^{Q}-s_{1}^{Q}t_{2}^{Q})\,,$
(6)

which is the polynomial corresponding to the directed graph in Figure 4,
according to the rule that we used in the construction of $S_{3}$.

$P_{2}$$P_{0}$$P_{1}$$Q_{2}$$Q_{0}$$Q_{1}$ Figure 4: Graph for the polynomial
from Equation 6.

Now we apply a result from [HMSV09, Section 2.2], the so-called _straightening
rules_ , to replace the second component of $\Psi_{G_{12}}\circ\Gamma$ with
another one without changing the map. This is obtained by decomposing the
graph from Figure 4 into ones whose edges do not intersect, as described in
Figure 5.

$P_{2}$$P_{0}$$P_{1}$$Q_{2}$$Q_{0}$$Q_{1}$=$P_{2}$$P_{0}$$P_{1}$$Q_{2}$$Q_{0}$$Q_{1}$+$P_{2}$$P_{0}$$P_{1}$$Q_{2}$$Q_{0}$$Q_{1}$=$P_{2}$$P_{0}$$P_{1}$$Q_{2}$$Q_{0}$$Q_{1}$+$P_{2}$$P_{0}$$P_{1}$$Q_{2}$$Q_{0}$$Q_{1}$+$P_{2}$$P_{0}$$P_{1}$$Q_{2}$$Q_{0}$$Q_{1}$+$P_{2}$$P_{0}$$P_{1}$$Q_{2}$$Q_{0}$$Q_{1}$
Figure 5: Straightening procedure for the graph from Figure 4.

This means that the polynomial from Equation 6 determines the same form on
$S_{3}$ as $-p_{X_{1}}-p_{X_{2}}+p_{Y_{1}}+p_{Y_{2}}$ (here we also used that
reversing the orientation of an edge in a graph changes the sign of the
corresponding polynomial). In other words, the second coordinate of the map
$\Psi_{G_{12}}$ is given by the linear form

$-x_{1}-x_{2}+y_{1}+y_{2}\,.$

Thus, the map

$\Psi_{G_{12}}\colon
S_{3}\dashrightarrow\overline{\mathscr{M}}_{0,2\\{1,2\\}}\cong\mathbb{P}^{1}\,,$

when $x_{0}\neq 0$ or $-x_{1}-x_{2}+y_{1}+y_{2}\neq 0$, is:

$(x_{0}:x_{1}:x_{2}:y_{1}:y_{2})\mapsto(x_{0}:-x_{1}-x_{2}+y_{1}+y_{2})\,.$
(7)

This map is not defined on the plane $x_{0}=-x_{1}-x_{2}+y_{1}+y_{2}=0$. On
the other hand, notice that the equation of $S_{3}$ can be written as

$\mathrm{det}\left(\begin{array}[]{cc}x_{0}&-x_{1}-x_{2}+y_{1}+y_{2}\\\
-y_{1}y_{2}&x_{1}x_{2}-y_{1}y_{2}\end{array}\right)=0\,.$

Therefore, in a suitable open subset of $S_{3}$, the map from Equation 7
coincides with

$(x_{0}:x_{1}:x_{2}:y_{1}:y_{2})\mapsto(-y_{1}y_{2}:x_{1}x_{2}-y_{1}y_{2})\,.$

In this way, we conclude that the map $\Psi_{G_{12}}$ is defined on the whole
$S_{3}$, except for the four nodes

$(0:0:1:0:1)\,,\;\;(0:0:1:1:0)\,,\;\;(0:1:0:1:0)\,,\;\;(0:1:0:0:1)\,.$ (8)

Notice that the first two and the last two nodes in Equation 8 are conjugated
under the real structure of $S_{3}$ described before. It is not possible to
further extend the map $\Psi_{G_{12}}$ to any of the four previous nodes,
since they are the images of the loci in
$\mathcal{U}\subset(\mathbb{P}^{1})^{6}$ where three out of the four points
$P_{1}$, $Q_{1}$, $P_{2}$, and $Q_{2}$ coincide, and the cross-ratio is not
defined for those four-tuples.

The fiber of $\Psi_{G_{12}}$ over a general point
$(\mu_{1}:\mu_{2})\in\mathbb{P}^{1}$ is the subvariety

$F_{\mu_{1},\mu_{2}}:=\bigl{\\{}\mu_{2}x_{0}-\mu_{1}(-x_{1}-x_{2}+y_{1}+y_{2})=\mu_{1}x_{1}x_{2}-(\mu_{1}-\mu_{2})y_{1}y_{2}=0\bigr{\\}}$
(9)

minus the four nodes from Equation 8. For a general $(\mu_{1}:\mu_{2})$, the
subvariety $F_{\mu_{1},\mu_{2}}$ is a smooth two-dimensional quadric, hence it
is isomorphic to $\mathbb{P}^{1}\times\mathbb{P}^{1}$. Recall now from
Equation 4 that blowing up $S_{3}$ at its nodes resolves the indeterminacies
of $\Psi_{G_{12}}$. Therefore, a general fiber of the morphism $\Phi_{G_{12}}$
is isomorphic to the blowup of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ at four
points. This proves Proposition 5.1. When this fiber is taken over a real
point, the variety $\mathscr{K}_{G_{12}}^{\Lambda_{12}}$ is endowed with a
real structure that swaps the two pairs of exceptional divisors of the blowup
and the two families of lines in $\mathbb{P}^{1}\times\mathbb{P}^{1}$.

##  6 Classes for spherical calligraphs

We now combine the results of Section 4 and Section 5 to define
$S^{2}$-classes of calligraphs and their product. Moreover, we prove that,
when we have a calligraphic split of a minimally rigid graph, the number of
complex realizations of such a graph on the sphere up to rotations, is the
product of the $S^{2}$-classes of the two calligraphs in the split.

From Section 4, we know that the degree of the class of the configuration
space of a graph $G$ with a split $G=G_{1}\cup G_{2}$ is

$\int_{\overline{\mathscr{M}}_{0,2V_{12}}}K_{G_{12}}\cdot(\eta_{1})_{\ast}(\widetilde{K}_{G_{1}})\cdot(\eta_{2})_{\ast}(\widetilde{K}_{G_{2}})\,.$

From Section 5, we know that $\mathscr{K}_{G_{12}}^{\Lambda_{12}}$ is a
nonsingular hypersurface in $\overline{\mathscr{M}}_{0,2V_{12}}$ for a general
$\Lambda_{12}\in\overline{\mathscr{M}}_{0,2\\{1,2\\}}$. Therefore, the
inclusion
$i\colon\mathscr{K}_{G_{12}}^{\Lambda_{12}}\longrightarrow\overline{\mathscr{M}}_{0,2V_{12}}$
is a regular embedding, so by [Ful98, Example 8.1.1]:

$K_{G_{12}}\cdot(\eta_{1})_{\ast}(\widetilde{K}_{G_{1}})\cdot(\eta_{2})_{\ast}(\widetilde{K}_{G_{2}})=i_{\ast}i^{\ast}\bigl{(}(\eta_{1})_{\ast}(\widetilde{K}_{G_{1}})\cdot(\eta_{2})_{\ast}(\widetilde{K}_{G_{2}})\bigr{)}\,.$

Since both $\mathscr{K}_{G_{12}}^{\Lambda_{12}}$ and
$\overline{\mathscr{M}}_{0,2V_{12}}$ are nonsingular, the map $i^{\ast}$ is a
ring homomorphism between the respective Chow rings, hence

$\int_{\overline{\mathscr{M}}_{0,2V_{12}}}K_{G_{12}}\cdot(\eta_{1})_{\ast}(\widetilde{K}_{G_{1}})\cdot(\eta_{2})_{\ast}(\widetilde{K}_{G_{2}})=\int_{\mathscr{K}_{G_{12}}^{\Lambda_{12}}}i^{\ast}(\eta_{1})_{\ast}(\widetilde{K}_{G_{1}})\cdot
i^{\ast}(\eta_{2})_{\ast}(\widetilde{K}_{G_{2}})\,.$

Summing up, so far we got that

$\int_{\overline{\mathscr{M}}_{0,2V}}K_{G}=\int_{\mathscr{K}_{G_{12}}^{\Lambda_{12}}}i^{\ast}(\eta_{1})_{\ast}(\widetilde{K}_{G_{1}})\cdot
i^{\ast}(\eta_{2})_{\ast}(\widetilde{K}_{G_{2}})\,.$ (10)

and to obtain this result we have not used yet that any of $G_{1}$ and $G_{2}$
are calligraphs. Now comes the time for using this hypothesis.

Let $H=(V,E)$ be a calligraph. If $\Lambda$ is a general choice of edge
lengths for $H$, then $\widetilde{\mathscr{K}}_{H}^{\widetilde{\Lambda}}$ from
Definition 3.4 is a smooth surface. Indeed, the variety
$\widetilde{\mathscr{K}}_{H}^{\widetilde{\Lambda}}$ is a fiber over a general
point of a morphism between smooth manifolds, so it is smooth by Sard’s
theorem. Moreover, since by definition adding $\\{0,1\\}$ or $\\{0,2\\}$ makes
$H$ rigid, the image of $\widetilde{\mathscr{K}}_{H}^{\widetilde{\Lambda}}$
under the forgetful morphism
$\eta\colon\overline{\mathscr{M}}_{0,2V}\longrightarrow\overline{\mathscr{M}}_{0,2\\{0,1,2\\}}$
is still a surface. By construction, this surface is different from
$\mathscr{K}_{G_{12}}^{\Lambda_{12}}$, therefore

$i^{\ast}\eta_{\ast}(\widetilde{K}_{H})\in
A^{1}(\mathscr{K}_{G_{12}}^{\Lambda_{12}})\,,$

where $A^{1}(\cdot)$ are the classes of codimension $1$ cycles in the Chow
ring. From Section 5, we know that $\mathscr{K}_{G_{12}}^{\Lambda_{12}}$ is
isomorphic to the blowup of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ at four
points, thus

$A(\mathscr{K}_{G_{12}}^{\Lambda_{12}})\cong\frac{\mathbb{Z}[F_{1},F_{2},E_{1},E_{2},E_{3},E_{4}]}{(F_{1}^{2},F_{2}^{2},F_{1}F_{2}-1,\\{F_{i}E_{j}\\}_{\begin{subarray}{c}i\in\\{1,2\\}\\\
j\in\\{1,\dotsc,4\\}\end{subarray}},\\{E_{j}^{2}+1\\}_{j\in\\{1,\dotsc,4\\}})}\,.$
(11)

In Equation 11, the isomorphism is the one induced by the structure of
$\mathscr{K}_{G_{12}}^{\Lambda_{12}}$ as a blowup of
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ described in Section 5, so the classes of
the two families of lines of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ are denoted
by $F_{1}$ and $F_{2}$, while the classes of the exceptional divisors are
denoted by $E_{1},\dotsc,E_{4}$. The real structure on
$\mathscr{K}_{G_{12}}^{\Lambda_{12}}$ described in Section 5 determines an
involution of $A(\mathscr{K}_{G_{12}}^{\Lambda_{12}})$ that swaps $F_{1}$ with
$F_{2}$, $E_{1}$ with $E_{2}$, and $E_{3}$ with $E_{4}$. Taking into account
that $\widetilde{\mathscr{K}}_{H}^{\widetilde{\Lambda}}$ and
$\mathscr{K}_{G_{12}}^{\Lambda_{12}}$ are real varieties when $\Lambda$ comes
from a real assignment of edge lengths, it follows that the quantity
$i^{\ast}\eta_{\ast}(\widetilde{K}_{H})$ can be uniquely expressed as

$a(F_{1}+F_{2})-b(E_{1}+E_{2})-c(E_{3}+E_{4})\,.$ (12)

where $a,b,c\in\mathbb{Z}$.

######  Definition 6.1.

Let $H$ be a calligraph. The _$S^{2}$ -class_ of $H$, denoted $[H]$, is the
triple $(a,b,c)\in\mathbb{Z}^{3}$ from Equation 12 that expresses
$i^{\ast}\eta_{\ast}(\widetilde{K}_{H})$ with respect to the generators of
$A(\mathscr{K}_{G_{12}}^{\Lambda_{12}})$.

######  Definition 6.2.

We define a quadratic form

$\begin{array}[]{rccc}\\_\cdot\\_\colon&\mathbb{Z}^{3}\times\mathbb{Z}^{3}&\longrightarrow&\mathbb{Z}\\\
&(a_{1},b_{1},c_{1}),(a_{2},b_{2},c_{3})&\mapsto&2(a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2})\end{array}\,.$

This quadratic form allows us to multiply two $S^{2}$-classes.

Now we have the tools to prove the first part of Theorem 2.6.

######  Proposition 6.3.

If $G$ is a minimally rigid graph and $(G_{1},G_{2}$) is a calligraphic split
of $G$, then the number of complex realizations of $G$ on the sphere up to
rotations is equal to the product of the classes of $G_{1}$ and $G_{2}$. In
other words, $\mathbf{c}_{S^{2}}(G)=[G_{1}]\cdot[G_{2}]$.

###### Proof.

If $G$ is minimally rigid and $(G_{1},G_{2}$) is a calligraphic split of $G$,
then by what we have proven so far and recalling Proposition 3.5, we have

$\mathbf{c}_{S^{2}}(G)=\int_{\mathscr{K}_{G_{12}}^{\Lambda_{12}}}i^{\ast}(\eta_{1})_{\ast}(\widetilde{K}_{G_{1}})\cdot
i^{\ast}(\eta_{2})_{\ast}(\widetilde{K}_{G_{2}})\,.$

Since both $G_{1}$ and $G_{2}$ are calligraphs, they have $S^{2}$-classes
$[G_{i}]=(a_{i},b_{i},c_{i})$. The intersection theory of the blowup of
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ at four points shows then that

$\int_{\mathscr{K}_{G_{12}}^{\Lambda_{12}}}i^{\ast}(\eta_{1})_{\ast}(\widetilde{K}_{G_{1}})\cdot
i^{\ast}(\eta_{2})_{\ast}(\widetilde{K}_{G_{2}})=\\\
\bigl{(}a_{1}(F_{1}+F_{2})-b_{1}(E_{1}+E_{2})-c_{1}(E_{3}+E_{4})\bigr{)}\\!\bigl{(}a_{2}(F_{1}+F_{2})-b_{2}(E_{1}+E_{2})-c_{2}(E_{3}+E_{4})\bigr{)}=\\\
2(a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2})\,,$

which, together with the previous discussion, concludes the proof. ∎

The second part of Theorem 2.6 follows from a concrete computation.

Let us compute the $S^{2}$-class of the calligraph $L$. To this end, we try to
understand $\widetilde{\mathscr{K}}_{L}^{\widetilde{\Lambda}}$ for a general

$\Lambda=(\Lambda_{01},\Lambda_{12})\in\overline{\mathscr{M}}_{0,2\\{0,1\\}}\times\overline{\mathscr{M}}_{0,2\\{1,2\\}}$

and its intersection with $\mathscr{K}_{G_{12}}^{\Lambda_{12}}$, which is
equal to $\mathscr{K}_{L}^{\Lambda}$. As in Section 5, we examine the
corresponding subvarieties in the Segre cubic $S_{3}$. Using the same
arguments that yield Equation 9, we express the blowdown of
$\mathscr{K}_{L}^{\Lambda}$ in $S_{3}$ as the intersection of the two loci:

$\displaystyle\mu_{2}x_{0}-\mu_{1}(x_{1}+x_{2}-y_{1}-y_{2})=\mu_{1}x_{1}x_{2}-(\mu_{1}+\mu_{2})y_{1}y_{2}$
$\displaystyle=0\quad\text{and}$
$\displaystyle\delta_{2}x_{2}-\delta_{1}(x_{0}+x_{1}-y_{1}-y_{2})=\delta_{1}x_{0}x_{1}-(\delta_{1}+\delta_{2})y_{1}y_{2}$
$\displaystyle=0\,,$

where $\Lambda_{12}=(\mu_{1}:\mu_{2})$ and
$\Lambda_{01}=(\delta_{1}:\delta_{2})$, once we identify both
$\overline{\mathscr{M}}_{0,2\\{1,2\\}}$ and
$\overline{\mathscr{M}}_{0,2\\{0,1\\}}$ with $\mathbb{P}^{1}$. A computer
algebra calculation shows that this intersection has equations

$\displaystyle\mu_{1}(\delta_{1}\mu_{1}-\delta_{2}\mu_{2})x_{2}^{2}-\delta_{1}\mu_{1}(\mu_{1}+\mu_{2})x_{2}y_{1}-\delta_{1}\mu_{1}(\mu_{1}+\mu_{2})x_{2}y_{2}+\delta_{1}(\mu_{1}+\mu_{2})^{2}y_{1}y_{2}$
$\displaystyle=0\,,$
$\displaystyle\delta_{1}(\mu_{1}+\mu_{2})x_{0}-(\mu_{1}\delta_{1}+\mu_{2}\delta_{2})x_{2}$
$\displaystyle=0\,,$
$\displaystyle\delta_{1}(\mu_{1}+\mu_{2})(x_{1}-y_{1}-y_{2})+(\mu_{1}\delta_{1}-\mu_{2}\delta_{2})x_{2}$
$\displaystyle=0\,,$

namely, it is a conic passing through the two nodes

$(0:1:0:1:0)\quad\text{and}\quad(0:1:0:0:1)$

(which are conjugated under the real structure of $S_{3}$) and no other node
of $S_{3}$. Therefore, the $S^{2}$-class of $L$ is $(1,1,0)$. By symmetry, the
$S^{2}$-class of $R$ is $(1,0,1)$.

Now that we know the $S^{2}$-classes of $L$ and $R$, we can constrain the
possibilities for $[C_{v}]$ from the knowledge of the number of realizations
of some simple graphs. In fact, both $L\cup C_{v}$ and $R\cup C_{v}$ are
isomorphic to the graph

whose number of realizations on the sphere, up to rotations, is $4$. This
means that $[L]\cdot[C_{v}]=[R]\cdot[C_{v}]=4$. Hence, if we write
$[C_{v}]=(a,b,c)$, then

$\left\\{\begin{array}[]{l}2(a-b)=4\,,\\\
2(a-c)=4\,,\end{array}\right.\quad\Rightarrow\quad\left\\{\begin{array}[]{l}b=c\,,\\\
a=b+2\,.\end{array}\right.$

Since $C_{v}\cup C_{w}$ is isomorphic to

we know that $[C_{v}]\cdot[C_{w}]=8$, so $a^{2}-b^{2}-c^{2}=4$. This, together
with the previous two relations, implies that there are only two possibilities
for $[C_{v}]$, namely

$(2,0,0)\quad\text{or}\quad(6,4,4)\,.$

To prove that $[C_{v}]$ is precisely $(2,0,0)$, we employ a technique that
relies on a deeper understanding of the workings of the divisors $D_{I,J}$.
Therefore, in order not to weigh down the exposition, we decided to place the
final part of this proof in an appendix (see Appendix A).

This last discussion, together with Proposition 6.3, concludes the proof of
Theorem 2.6.

We end the section by showing that, for our purposes, it is actually
irrelevant whether $[C_{v}]=(2,0,0)$ or $[C_{v}]=(6,4,4)$; see Remark 2.7. In
fact, there is an automorphism of $\mathbb{Z}$-modules of $\mathbb{Z}^{3}$
into itself that maps $(1,1,0)$ and $(1,0,1)$ to themselves and sends
$(2,0,0)$ to $(6,4,4)$. This map is given by the matrix

$O=\begin{pmatrix}3&-2&-2\\\ 2&-1&-2\\\ 2&-2&-1\end{pmatrix}\,.$

One can then check that if $A$ is the matrix of the quadratic form in Theorem
2.6, namely

$A=\begin{pmatrix}2&0&0\\\ 0&-2&0\\\ 0&0&-2\end{pmatrix}\,,$

then $OAO^{t}=A$, i. e., the transformation $O$ preserves the quadratic form
of $A$. Hence, the matrix $O$ determines an automorphism of the subring of
real classes in the Chow ring of $\mathscr{K}_{G_{12}}^{\Lambda_{12}}$ that
keeps the $S^{2}$-classes of $L$ and $R$ invariant, and swaps $(2,0,0)$ and
$(6,4,4)$. Therefore, any of these two triples could be chosen as the
$S^{2}$-class of $C_{v}$.

######  Remark 6.4.

Define the _coupler curve_ and _coupler multiplicity_ for calligraphs as in
[GEHL22b, Section 2.2], but for the sphere instead of the plane. If $(2,0,0)$
is chosen as the $S^{2}$-class of $C_{v}$, then the first integer of the
$S^{2}$-class of a calligraph equals (up to coupler multiplicity) the degree
of a general coupler curve $D\subset S^{2}_{\mathbb{C}}$ of this calligraph.
The second number corresponds to the multiplicity of $D$ at those complex
conjugate points on the conic at infinity, such that the planes passing
through these points are parallel to the tangent plane of $S^{2}_{\mathbb{C}}$
at the realization of vertex $1$. Similarly for the third number, but with
vertex $1$ replaced by vertex $2$.

##  7 The recursion

In this section, we use the theory we have been developing so far, in
particular Theorem 2.6, to synthesize a recursive procedure for the
computation of the number of realizations on the sphere, up to rotations, of a
minimally rigid graph. This can be considered as an “enhanced” version of the
algorithm developed in [GGS20].

Say that we are given a minimally rigid graph $G$ and we want to compute
$\mathbf{c}_{S^{2}}(G)$. If $(G_{1},G_{2})$ is a calligraphic split of $G$,
then by Theorem 2.6 we know that

$\mathbf{c}_{S^{2}}(G)=(a_{1},b_{1},c_{1})\cdot(a_{2},b_{2},c_{2})\,,$

where $(a_{i},b_{i},c_{i})$ is the $S^{2}$-class of $G_{i}$. To compute these
two $S^{2}$-classes we employ the “basic calligraphs” from Definition 2.5,
whose $S^{2}$-classes are already known from Theorem 2.6.

Let us write $H$ instead of $G_{1}$ to refresh the notation (the same
procedure shall be applied to $G_{2}$). If $H=(V,E)$, we construct the
following three graphs:

$H_{L}:=(V,E\cup E_{L})\,,\quad H_{R}:=(V,E\cup E_{R})\,,\quad
H_{C_{v}}:=(V\cup\\{v\\},E\cup E_{C_{v}})\,,$

where $v$ is a vertex not in $H$ and $E_{L}$, $E_{R}$, and $E_{C_{v}}$ are the
edges of $L$, $R$, and $C_{v}$, respectively. Notice that $(H,L)$, $(H,R)$,
and $(H,C_{v})$ are calligraphic splits of $H_{L}$, $H_{R}$, and $H_{C_{v}}$,
respectively.

Suppose that all $H_{L}$, $H_{R}$, and $H_{C_{v}}$ are minimally rigid. Let us
denote $[H]=(a,b,c)$. Theorem 2.6 implies that

$\displaystyle(a,b,c)\cdot(1,1,0)$
$\displaystyle=2(a-b)=\mathbf{c}_{S^{2}}(H_{L})\,,$ (13)
$\displaystyle(a,b,c)\cdot(1,0,1)$
$\displaystyle=2(a-c)=\mathbf{c}_{S^{2}}(H_{R})\,,$
$\displaystyle(a,b,c)\cdot(2,0,0)$
$\displaystyle=4a=\mathbf{c}_{S^{2}}(H_{C_{v}})\,.$

If the right hand sides of Equation 13 are known, then we can solve for
$(a,b,c)$ and obtain the $S^{2}$-class of $H$. To get the values
$\mathbf{c}_{S^{2}}(H_{L})$, $\mathbf{c}_{S^{2}}(H_{R})$, and
$\mathbf{c}_{S^{2}}(H_{C_{v}})$ we can then reiterate the procedure by finding
calligraphic splits of each of these graphs.

Here there is the possibility that one of $H_{L}$, $H_{R}$, or $H_{C_{v}}$
equals the initial minimally rigid graph $G$. This would start an infinite
recursion, which hence would not result in an algorithm for the computation of
$\mathbf{c}_{S^{2}}(G)$. To prevent this from happening, we ask that the
initial calligraphic split $(G_{1},G_{2})$ is _non-trivial_ , namely that both
$G_{1}$ and $G_{2}$ have at least $5$ vertices. In this way, we ensure that
the iteration of the procedure does not loop back to the initial situation.

It may happen, however, that one of $H_{L}$, $H_{R}$, and $H_{C_{v}}$, for
example $H_{L}$, is not minimally rigid. Then, since $H$ is a calligraph, this
means that $H_{L}$ has still one degree of freedom or its configuration space
is empty. In these cases the degree of $K_{H_{L}}$ is zero, so Equation 10
implies that $(a,b,c)\cdot(1,1,0)=0$, namely $a=b$. We get the equations
$a=c$, respectively $a=0$, if $H_{R}$, respectively $H_{C_{v}}$, is not
minimally rigid.

Therefore, whether the graphs $H_{L}$, $H_{R}$, and $H_{C_{v}}$ are minimally
rigid or not, they can be used to determine the $S^{2}$-class of $H$. Notice
that, in order for this recursive procedure to become an algorithm, it is
crucial that the graphs $H_{L}$, $H_{R}$, and $H_{C_{v}}$, when minimally
rigid, are smaller than the graph $G$ we started with. However, it is not
always possible to find a calligraphic split of $G$ that has this property. In
this case, we fall back and use the algorithm from [GGS20] to compute the
number of realizations.

We summarize in Algorithm CountByCalligraphicSplit the procedure we described
in this section.

Algorithm CountByCalligraphicSplit

1:A minimally rigid graph $G$.

2:The number $\mathbf{c}_{S^{2}}(G)$ of realizations of $G$ on the complex
sphere, up to $\mathrm{SO}_{3}(\mathbb{C})$.

3:

4:If there is no non-trivial calligraphic split of $G$ Then

5: Return $\mathbf{c}_{S^{2}}(G)$ using the algorithm from [GGS20].

6:End If

7:Construct a non-trivial calligraphic split $(G_{1},G_{2})$ of $G$.

8:Return QuadForm(S2-Class($G_{1}$), S2-Class($G_{2}$)).

Algorithm QuadForm

1:Two integer vectors $(a_{1},b_{1},c_{1})$ and $(a_{2},b_{2},c_{2})$.

2:Their product according to the quadric form from Theorem 2.6.

3:

4:Return $2(a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2})$.

Algorithm S2-Class

1:A calligraph $H$.

2:The $S^{2}$-class $[H]$.

3:

4:If $H=L$ Then

5: Return $(1,1,0)$.

6:Else If $H=R$ Then

7: Return $(1,0,1)$.

8:Else If $H=C_{v}$ Then

9: Return $(2,0,0)$.

10:End If

11:Initialize eqs := empty list.

12:For $F\in\\{L,R,{C_{v}}\\}$ Do

13: If $H_{F}$ is minimally rigid Then

14: Compute $r:=\text{CountByCalligraphicSplit}(H_{F})$.

15: Append QuadForm(S2-Class($F$), $(a,b,c)$) $=r$ to eqs.

16: Else

17: Append QuadForm(S2-Class($F$), $(a,b,c)$) $=0$ to eqs.

18: End If

19:End For

20:Solve the equations in eqs with respect to the variables $a$, $b$, and $c$.

21:Return $(a,b,c)$.

######  Example 7.1.

Taking the example from [GEHL22b, Appendix B] we have a graph with a
calligraphic split (see Figure 6). Similarly to the plane the realization
count on the sphere cannot be computed by previous algorithms but it can by
our recursion. The two realization counts in the plane and on the sphere
differ due to the leaves in the recursion tree but the algorithm is the very
same. We get

$\mathbf{c}_{S^{2}}(G)=491520=(384,0,0)\cdot(640,256,384).$

The classes of the component graphs can be computed recursively as described
in Algorithm S2-Class using the calligraphic splits from [GEHL22b].

Figure 6: A calligraphic split of a graph $G=G_{1}\cup G_{2}$.

The implementation [GGS22] of the algorithm from [GGS20] for computing
realization counts on the sphere is much slower than the planar implementation
[CGG+18a] for [CGG+18b]. This is partially for implementation reasons but
mainly due to the algorithms themselves. For this reason the algorithm
described in the current paper can take advantage already for smaller graphs.
A timing comparison can be seen in Figure 7.

%#vertices$0$$50$$100$$150$$200$$8$$9$$10$$11$ Figure 7: The timing for
Algorithm CountByCalligraphicSplit relative to Algorithm [GGS20] using the set
of all minimally rigid graphs that have a non-trivial calligraphic split. The
$100\%$ line represents the timing for Algorithm [GGS20]. We see that using
the splitting we are already faster for graphs with at least 9 vertices.

We indicated throughout the paper that the recursive algorithm presented here
is indeed the same as the one from [GEHL22b] just the fallback algorithm is
different, i. e. the algorithm that is used when there is no calligraphic
split. With that respect the implementation [GEHL22a] in its Version 1 could
be immediately used together with [Gra22] for doing computations on the
sphere. We updated the implementation to make the workflow simpler.

##  Appendix A The $S^{2}$-class of $C_{v}$

The goal of this appendix is to prove that the $S^{2}$-class of $C_{v}$ is
$(2,0,0)$ starting from what we have proven in Section 6. Since we have
already showen that there are only two possibilities for $[C_{v}]$, namely
$(2,0,0)$ and $(6,4,4)$, it is enough to show that $[C_{v}]$ is different from
$(6,4,4)$.

First of all, we describe the exceptional locus of the blowup map
$\overline{\mathscr{M}}_{0,2V_{12}}\rightarrow S_{3}$ defined in Section 5.

######  Proposition A.1.

The exceptional locus of the blowup map
$\overline{\mathscr{M}}_{0,2V_{12}}\rightarrow S_{3}$ is the union of the
supports of the divisors $D_{I,J}$ such that
$\left|I\right|=\left|J\right|=3$.

###### Proof.

In this proof we use some concepts from the theory of quotients of algebraic
varieties, i. e., invariant theory. Specifically, we use the notion of
_stable_ and _semistable_ elements; for a reference, see [Dol03, Section 8.1].

As we clarified in Section 5, the variety $S^{3}$ is a moduli space of
$6$-tuples of points in $\mathbb{P}^{1}$ with at most three coinciding points.
In particular, there is a rational map
$\left(\mathbb{P}^{1}\right)^{6}\dasharrow S^{3}$ whose general fibers are the
orbits of $6$-tuples of different points under the action of
$\mathbb{P}\mathrm{GL}_{2}$. This map factors through the map
$\overline{\mathscr{M}}_{0,2V_{12}}\rightarrow S^{3}$ that is the blowup at
the $10$ nodes of $S^{3}$, mentioned at the beginning of Section 5, since
$\overline{\mathscr{M}}_{0,2V_{12}}$ is the moduli space of stable curves with
$6$ marked points. From [Dol03, Example 11.6], we know that the preimages
under $\left(\mathbb{P}^{1}\right)^{6}\rightarrow S_{3}$ of the nodes of
$S_{3}$ are constituted of the elements that are semistable but not stable.
Now, [Dol03, Theorem 11.2] states that the latter elements are the $6$-tuples
of points in $\mathbb{P}^{1}$ such that there are exactly three points
coinciding: in fact, using the reference’s notation, we have $n=1$, $m=6$,
$k_{i}=1$ for all $i\in\\{1,\dotsc,6\\}$ and $W$ can only be a point of
$\mathbb{P}^{1}$, so the condition for semistability is that the number of
points in a tuple that coincides with $W$ is less than or equal to the
quantity

$\frac{\dim(W)+1}{n+1}\left(\sum_{i=1}^{m}k_{i}\right)=\frac{1}{2}\cdot
6=3\,.$

The image in $\overline{\mathscr{M}}_{0,2V_{12}}$ of the set of these
$6$-tuples is precisely the union of the supports of the divisors described in
the statement. In fact, as explained in [GGS20, Definition 9] where the
notation is introduced (slightly modifying the one used in [Kee92]), the
elements of the support of a divisor $D_{I,J}$ are the stable rational curves
that have two components, one with $|I|$ marked points and the other with
$|J|$ marked points. ∎

We now show that $[C_{v}]=(2,0,0)$. To do so, it is enough to show that, when
$\Lambda\in\overline{\mathscr{M}}_{0,2\\{1,2\\}}\times\overline{\mathscr{M}}_{0,2\\{1,v\\}}\times\overline{\mathscr{M}}_{0,2\\{2,v\\}}\times\overline{\mathscr{M}}_{0,2\\{0,v\\}}$
is general and
$\widetilde{\Lambda}:=\bigl{(}\Lambda_{1v},\Lambda_{2v},\Lambda_{0v}\bigr{)}$,
the curve

$\eta(\widetilde{\mathscr{K}}_{C_{v}}^{\widetilde{\Lambda}})\cap\mathscr{K}_{G_{12}}^{\Lambda_{12}}$

does not intersect any divisor $D_{I,J}$ of
$\overline{\mathscr{M}}_{0,2V_{12}}$ with $|I|=|J|=3$, where
$\eta\colon\overline{\mathscr{M}}_{0,2V_{C_{v}}}\rightarrow\overline{\mathscr{M}}_{0,2V_{12}}$
is the forgetful morphism.

Indeed, in light of Proposition A.1, this means that the image of that curve
in $S_{3}$ does not pass through any of the nodes of the cubic hypersurface
$S_{3}$, which in turn by Definition 6.1 means that the second and third entry
of the $S^{2}$-class $[C_{v}]$ must be zero, ruling out the possibility
$[C_{v}]=(6,4,4)$. Let us first understand which divisors
$D_{I,J}\subset\overline{\mathscr{M}}_{0,2V_{12}}$ with $|I|=|J|=3$ intersect
$\mathscr{K}_{G_{12}}^{\Lambda_{12}}$. From [Kee92, Fact 2], we know that, for
any divisor $D_{I,J}$ with $|I\cap\\{P_{1},Q_{1},P_{2},Q_{2}\\}|=2$, the edge
forgetful morphism
$\pi_{\\{1,2\\}}\colon\overline{\mathscr{M}}_{0,2V_{12}}\rightarrow\overline{\mathscr{M}}_{0,2\\{1,2\\}}$
is constant and equal to one of the three special values $(1:0)$, $(0:1)$, or
$(1:1)$. Notice that the convention we follow is that, for a divisor $D_{I,J}$
in $\overline{\mathscr{M}}_{0,2V}$, where $G=(V,E)$ is a graph, the sets $I$
and $J$ are a partition of the set $\\{P_{w},Q_{w}\\}_{w\in E}$ of marked
points, while [Kee92] denotes these divisors by $D^{T}$, where $T$ is a subset
of the set $\\{P_{w},Q_{w}\\}_{w\in E}$, so one can think that $T$ plays the
role of $I$ and the complement of $T$ in $\\{P_{w},Q_{w}\\}_{w\in E}$ plays
the role of $J$. However, the variety $\mathscr{K}_{G_{12}}^{\Lambda_{12}}$ is
by definition a fiber of $\pi_{\\{1,2\\}}$ over a general point of
$\overline{\mathscr{M}}_{0,2V_{12}}$, hence it cannot intersect divisors
$D_{I,J}$ such that $|I\cap\\{P_{1},Q_{1},P_{2},Q_{2}\\}|=2$. Then, the only
divisors $D_{I,J}$ with $|I|=|J|=3$ that intersect
$\mathscr{K}_{G_{12}}^{\Lambda_{12}}$ are those such that
$|I\cap\\{P_{1},Q_{1},P_{2},Q_{2}\\}|\not=2$. There are only four
possibilities (together with the other four conjugate divisors obtained by
swapping $I$ with $J$), namely:

$I=\\{P_{0},Q_{0},P_{1}\\}\quad\text{or}\quad
I=\\{P_{0},Q_{0},Q_{1}\\}\quad\text{or}\quad
I=\\{P_{0},Q_{0},P_{2}\\}\quad\text{or}\quad I=\\{P_{0},Q_{0},Q_{2}\\}\,.$

To conclude the computation, notice that, for the curve
$\eta(\widetilde{\mathscr{K}}_{C_{v}}^{\widetilde{\Lambda}})\cap\mathscr{K}_{G_{12}}^{\Lambda_{12}}$
to intersect one of the divisors $D_{I,J}$ above, it must happen that
$\widetilde{\mathscr{K}}_{C_{v}}^{\widetilde{\Lambda}}$ intersects
$\eta^{-1}(D_{I,J})$. We focus on $D_{I,J}$ with $I=\\{P_{0},Q_{0},P_{1}\\}$,
since the other three cases are completely analogous. The preimage
$\eta^{-1}(D_{I,J})$ is the union of the supports four divisors in
$\overline{\mathscr{M}}_{0,2V_{C_{v}}}$.They are also of the form $D_{M,N}$,
where

$\displaystyle M$ $\displaystyle=\\{P_{0},Q_{0},P_{1},P_{v},Q_{v}\\}\,,$
$\displaystyle N$ $\displaystyle=\\{Q_{1},P_{2},Q_{2}\\}\,,$ $\displaystyle M$
$\displaystyle=\\{P_{0},Q_{0},P_{1}\\}\,,$ $\displaystyle N$
$\displaystyle=\\{Q_{1},P_{2},Q_{2},P_{v},Q_{v}\\}\,,$ $\displaystyle M$
$\displaystyle=\\{P_{0},Q_{0},P_{1},P_{v}\\}\,,$ $\displaystyle N$
$\displaystyle=\\{Q_{1},P_{2},Q_{2},Q_{v}\\}\,,$ $\displaystyle M$
$\displaystyle=\\{P_{0},Q_{0},P_{1},Q_{v}\\}\,,$ $\displaystyle N$
$\displaystyle=\\{Q_{1},P_{2},Q_{2},P_{v}\\}\,.$

We denote them by $D_{1}$, $D_{2}$, $D_{3}$, and $D_{4}$. With an argument
analogous to the one just used for $\mathscr{K}_{G_{12}}^{\Lambda_{12}}$, the
fact that $\Lambda_{0v}$ is general implies that
$\widetilde{\mathscr{K}}_{C_{v}}^{\widetilde{\Lambda}}$ does not intersect
$D_{2}$ (since $|M\cap\\{P_{0},Q_{0},P_{v},Q_{v}\\}|=2$); the fact that
$\Lambda_{1v}$ is general implies that
$\widetilde{\mathscr{K}}_{C_{v}}^{\widetilde{\Lambda}}$ does not intersect
$D_{3}$ and $D_{4}$ (since $|M\cap\\{P_{1},Q_{1},P_{v},Q_{v}\\}|=2$); the fact
that $\Lambda_{2v}$ is general implies that
$\widetilde{\mathscr{K}}_{C_{v}}^{\widetilde{\Lambda}}$ does not intersect
$D_{1}$ (since $|M\cap\\{P_{2},Q_{2},P_{v},Q_{v}\\}|=2$). This concludes the
proof that $[C_{v}]=(2,0,0)$.

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(MG) University of Trieste, Department of Mathematics and Geosciences, Via
Valerio 12/1, 34127 Trieste, Italy
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(GG, NL) Johann Radon Institute for Computational and Applied Mathematics
(RICAM), Austrian Academy of Sciences
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(JS) Johannes Kepler University Linz, Research Institute for Symbolic
Computation (RISC)
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