# The field of moduli of varieties with a structure

Giulio Bresciani  Scuola Normale Superiore
Piazza dei Cavalieri 7
56126 Pisa
Italy<EMAIL_ADDRESS>
###### Abstract.

If $X$ is a variety with an additional structure $\xi$, such as a marked
point, a divisor, a polarization, a group structure and so forth, then it is
possible to study whether the pair $(X,\xi)$ is defined over the field of
moduli. There exists a precise definition of “algebraic structures” which
covers essentially all of the obvious concrete examples. We prove several
formal results about algebraic structures. There are immediate applications to
the study of fields of moduli of curves and finite sets in $\mathbb{P}^{2}$,
but the results are completely general.

Fix $G$ a finite group of automorphisms of $X$, a $G$-structure is an
algebraic structure with automorphism group equal to $G$. First, we prove that
$G$-structures on $X$ are in a $1:1$ correspondence with twisted forms of
$X/G\dashrightarrow\mathcal{B}G$. Secondly we show that, under some
assumptions, every algebraic structure on $X$ is equivalent to the structure
given by some $0$-cycle. Third, we give a cohomological criterion for checking
the existence of $G$-structures not defined over the field of moduli. Fourth,
we identify geometric conditions about the action of $G$ on $X$ which ensure
that every $G$-structure is defined over the field of moduli.

##  1\. Introduction

Let $K/k$ be a possibly infinite Galois extension of fields. Consider a
variety $X$ over $K$ with some additional structure $\xi$: for instance, $\xi$
might be a preferred point $x\in X(K)$, or a polarization, or a group scheme
structure (see [BV22, §5] for a precise definition of “structures”).

Given a Galois element $\sigma\in\operatorname{Gal}(K/k)$, consider the twist
$\sigma^{*}(X,\xi)$: if $X$ is defined by polynomials, this corresponds to
applying $\sigma$ to the coefficients of the polynomials. It can be shown that
the subgroup $H\subset\operatorname{Gal}(K/k)$ of elements $\sigma$ such that
$\sigma^{*}(X,\xi)\simeq(X,\xi)$ is open, and the field of elements of $K$
fixed by $H$ is called _the field of moduli_ of $(X,\xi)$. If $(X,\xi)$
descends to some subextension $K/k^{\prime}/k$, then $k^{\prime}$ contains the
field of moduli. The following is the basic question.

######  Question.

Is $(X,\xi)$ defined over its field of moduli?

One of the oldest known results regarding this question is the fact that an
elliptic curve $E$ over $\bar{\mathbb{Q}}$ is defined over $\mathbb{Q}(j_{E})$
where $j_{E}$ is the $j$-invariant of $E$; this result predates the concept of
field of moduli by several decades. Fields of moduli were introduced by
Matsusaka [Mat58] in 1958 and later clarified by Shimura [Shi59], who also
proved that a generic, principally polarized abelian variety of odd dimension
in characteristic $0$ is defined over its field of moduli [Shi72]. They have
been studied intensively over the years, mainly for curves and abelian
varieties, see for instance [Koi72] [Mur96] [DD97] [DE99] [Hug07] [CQ05]
[Kon09] [Mar13].

One main reason why results are restricted to curves and abelian varieties is
the lack of appropriate technology. In particular, a lot of results about
curves rely on results by Dèbes, Douai and Emsalem contained in [DD97] [DE99]
which until recently were only available in dimension $1$. In our joint
article with A. Vistoli [BV22], we have generalized and clarified their
methods and results: there are now a general framework and general techniques
for studying fields of moduli of varieties with a structure in arbitrary
dimension. As an example of our new techniques, we reproved and generalized
Shimura’s result about abelian varieties in a much more theoretical fashion
[BV22, Corollary 6.25].

The study of fields of moduli of higher dimensional varieties is thus a
largely unexplored topic. As a first application of our joint work with
Vistoli [BV22] to an open problem, in [Bre23b] [Bre23c] [Bre23d] we study the
field of moduli of curves and finite subsets of $\mathbb{P}^{2}$. Among other
things, we prove that every smooth plane curve of degree prime with $6$ is
defined over the field of moduli, and that odd degree is sufficient if the
base field is $\mathbb{R}$.

This brief note is an expansion of the general technology constructed in
[BV22]. The results contained here are crucial for our work on
$\mathbb{P}^{2}$, but completely general in nature, and will be applied in
more forthcoming works about fields of moduli of higher dimensional varieties.

### Acknowledgements

I would like to thank an anonymous referee for providing useful remarks.

### Notation

We use $\mathop{\underline{\mathrm{Aut}}}\nolimits$ for sheaves of groups of
automorphisms, and $\operatorname{Aut}$ for set-theoretic groups of
automorphisms.

##  2\. Contents of the paper

For the rest of the paper, $k$ is a field with separable closure $K$, and $X$
is a separated, connected, normal algebraic space of finite type over $K$.

In order to study the problem of the field of moduli of the space $X$ with
some additional structure $\xi$, such as a group structure, marked points,
effective divisors, polarizations and so forth we need to define precisely
what “an algebraic structure” is. We have given such a formalization using
stacks in [BV22, §5], and we recall it here in §3.

In our intentions, the concept of algebraic structure was nothing more than a
unifying definition. It turns out that studying algebraic structures as
independent, abstract objects not necessarily tied to some geometric meaning
leads downstream to insights about the original problem of fields of moduli of
actual, geometric objects. Let us give an example.

In [Bre23b] we prove that, for the large majority of finite subgroups
$G\subset\mathrm{PGL}_{3}(\bar{\mathbb{Q}})$, _every_ algebraic structure on
$\mathbb{P}^{2}_{\bar{\mathbb{Q}}}$ with automorphism group equal to $G$ is
defined over its field of moduli. This is completely independent of the
geometric origin of the structure: the result holds for sets of points, for
smooth curves, for cycles and so forth. The general techniques which we give
here have nothing to do with $\mathbb{P}^{2}$, though, so the same kind of
analysis can be done for other varieties.

Fix $G\subset\operatorname{Aut}_{K}(X)$ is a finite group of automorphisms of
$X$ of order prime with $\operatorname{char}k$. A $G$-structure is an
algebraic structure $\xi$ on $X$ such that the group scheme of automorphism
$\mathop{\underline{\mathrm{Aut}}}\nolimits(X,\xi)$ is constant and equal to
$G$.

###  2.1. Equivalent structures

Given a $G$-structure $\xi$ on $X$, the concept of field of moduli $k_{\xi}$
of $(X,\xi)$ is defined, and there is a finite étale gerbe $\mathcal{G}_{\xi}$
over the field of moduli $k_{\xi}$ called the _residual gerbe_ [BV22, §3.1]
which parametrizes twisted forms of $(X,\xi)$, and a universal family
$\mathcal{X}_{\xi}\to\mathcal{G}_{\xi}$ whose fibers are the corresponding
twists of $X$ [BV22, §5]. In particular, $\xi$ is defined over $k_{\xi}$ if
and only if $\mathcal{G}_{\xi}(k_{\xi})\neq\emptyset$. The coarse moduli space
$\mathbf{X}_{\xi}$ of $\mathcal{X}_{\xi}$ is called the _compression_ of
$\xi$, and there is an induced rational map
$\mathbf{X}_{\xi}\dashrightarrow\mathcal{G}_{\xi}$.

We regard two $G$-structures as equivalent if they have the same field of
moduli and isomorphic universal families over isomorphic residual gerbes.
Loosely speaking, two $G$-structures are equivalent if they contain
essentially the same data and hence their respective twisted forms are in a
$1:1$ correspondence.

###  2.2. Twisted $G$-quotients

Our first result is a classification of algebraic structures up to
equivalence.

######  Definition 1.

A _twisted $G$-quotient_ of $X$ over $k$ is an algebraic space $Y$ over $k$
with an isomorphism $Y_{K}\simeq X/G$ such that for every
$\sigma\in\operatorname{Gal}(K/k)$ there exists a $\sigma$-linear automorphism
of $X$ which commutes with the composition $X\to X/G\simeq Y_{K}\to Y$.
Equivalently (see Corollary 10), a twisted $G$-quotient is a rational map
$Y\dashrightarrow\mathcal{G}$ which is a twisted form over $k$ of
$X/G\dashrightarrow\mathcal{B}G$.

We say that two twisted $G$-quotients are equivalent if there is an
isomorphism which respects the identification with $X/G$.

######  Theorem 2.

Mapping an algebraic structure $\xi$ with field of moduli $k$ to the
compression $\mathbf{X}_{\xi}\dashrightarrow\mathcal{G}_{\xi}$ defines a one-
to-one correspondence between $G$-structures on $X$ and twisted $G$-quotients
of $X$ over $k$, up to equivalence.

The structure $\xi$ is defined over the field of moduli if and only if the
projection $X\to X/G$ descends to a ramified covering of the compression
$\mathbf{X}_{\xi}$.

Thanks to Theorem 2, one can study twisted $G$-quotients directly, forgetting
about the original structure. If one can prove that, for a given $G$, every
twisted $G$-quotient $Y$ has a smooth (more generally, liftable cf. [BV22,
Definition 6.6]) rational point, then we automatically get that every
$G$-structure $\xi$ is defined over its field of moduli by the Lang-Nishimura
theorem for tame stacks [BV23, Theorem 4.1] applied to
$\mathbf{X}_{\xi}\dashrightarrow\mathcal{G}_{\xi}$, regardless of whether
$\xi$ was the datum of a point, a divisor, a group structure or anything else.
In fact, this is the case for most finite subgroups of both
$\mathop{\underline{\mathrm{Aut}}}\nolimits(\mathbb{P}^{1})=\mathrm{PGL}_{2}$
[Bre22] and
$\mathop{\underline{\mathrm{Aut}}}\nolimits(\mathbb{P}^{2})=\mathrm{PGL}_{3}$
[Bre23b].

###  2.3. Interpretation as cycle-structures

If, on the other hand, we find a twisted $G$-quotient $Y$ over $k$ whose
associated structure is not defined over $k$, we can search for a meaningful
structure whose compression is $Y$. This is the technique we used to construct
examples of cycles on $\mathbb{P}^{1}$ [Bre22] and on $\mathbb{P}^{2}$
[Bre23b] [Bre23c] not defined over their field of moduli. The next result says
that, under some assumptions, it is always possible to give such an
interpretation using $0$-cycles.

######  Theorem 3.

Assume that $\operatorname{char}k=0$, that
$\mathop{\underline{\mathrm{Aut}}}\nolimits(X)$ is of finite type over $k$,
and that there exists some finite extension $k^{\prime}/k$ with a model
$\mathfrak{X}^{\prime}$ over $k^{\prime}$ such that
$\mathfrak{X}^{\prime}(k^{\prime})$ is dense.

For every algebraic structure $\xi$ on $X$, there exists a $0$-cycle $Z$ on
$X$ such that $(X,Z)$ is equivalent to $(X,\xi)$.

###  2.4. Cohomological criterion

Using non-abelian cohomology, we give a criterion to study the existence of
$G$-structures not defined over their field of moduli. Suppose that $K$ is
separably closed, let $\mathfrak{X}$ be an integral algebraic space of finite
type over $k$ with $\mathfrak{X}_{K}=X$, and $\mathfrak{G}$ a group scheme
finite étale over $k$ with a faithful action on $\mathfrak{X}$ such that
$\mathfrak{G}_{K}=G\subset\mathop{\underline{\mathrm{Aut}}}\nolimits(X)$. Let
$\mathfrak{N}\subset\mathop{\underline{\mathrm{Aut}}}\nolimits(\mathfrak{X})$
a subgroup sheaf which normalizes $\mathfrak{G}$, and
$\mathfrak{Q}=\mathfrak{N}/\mathfrak{G}$ the quotient.

######  Theorem 4.

If the natural map
$\operatorname{H}^{1}(k,\mathfrak{N})\to\operatorname{H}^{1}(k,\mathfrak{Q})$
is not surjective, then there exists a $G$-structure on $X$ with field of
moduli $k$ which does not descend to $k$. If $\mathfrak{N}$ is the entire
normalizer of $\mathfrak{G}$, the converse holds.

###  2.5. Rational points on twisted quotients

While Theorem 4 is mostly useful to construct counterexamples by exhibiting an
actual element of $\operatorname{H}^{1}(k,\mathfrak{Q})$ which does not lift
to $\mathfrak{N}$, it is harder to apply it in the other direction, namely
showing that
$\operatorname{H}^{1}(k,\mathfrak{N})\to\operatorname{H}^{1}(k,\mathfrak{Q})$
is surjective when $\mathfrak{N}$ is the entire normalizer of $\mathfrak{G}$.
Usually, there is a more fruitful strategy for trying to show that, for fixed
$G$, an arbitrary $G$-structure is defined over the field of moduli. Assume
that $|G|$ is prime with $\operatorname{char}k$.

Let $Y\dashrightarrow\mathcal{G}$ be a twisted $G$-quotient over $k$. If we
can find a smooth rational point $y\in Y(k)$, since $|G|$ is prime with
$\operatorname{char}p$ then by the Lang-Nishimura theorem for tame stacks
[BV23, Theorem 4.1] we get that $\mathcal{G}(k)\neq\emptyset$, hence the
associated structure is defined over the field of moduli. Furthermore, the
smoothness assumption can be relaxed, see [BV22, §6] [Bre23a]. It is then
important to clarify under which conditions a closed subset $Z\subset X$
descends to a rational point (or more generally to a subspace) of every
twisted $G$-quotient. We identify and study such conditions in §7.

##  3\. Algebraic structures

Let us recall how we formalized the concept of algebraic structures in [BV22].

Denote by $(\mathrm{FAS}/k)$ the fibered category over $(\mathrm{Aff}/k)$
whose objects over an affine scheme $S$ over $k$ are flat, finitely presented
morphisms $Y\to S$, where $Y$ is an algebraic space. A _category of structured
spaces_ over $k$ is a locally finitely presented fppf stack
$\mathcal{M}\to(\mathrm{Aff}/k)$ with a faithful cartesian functor
$\mathcal{M}\to(\mathrm{FAS}/k)$.

The following definition is implicit in [BV22].

######  Definition 5.

An _algebraic structure_ over $X$ is the datum of a category of structured
spaces $\mathcal{M}\to(\mathrm{Aff}/k)$ with an algebraic (cf. [BV22,
Definition 3.6]) morphism $\operatorname{Spec}K\to\mathcal{M}$ whose
composition with $\mathcal{M}\to(\mathrm{FAS}/k)$ is the $K$-point of
$(\mathrm{FAS}/k)$ corresponding to $X$.

Given a structure $\xi$ on $X$, the field of moduli $k_{\xi}$ of $(X,\xi)$ is
well defined [BV22, Definition 3.11], and there is an algebraic stack
$\mathcal{G}_{\xi}$ together with a universal family
$\mathcal{X}_{\xi}\to\mathcal{G}_{\xi}$ which parametrizes twisted forms of
$(X,\xi)$. The stack $\mathcal{G}_{\xi}$ is a gerbe of finite type over
$k_{\xi}$ [BV22, §3.1], and is called the residual gerbe of $(X,\xi)$. The
residual gerbe is naturally equipped with a fully faithful morphism
$\mathcal{G}_{\xi}\to\mathcal{M}$.

As we have said in the introduction, we regard two algebraic structures on $X$
as equivalent if they have the same field of moduli and isomorphic universal
families over isomorphic residual gerbes.

Since the residual gerbe together with the composition
$\mathcal{G}_{\xi}\to\mathcal{M}\to(\mathrm{FAS}/k)$ is itself a category of
structured spaces, we get the following fact, which can be regarded as a
shortcut for the definition of algebraic structure. The proof is
straightforward.

######  Lemma 6.

Mapping an algebraic structure $\xi$ to its universal family
$\mathcal{X}_{\xi}\to\mathcal{G}_{\xi}$ defines a bijection between

  * •

algebraic structures on $X$ up to equivalence, and

  * •

pair of morphisms $p:\operatorname{Spec}K\to\mathcal{G}$,
$\mathcal{X}\to\mathcal{G}$ with an isomorphism
$\mathcal{X}\times_{\mathcal{G}}\operatorname{Spec}K\xrightarrow{\sim}X$ where
$\mathcal{G}$ is a finite gerbe over a finite subextension of $K/k$ such that
the induced action of
$\mathop{\underline{\mathrm{Aut}}}\nolimits_{\mathcal{G}}(p)$ on $X$ is
faithful, up to equivalence.

##  4\. Proof of Theorem 2

Consider the groups $\operatorname{Aut}_{k}(X),\operatorname{Aut}_{K}(X)$ of
automorphisms of $X$ over $k$ and $K$ respectively. A $k$-automorphism of $X$
induces a $k$-automorphism of $K\subset\operatorname{H}^{0}(X,\mathcal{O})$,
since $K$ is the separable closure of $k$ in
$\operatorname{H}^{0}(X,\mathcal{O})$. Hence, there is a short exact sequence

$1\to\operatorname{Aut}_{K}(X)\to\operatorname{Aut}_{k}(X)\to\operatorname{Gal}(K/k)$

where the image of the right arrow is the Galois group of the field of moduli
of $X$, see [BV22, Proposition 3.13].

For ease of notation, from now on we always assume that twisted quotients are
defined over $k$, and that structures have field of moduli equal to $k$: by
base changing, it is always possible to reduce to this case.

Let $G\subset\operatorname{Aut}_{K}(X)$ be a finite group and $Y$ a twisted
$G$-quotient of $X$ over $k$ with natural projection $q:X\to X/G\simeq
Y_{K}\to Y$.

######  Definition 7.

The _distinctive subgroup_ $\mathcal{N}_{Y}\subset\operatorname{Aut}_{k}(X)$
of $Y$ is the subgroup of automorphisms $\phi$ of $X$ over $k$ such that the
diagram

${X}$${X}$${Y}$$\scriptstyle{q}$$\scriptstyle{\phi}$$\scriptstyle{q}$

commutes. If $\xi$ is an algebraic structure on $X$, the distinctive subgroup
of $\xi$ is the distinctive subgroup of its compression, and we simply write
$\mathcal{N}_{\xi}$.

The composition
$\mathcal{N}_{Y}\subset\operatorname{Aut}_{k}(X)\to\operatorname{Gal}(K/k)$ is
surjective by definition of twisted quotient. Furthermore, since $X$ is
separated and integral, any automorphism of $X$ which commutes with $X\to X/G$
is an element of $G$. We thus get a short exact sequence

$1\to G\to\mathcal{N}_{Y}\to\operatorname{Gal}(K/k)\to 1.$

######  Proposition 8.

Let $Y$ be a twisted $G$-quotient of $X$ over $k$. The quotient stack
$[\operatorname{Spec}K/\mathcal{N}_{Y}]$ is a finite étale gerbe over $k$ and
$[X/\mathcal{N}_{Y}]\to[\operatorname{Spec}K/\mathcal{N}_{Y}]$ together with
the tautological morphism
$\operatorname{Spec}K\to[\operatorname{Spec}K/\mathcal{N}_{Y}]$ defines an
algebraic structure with field of moduli equal to $k$ and compression equal to
$Y$.

###### Proof.

The morphism $[\operatorname{Spec}K/\mathcal{N}_{Y}]\to\operatorname{Spec}k$
is induced by $\operatorname{Spec}K\to\operatorname{Spec}k$, which is by
construction $\mathcal{N}_{Y}$-invariant. Notice that
$[\operatorname{Spec}K/\mathcal{N}_{Y}]_{K}\simeq[\operatorname{Spec}K/G]=\mathcal{B}G$,
hence $[\operatorname{Spec}K/\mathcal{N}_{Y}]$ is a finite étale gerbe.
Furthermore, we have

$\operatorname{Spec}K\times_{[\operatorname{Spec}K/\mathcal{N}_{Y}]}[X/\mathcal{N}_{Y}]\simeq\operatorname{Spec}K\times_{[\operatorname{Spec}K/G]}[X/G]\simeq
X.$

We conclude by applying Lemma 6. ∎

######  Corollary 9.

Two algebraic structures $\xi,\xi^{\prime}$ on $X$ with étale automorphism
groups are equivalent if and only if their distinctive subgroups are equal. ∎

######  Corollary 10.

The datum of a twisted $G$-quotient of $X$ over $k$ is equivalent to the datum
of a twisted form $Y\dashrightarrow\mathcal{G}$ of
$X/G\dashrightarrow\mathcal{B}G$ over $k$.

###### Proof.

If $Y$ is a twisted $G$-quotient over $k$,
$X/\mathcal{N}_{Y}\dashrightarrow[\operatorname{Spec}K/\mathcal{N}_{Y}]$ is a
twisted form of $X/G\dashrightarrow\mathcal{B}G$ over $k$.

Conversely, assume that $Y\dashrightarrow\mathcal{G}$ is a twisted form of
$X/G\dashrightarrow\mathcal{B}G$ over $k$; let
$\sigma\in\operatorname{Gal}(K/k)$ be a Galois automorphism of $K/k$ and
$V\subset Y$ be an open subset such that $V\to\mathcal{G}$ is a morphism,
write $U\subset X$ for the inverse image of $V$.

Any two morphisms $\operatorname{Spec}K\to\mathcal{G}$ are isomorphic since
$\mathcal{G}$ is a finite étale gerbe and $K$ is separably closed. This
implies that $V\times_{\mathcal{G}}\operatorname{Spec}K$ is isomorphic to both
$U$ and $\sigma^{*}U$, hence we get an isomorphism $U\simeq\sigma^{*}U$ over
$V$, or equivalently a $\sigma$-linear automorphism of $U$ over $V$. Since $X$
is normal, it coincides with the integral closure of $V$ in $U$; we thus get
an induced $\sigma$-linear automorphism of $X$ over $Y$. ∎

The first part of Theorem 2 is a direct consequence of Lemma 6, Proposition 8
and Corollary 10.

If $Y\dashrightarrow\mathcal{G}$ is a twisted form of
$X/G\dashrightarrow\mathcal{B}G$, a direct way of constructing the universal
family
$[X/\mathcal{N}_{Y}]\to\mathcal{G}=[\operatorname{Spec}K/\mathcal{N}_{Y}]$
without using distinctive subgroups is to consider the relative normalization
of $Y\times\mathcal{G}$ with respect to $\operatorname{Spec}k(Y)\to
Y\times\mathcal{G}$.

Let us now look at the second part of Theorem 2. If
$\operatorname{Spec}k\to\mathcal{G}_{\xi}$ is a rational section, then
$E\mathrel{\smash{\overset{\mathrm{\scriptscriptstyle
def}}{=}}}\operatorname{Spec}k\times_{\mathcal{G}_{\xi}}\mathcal{X}_{\xi}$
with the composition $E\to\mathcal{X}_{\xi}\to\mathbf{X}_{\xi}$ defines a
twisted form of $X\to X/G$ over $k$.

On the other hand, if $E\to\mathbf{X}_{\xi}$ is a twisted form of $X\to X/G$,
then $\operatorname{Aut}(X/E)=\operatorname{Gal}(K/k)\subset\mathcal{N}_{\xi}$
defines a section of $\mathcal{N}_{\xi}\to\operatorname{Gal}(K/k)$, which in
turn induces a morphism
$\operatorname{Spec}k\to\mathcal{G}_{\xi}=[\operatorname{Spec}K/\mathcal{N}_{\xi}]$.

##  5\. Proof of Theorem 3

Let us construct the algebraic structure associated with a cycle $Z$ on $X$
with finite automorphism group scheme.

If $C\subset X$ is a reduced, irreducible closed subscheme and $S$ is a scheme
over $k$, a twisted form of $C$ over $S$ is a flat, locally of finite type
morphism $\mathcal{X}\to S$ with a closed subscheme
$\mathcal{C}\subset\mathcal{X}$ such that there exists a finite subextension
$K/k^{\prime}/k$, a scheme $S^{\prime}$ over $k^{\prime}$ and an fppf covering
$S^{\prime}\to S$ such that

$(\mathcal{X}|_{S^{\prime}}\times_{k^{\prime}}K,\mathcal{C}|_{S^{\prime}}\times_{k^{\prime}}K)\simeq(X\times_{K}S^{\prime}_{K},C\times_{K}S^{\prime}_{K}).$

As expected, $(X,C)$ defines a twisted form of $(X,C)$: since $(X,C)$ descends
to some finite subextension $K/k^{\prime}/k$, then
$(X\times_{k^{\prime}}K,C\times_{k^{\prime}}K)$ is isomorphic to
$(X\times_{K}\operatorname{Spec}K\otimes_{k^{\prime}}K,C\times_{K}\operatorname{Spec}K\otimes_{k^{\prime}}K)$
and hence we may take $S^{\prime}=S=\operatorname{Spec}K$.

If $Z=\sum_{i}n_{i}C_{i}$ is a cycle, a twisted form of $Z$ over $S$ is a
flat, locally of finite type morphism $\mathcal{X}\to S$ with a formal sum
$\sum_{i}n_{i}\mathcal{C}_{i}$ where $\mathcal{C}_{i}\subset\mathcal{X}$ is a
twist of $C_{i}$.

The residual gerbe $\mathcal{G}_{Z}$ is the functor $S\mapsto${twisted forms
of $Z$}; if the group scheme of automorphisms
$\mathop{\underline{\mathrm{Aut}}}\nolimits(X,Z)$ is finite then
$\mathcal{G}_{Z}$ is an algebraic stack which is a gerbe by [BV22, Proposition
3.9] and $\mathcal{X}_{Z}\to\mathcal{G}_{Z}$ is the corresponding universal
family whose $S$-points are twisted forms $(\mathcal{X},\mathcal{Z})$ of $Z$
over $S$ with a section $S\to\mathcal{X}$. The coarse moduli space of
$\mathcal{G}_{Z}$ is the spectrum of the field of moduli $k_{Z}$ of $Z$.

######  Lemma 11.

If
$\mathop{\underline{\mathrm{Aut}}}\nolimits(X,Z)=\operatorname{Aut}_{K}(X,Z)$
is finite étale, then
$\mathcal{N}_{Z}=\operatorname{Aut}_{k}(X,Z)\subset\operatorname{Aut}_{k}(X)$.

###### Proof.

Let $\mathbf{X}_{Z}$ be the compression, i.e. the coarse moduli space of
$\mathcal{X}_{Z}$, and denote by $c$ the morphism $X\to\mathcal{X}_{Z}$.
First, let us show that $\mathcal{X}_{Z}\to\mathbf{X}_{Z}$ is birational.
Since $X$ is integral, there exists an open subset $U\subset X$ which is
$\operatorname{Aut}_{K}(X,Z)$-invariant and such that the action on $U$ is
free; this induces an open subset $\mathcal{U}\subset\mathcal{X}_{Z}$. A
morphism $s:S\to\mathcal{U}$ corresponds to a twisted form
$(\mathcal{X},\mathcal{Z})$ of $(X,Z)$ with a section $S\to\mathcal{X}$ which
lands in the subspace of $\mathcal{X}$ corresponding to $U$. An automorphism
of $s$ is an automorphism of the twisted form $(\mathcal{X},\mathcal{Z})$
which maps $s$ to itself. Since $s$ lands in the locus corresponding to $U$,
and the action on $U$ is free, it is straightforward to check that such an
automorphism is the identity. This implies that $\mathcal{U}$ is an algebraic
space, and thus $\mathcal{X}_{Z}\to\mathbf{X}_{Z}$ is birational.

Let us now prove the main statement. By definition, $\mathcal{N}_{Z}$ is the
group of $k$-automorphisms of $X$ which commute with the composition
$X\to\mathcal{X}_{Z}\to\mathbf{X}_{Z}$. If $g$ is a $k$-linear automorphism of
$(X,Z)$, then by definition of $\mathcal{X}_{Z}$ the composition $c\circ g$ is
$2$-isomorphic to $c$, so in particular $g\in\mathcal{N}_{Z}$. On the other
hand, if $g\in\mathcal{N}_{Z}$, the two compositions of $c\circ g$ and $c$
with $\mathcal{X}_{Z}\to\mathbf{X}_{Z}$ are equal. Since
$\mathcal{X}_{Z}\to\mathbf{X}_{Z}$ is birational and $c$ is surjective, we get
that $c\circ g$ and $c$ are isomorphic on a dense open subset. Since
$\mathcal{X}_{Z}$ is separated (which follows from the fact that $X$ is
separated) then $c\circ g$ and $c$ are isomorphic globally. By definition of
$\mathcal{X}_{Z}$, this implies that $g^{*}Z=Z$, hence
$g\in\operatorname{Aut}_{k}(X,Z)$. ∎

As a consequence, another way of constructing
$\mathcal{X}_{Z}\to\mathcal{G}_{Z}$ is by considering the quotient stacks
$[X/\operatorname{Aut}_{k}(X,Z)]\to[\operatorname{Spec}K/\operatorname{Aut}_{k}(X,Z)]$
as in Proposition 8.

######  Lemma 12.

Let $Y$ be a twisted $G$-quotient of $X$ over $k$ and $U\subset X$ a
$\mathcal{N}_{Y}$-invariant non-empty open subset, i.e. the inverse image of
an open subset of $Y$. Assume that there exists a finite subextension
$K/k^{\prime}/k$ such that $Y(k^{\prime})$ is dense. For every
$\tau\in\operatorname{Aut}_{k}(X)\smallsetminus\mathcal{N}_{Y}$, there exists
a $\mathcal{N}_{Y}$-invariant finite subset $Z\subset U(K)$ such that
$\tau(Z)\neq Z$.

###### Proof.

Let $q:X\to X/G=Y_{K}\to Y$ be the composition, by hypothesis
$q\circ\tau\neq\tau$. Up to enlarging $k^{\prime}$, we may assume that
$k^{\prime}/k$ is Galois. We may find a finite _subset_
$H\subset\mathcal{N}_{Y}$ such that $H\to\operatorname{Gal}(k^{\prime}/k)$ is
surjective. Notice that we cannot impose that $H$ is a subgroup, but
fortunately this is not necessary.

If $x:\operatorname{Spec}K\to U$ is a point such that $q\circ
x:\operatorname{Spec}K\to Y$ is $k^{\prime}$-rational and
$\phi\in\mathcal{N}_{Y}$, $h\in H\subset\mathcal{N}_{Y}$ are elements with
equal images in $\operatorname{Gal}(k^{\prime}/k)$, then clearly

$q\circ\phi\circ x=q\circ h\circ x:\operatorname{Spec}K\to Y.$

Since $\tau$ is not in $\mathcal{N}_{Y}$, I claim that there exists a point
$x:\operatorname{Spec}K\to U$ whose image in $Y$ is $k^{\prime}$-rational and
such that $\tau\circ x\neq\phi\circ x$ for every $\phi\in\mathcal{N}_{Y}$. If
by contradiction this is false, then for every $x\in U(K)$ whose image in $Y$
is $k^{\prime}$-rational we may choose $h_{x}\in H$ such that $q\circ\tau\circ
x=q\circ h_{x}\circ x$. Since $H$ is finite and $k^{\prime}$-rational points
are dense, this implies that there exists an element $h\in H$ such that
$q\circ\tau=q\circ h$. Since $h\in H\subset\mathcal{N}_{Y}$, then $q\circ h=q$
and thus $q\circ\tau=q$, which is absurd.

Let $x:\operatorname{Spec}K\to U$ be a point whose image in $Y$ is
$k^{\prime}$-rational and such that $\tau\circ x\neq\phi\circ x$ for every
$\phi\in\mathcal{N}_{Y}$, and denote by $Z$ its $\mathcal{N}_{Y}$-orbit.
Clearly, $Z$ is $\mathcal{N}_{Y}$-invariant and $\tau(Z)\neq Z$ since
$\tau\circ x\in\tau(Z)$ and $\tau\circ x\not\in Z$. ∎

Let us now prove Theorem 3. Thanks to Corollary 9 and since we are in
characteristic $0$, it is enough to find a $0$-cycle $Z$ on $X$ such that
$\operatorname{Aut}_{k}(X,Z)=\mathcal{N}_{\xi}$.

Let $Z_{0}$ be the empty $0$-cycle, and define a $\mathcal{N}_{\xi}$-invariant
cycle $Z_{i}$ by recursion as follows. If
$\operatorname{Aut}_{k}(X,Z_{i})=\mathcal{N}_{\xi}$, then $Z_{i+1}=Z_{i}$.
Otherwise, choose
$\tau\in\operatorname{Aut}_{k}(X,Z_{i})\smallsetminus\mathcal{N}_{\xi}$ and
define

$Z_{i+1}=Z_{i}+(i+1)Z_{\tau},$

where $Z_{\tau}\subset X(K)$ is a $\mathcal{N}_{\xi}$-invariant subset such
that $\tau(Z_{\tau})\neq Z_{\tau}$ and such that the support of $Z_{\tau}$ is
disjoint from the support of $Z_{i}$, it exists by Lemma 12. By construction,
the coefficients of $Z_{i}$ are at most $i$ while the coefficients of
$Z_{\tau}$ are equal to $i+1$. This implies that

$\operatorname{Aut}_{k}(X,Z_{i+1})=\operatorname{Aut}_{k}(X,Z_{i})\cap\operatorname{Aut}_{k}(X,Z_{\tau}).$

In particular, $\mathcal{N}_{\xi}\subset\operatorname{Aut}_{k}(X,Z_{i+1})$ and
$\tau\not\in\operatorname{Aut}_{k}(X,Z_{i+1})$, hence
$\operatorname{Aut}_{k}(X,Z_{i+1})\subsetneq\operatorname{Aut}_{k}(X,Z_{i})$
if $\operatorname{Aut}_{k}(X,Z_{i})$ is different from $\mathcal{N}_{\xi}$.

The group $\operatorname{Aut}_{k}(X)$ is an extension of
$\operatorname{Gal}(K/k)$ by
$\operatorname{Aut}_{K}(X)=\mathop{\underline{\mathrm{Aut}}}\nolimits(X)(K)$.
The former is a compact group with respect to the pro-finite topology, while
the latter is compact with respect to the Zariski topology since
$\mathop{\underline{\mathrm{Aut}}}\nolimits(X)$ is noetherian. Notice that the
subgroups
$\operatorname{Aut}_{k}(X,Z_{i})\cap\operatorname{Aut}_{K}(X)=\operatorname{Aut}_{K}(X,Z_{i})$
and
$\operatorname{im}(\operatorname{Aut}_{k}(X,Z_{i})\to\operatorname{Gal}(K/k))=\operatorname{Gal}(K/k_{Z_{i}})$
are both closed. This implies that the chain
$\operatorname{Aut}_{k}(X,Z_{i})\supseteq\operatorname{Aut}_{k}(X,Z_{i})$ is
eventually stable, i.e. $\operatorname{Aut}_{k}(X,Z_{i})=\mathcal{N}_{\xi}$
for some $i>>0$. This concludes the proof of Theorem 3.

##  6\. Proof of Theorem 4

Let $\mathfrak{G}$ be a finite, étale group scheme over $k$ which acts
faithfully on an algebraic space $\mathfrak{X}$ of finite type over $k$. Write
$X=\mathfrak{X}_{K}$, $G=\mathfrak{G}_{K}$. We are going to show that twisted
$G$-quotients of $X$ are obtained by twisting
$\mathfrak{X}/\mathfrak{G}\dashrightarrow\mathcal{B}_{k}\mathfrak{G}$ with
some torsor.

Let $\mathop{\underline{\mathrm{Aut}}}\nolimits(\mathfrak{X})$ be the sheaf of
automorphisms of $\mathfrak{X}$. If $\mathfrak{X}$ is projective, this is
representable, but we don’t need this assumption. Let
$\mathfrak{N}\subset\mathop{\underline{\mathrm{Aut}}}\nolimits(\mathfrak{X})$
be a subgroup sheaf which normalizes $\mathfrak{G}$, and write $\mathfrak{Q}$
for the quotient $\mathfrak{N}/\mathfrak{G}$.

######  Lemma 13.

If $\mathfrak{N}$ is the entire normalizer of $\mathfrak{G}$, the sheaf of
groups $\mathfrak{Q}$ is isomorphic to the fppf sheaf $\mathfrak{F}$ of
automorphisms of $\mathfrak{X}/\mathfrak{G}$ for which, fppf locally, there
exists an automorphism of $\mathcal{B}_{k}\mathfrak{G}$ making the obvious
diagram $2$-commutative.

Notice that the particular automorphism of $\mathcal{B}_{k}\mathfrak{G}$ is
_not_ part of the definition of $\mathfrak{F}$, we only require existence.

###### Proof.

There is an obvious injective map $\mathfrak{Q}\to\mathfrak{F}$, let us show
that it is surjective. If $S$ is a scheme over $k$ and $f$ is an automorphism
of $\mathfrak{X}_{S}/\mathfrak{G}_{S}$ in $\mathfrak{F}(S)$, the hypothesis
implies that there exists an fppf covering $S^{\prime}\to S$ with an
automorphism $h$ of $\mathfrak{X}_{S^{\prime}}$ which lifts $f_{S^{\prime}}$.
The fact that $h$ lifts an automorphism of
$\mathfrak{X}_{S^{\prime}}/\mathfrak{G}_{S^{\prime}}$ implies that
$h\in\mathfrak{N}(S^{\prime})$. While the class of $h$ is not well defined and
depends on a choice, its image $q_{0}\in\mathfrak{Q}(S^{\prime})$ is unique.
The uniqueness of $q_{0}$ implies that it descends to an element
$q\in\mathfrak{Q}(S)$. It is straightforward to check that the image of $q$ in
$\mathfrak{F}$ is $f$. ∎

Let $T\to\operatorname{Spec}k$ be a $\mathfrak{Q}$-torsor over $k$, it
corresponds to a morphism
$\operatorname{Spec}k\to\mathcal{B}_{k}\mathfrak{Q}$. Define a twisted form
$\mathcal{X}\to\mathcal{G}$ of $[X/G]\to\mathcal{B}_{K}G$ as the fibered
products

${\mathcal{X}}$${\mathcal{G}}$${\operatorname{Spec}k}$${\left[\mathfrak{X}/\mathfrak{N}\right]}$${\mathcal{B}_{k}\mathfrak{N}}$${\mathcal{B}_{k}\mathfrak{Q}.}$$\scriptstyle{T}$

Since $T$ does not lift to $\mathfrak{N}$, then $\mathcal{G}(k)=\emptyset$ and
hence the corresponding structure is not defined over $k$.

Let us give another construction. There are actions of $\mathfrak{Q}$ on both
$[\mathfrak{X}/\mathfrak{G}]$ and on
$\mathcal{B}_{k}\mathfrak{G}=[\operatorname{Spec}k/\mathfrak{G}]$, and using
Romagny’s theory of group actions of stacks [Rom05] it is possible to define
twists

$[\mathfrak{X}/\mathfrak{G}]\times^{\mathfrak{Q}}T=[([\mathfrak{X}/\mathfrak{G}]\times
T)/\mathfrak{Q}],~{}\mathcal{B}_{k}\mathfrak{G}\times^{\mathfrak{Q}}T=[(\mathcal{B}_{k}\mathfrak{G}\times
T)/\mathfrak{Q}],$

with an induced morphism

$[\mathfrak{X}/\mathfrak{G}]\times^{\mathfrak{Q}}T\to\mathcal{B}_{k}\mathfrak{G}\times^{\mathfrak{Q}}T$

whose coarse moduli space is a twisted $G$-quotient.

By definition, $\mathcal{B}_{k}\mathfrak{G}\times^{\mathfrak{Q}}T$ is the
gerbe of liftings of $T$ to $\mathfrak{N}$. If $T$ is trivial, it is immediate
to check that
$[\mathfrak{X}/\mathfrak{G}]\times^{\mathfrak{Q}}T\simeq[X/\mathfrak{G}]$,
hence in general $[\mathfrak{X}/\mathfrak{G}]\times^{\mathfrak{Q}}T$ is a
twisted form of $[X/\mathfrak{G}]$.

If $T$ does not lift to $\mathfrak{N}$, the coarse moduli space of
$[\mathfrak{X}/\mathfrak{G}]\times^{\mathfrak{Q}}T$ is a twisted $G$-quotient
whose associated gerbe has no sections, this proves the first part of Theorem
4. The second part is a direct consequence of the following Proposition 14.

######  Proposition 14.

Assume that $\mathfrak{N}$ is the entire normalizer of $\mathfrak{G}$. Every
twisted $G$-quotient is the coarse moduli space of a twist

$[\mathfrak{X}/\mathfrak{G}]\times^{\mathfrak{Q}}T\to\mathcal{B}_{k}\mathfrak{G}\times^{\mathfrak{Q}}T$

for some $\mathfrak{Q}$-torsor $T\to\operatorname{Spec}k$.

###### Proof.

Let $Y\dashrightarrow\mathcal{G}$ be a twisted $G$-quotient over $k$. Define a
category fibered in sets $T$ as follows. If $S$ is a scheme over $k$, $T(S)$
is the set of isomorphisms

$\mathfrak{X}_{S}/\mathfrak{G}_{S}\xrightarrow{\sim}Y_{S}$

for which there exists an fppf covering $S^{\prime}\to S$ and a
$2$-commutative diagram

${\mathfrak{X}_{S^{\prime}}/\mathfrak{G}_{S^{\prime}}}$${\mathcal{B}_{S^{\prime}}\mathfrak{G}_{S^{\prime}}}$${Y_{S^{\prime}}}$${\mathcal{G}_{S^{\prime}}}$

where the vertical arrows are isomorphisms. We stress that the fppf covering
and the $2$-commutative diagram are _not_ part of the datum, we are only
selecting the isomorphisms for which such a diagram exists. It is clear that
$T$ is a subsheaf of the sheaf of isomorphisms between
$\mathfrak{X}/\mathfrak{G}$ and $Y$. The action of $\mathfrak{Q}$ on
$\mathfrak{X}/\mathfrak{G}$ induces an action on $T$, and Lemma 13 implies
that $T$ is a $\mathfrak{Q}$-torsor. It is straightforward to check that the
given $G$-quotient is the twist of
$\mathfrak{X}/\mathfrak{G}\dashrightarrow\mathcal{B}_{k}\mathfrak{G}$ by $T$.

∎

Sometimes, it is interesting to study whether the structure actually descends
to $\mathfrak{X}$, as opposed to a twist of $\mathfrak{X}$. For instance, if
$\mathfrak{X}=\mathbb{P}^{1}_{k}$ and the structure is a divisor
$D\subset\mathbb{P}^{1}_{K}$, one may want to study whether $D$ descends to
$\mathbb{P}^{1}_{k}$ [Mar13] [Bre22], but Theorem 2 only tells us whether $D$
descends to a Brauer-Severi variety of dimension $1$. Similarly, one might
want to study if the embedding of a curve in $\mathbb{P}^{2}$ is defined over
the field of moduli, and not only if the curve embeds in a Brauer-Severi
surface over the field of moduli.

######  Definition 15.

Let $X,G$ be as above, and $\mathfrak{X}$ a model of $X$ over $k$. A
$G$-structure $\mathcal{X}\to\mathcal{G}$ of $X$ with field of moduli $k$ is
_$\mathfrak{X}$ -neutral_ if there exists a $2$-cartesian diagram

${\mathfrak{X}}$${\mathcal{X}}$${\operatorname{Spec}k}$${\mathcal{G}}$$\scriptstyle{p}$

such that the induced embedding
$\mathop{\underline{\mathrm{Aut}}}\nolimits_{\mathcal{G}}(p)\subset\mathop{\underline{\mathrm{Aut}}}\nolimits(\mathfrak{X})$
identifies $\mathop{\underline{\mathrm{Aut}}}\nolimits_{\mathcal{G}}(p)(K)$
with $G$.

Consider the quotient sheaf
$\mathop{\underline{\mathrm{Aut}}}\nolimits(\mathfrak{X})/\mathfrak{N}$.
Taking the fibers of
$\mathop{\underline{\mathrm{Aut}}}\nolimits(\mathfrak{X})\to\mathop{\underline{\mathrm{Aut}}}\nolimits(\mathfrak{X})/\mathfrak{N}$
defines a function
$(\mathop{\underline{\mathrm{Aut}}}\nolimits(\mathfrak{X})/\mathfrak{N})(k)\to\operatorname{H}^{1}(k,\mathfrak{N})$.
Notice that these are sets, not groups. Still, they have a preferred object
and it makes sense to consider the kernel

$\mathrm{K}=\ker(\operatorname{H}^{1}(k,\mathfrak{N})\to\operatorname{H}^{1}(k,\mathop{\underline{\mathrm{Aut}}}\nolimits(\mathfrak{X}))),$

which is the image of
$(\mathop{\underline{\mathrm{Aut}}}\nolimits(\mathfrak{X})/\mathfrak{N})(k)$,
i.e. we have a long exact sequence for non-abelian cohomology. The elements of
$\mathrm{K}$ are the $\mathfrak{N}$-torsor which define the trivial twist of
$\mathfrak{X}$.

The following version of Theorem 4 is, again, a direct consequence of
Proposition 14

######  Theorem 16.

If the composition

$(\mathop{\underline{\mathrm{Aut}}}\nolimits(\mathfrak{X})/\mathfrak{N})(k)\twoheadrightarrow\mathrm{K}\hookrightarrow\operatorname{H}^{1}(k,\mathfrak{N})\to\operatorname{H}^{1}(k,\mathfrak{Q})$

is not surjective, there exists a non-$\mathfrak{X}$-neutral $G$-structure
with field of moduli equal to $k$. If $\mathfrak{N}$ is the entire normalizer
of $\mathfrak{G}$ in
$\mathop{\underline{\mathrm{Aut}}}\nolimits(\mathfrak{X})$ the converse holds.

##  7\. Distinguished subsets

As we have seen above, $G$-structures with field of moduli $k$ correspond to
twisted $G$-quotients of $X$ over $k$. If $Y$ is a twisted $G$-quotient over
$k$ there is a rational map $Y\dashrightarrow\mathcal{G}$ where $\mathcal{G}$
is the residual gerbe and the structure is defined over the field of moduli if
and only if $\mathcal{G}(k)\neq\emptyset$.

If $y\in Y(k)$ is a smooth rational point and $G$ has degree prime with
$\operatorname{char}k$, then by the Lang-Nishimura theorem for tame stacks
[BV23, Theorem 4.1] we have that $\mathcal{G}(k)\neq\emptyset$ and hence the
$G$-structure is defined over $k$. The smoothness assumption on $y$ can be
relaxed, see [BV22, §6], [Bre23a]. Because of this, it is important to have a
framework to construct rational points (or, more generally, subspaces) of
arbitrary twisted $G$-quotients. The proof of the following is
straightforward.

######  Lemma 17.

A closed subscheme $Z\subset X$ descends to a twisted $G$-quotient $Y$ if and
only if $\phi(Z)=Z$ for every $\phi\in\mathcal{N}_{Y}$.∎

Write $\mathcal{N}_{X/k,G}\subset\operatorname{Aut}_{k}(X)$ for the normalizer
of $G$ in $\operatorname{Aut}_{k}(X)$.

######  Lemma 18.

We have $\mathcal{N}_{Y}\subset\mathcal{N}_{X/k,G}$.

###### Proof.

If $\phi\in\mathcal{N}_{Y}$ and $g\in G$, then clearly $\phi^{-1}\circ
g\circ\phi\in\mathcal{N}_{Y}\cap\operatorname{Aut}_{K}(X)=G$, so
$\phi\in\mathcal{N}_{X/k,G}$. ∎

######  Definition 19.

A closed subset $Z\subset X$ is a _distinguished subset_ if, for every
$\tau\in\mathcal{N}_{X/k,G}$, $\tau(Z)=Z$.

A distinguished subset is $G$-invariant since $G\subset\mathcal{N}_{X/k,G}$,
but the converse is false. Write $\pi$ for the projection $X\to X/G$.

######  Lemma 20.

Let $Z\subset X$ be a distinguished subset and $Y$ a twisted $G$-quotient over
$k$. Then $\pi(Z)\subset X/G$ descends to a closed subset of $Y$.

###### Proof.

Follows directly from Lemma 17 and Lemma 18. ∎

Conjugation defines a homomorphism
$\mathcal{N}_{X/k,G}\to\operatorname{Aut}(G)$, let
$\mathcal{A}_{X/k,G}\subset\operatorname{Aut}(G)$ be its image. Clearly,
$\mathcal{A}_{X/k,G}$ contains every inner automorphism of $G$.

While distinguished subsets are naturally defined in terms of the group
$\mathcal{N}_{X/k,G}\subset\operatorname{Aut}_{k}(X)$, it is often sufficient
to have knowledge about $\mathcal{A}_{X/k,G}\subset\operatorname{Aut}(G)$. Let
us give some examples of this.

######  Example 21.

If $H\subset G$ is a subgroup and $Z\subset X$ a subspace, we say that $H$
stabilizes (resp. fixes) $Z$ if the elements of $H$ restrict to automorphisms
(resp. to the identity) on $Z$.

Let $H\subset G$ be a subgroup, and let $S_{H}$ be the set of subgroups of $G$
of the form $\phi(H)$ for some $\phi\in\mathcal{A}_{X/k,G}$. The following are
distinguished subsets.

  * •

The union and the intersection of the fixed loci of elements of $S_{H}$.

  * •

Let $n$ be an integer, and suppose that $H$ stabilizes (resp. fixes) a finite
number of irreducible closed subsets of dimension $n$, write $C_{n,H}$ for
their union and similarly $C_{n,H^{\prime}}$ for $H^{\prime}\in S_{H}$. Then
$\bigcup_{H^{\prime}\in S_{H}}C_{n,H^{\prime}}$ and $\bigcap_{H^{\prime}\in
S_{H}}C_{n,H^{\prime}}$ are distinguished subsets.

  * •

Suppose that $X$ is smooth and proper and that it has a line bundle $L$ whose
class in the Néron-Severi group is $\operatorname{Aut}_{k}(X)$-invariant, e.g.
$X=\mathbb{P}^{n}$ and $L=\mathcal{O}(1)$. Then we can repeat the previous
point but restricted to irreducible closed subset of a fixed degree $d$.

Because of Example 21 and the many other similar examples that can be given,
we want to understand the group
$\mathcal{A}_{X/k,G}\subset\operatorname{Aut}(G)$. We have inclusions

$\operatorname{Inn}(G)\subset\mathcal{A}_{X/K,G}\subset\mathcal{A}_{X/k,G}\subset\operatorname{Aut}(G).$

Usually, the subgroup $\mathcal{A}_{X/K,G}$ is easy to understand, since it is
defined in terms of “geometric” $K$-automorphisms of $X$ and it is normal in
$\mathcal{A}_{X/k,G}$. It remains to study its cokernel.

Suppose that $X,G$ descend to $\mathfrak{X},\mathfrak{G}$ over $k$ with
$\mathfrak{G}$ a group scheme, and that the action of $G$ on $X$ descends to
an action of $\mathfrak{G}$ on $\mathfrak{X}$. In particular, we have an
action of $\operatorname{Gal}(K/k)$ on $X$ and hence a section
$\operatorname{Gal}(K/k)\to\operatorname{Aut}_{k}(X)$ of
$\operatorname{Aut}_{k}(X)\to\operatorname{Gal}(K/k)$. Furthermore, we have an
action $\operatorname{Gal}(K/k)\to\operatorname{Aut}(G)$,
$\sigma\mapsto\phi_{\sigma}$ on $G=\mathfrak{G}(K)$.

######  Proposition 22.

The image of $\operatorname{Gal}(K/k)\to\operatorname{Aut}_{k}(X)$ is
contained in $\mathcal{N}_{X/k,G}$, and $\mathcal{N}_{X/k,G}$ is generated by
$\mathcal{N}_{X/K,G}$ and $\operatorname{Gal}(K/k)$.

###### Proof.

Choose an element $\sigma\in\operatorname{Gal}(K/k)$, let $\phi_{\sigma}$ be
the induced automorphism of $G=\mathfrak{G}(K)$. Since the action
$\rho:G\times X\to X$ descends to an action
$\mathfrak{G}\times\mathfrak{X}\to\mathfrak{X}$ over $k$, we have a
commutative diagram

${G\times X}$${X}$${G\times
X}$${X}$$\scriptstyle{\rho}$$\scriptstyle{\phi_{\sigma}\times\sigma^{*}}$$\scriptstyle{\sigma^{*}}$$\scriptstyle{\rho}$

where the vertical arrows are $\sigma$-equivariant. It follows that
$\sigma^{*}:X\to X$ is $\phi_{\sigma}$-equivariant, and hence
$\sigma^{*}\in\mathcal{N}_{X/k,G}$.

Now consider an element
$\tau\in\mathcal{N}_{X/k,G}\subset\operatorname{Aut}_{k}(X)$ normalizing $G$,
and let $\sigma$ be its image in $\operatorname{Gal}(K/k)$. Since
$\sigma^{*}\in\mathcal{N}_{X/k,G}$, then $\tau\circ\sigma^{*-1}$ is an element
of $\mathcal{N}_{X/k,G}$ which maps to the identity in
$\operatorname{Gal}(K/k)$, i.e. $\tau\circ\sigma^{*-1}\in\mathcal{N}_{X/K,G}$.
∎

######  Corollary 23.

The subgroup $\mathcal{A}_{X/k,G}\subset\operatorname{Aut}(G)$ is generated by
$\mathcal{A}_{X/K,G}$ and by the image of
$\operatorname{Gal}(K/k)\to\operatorname{Aut}(G)$. In particular, if
$\mathfrak{G}$ is an inner form of $G$ then
$\mathcal{A}_{X/k,G}=\mathcal{A}_{X/K,G}$.

## References

  * [Bre22] Giulio Bresciani, _The field of moduli of a divisor on a rational curve_ , arxiv:2211.03438, 2022.
  * [Bre23a] by same author, _The arithmetic of tame quotient singularities in dimension 2_ , Int. Math. Res. Not. (2023), https://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rnad079/7142915.
  * [Bre23b] by same author, _The field of moduli of plane curves_ , arxiv:2303.01454, 2023.
  * [Bre23c] by same author, _The field of moduli of sets of points in $\mathbb{P}^{2}$_, arxiv:2303.01408, 2023.
  * [Bre23d] by same author, _Real versus complex plane curves_ , arxiv:2309.12192, 2023.
  * [BV22] Giulio Bresciani and Angelo Vistoli, _Fields of moduli and the arithmetic of tame quotient singularities_ , arxiv:2210.04789, 2022, to appear in Compositio Mathematica.
  * [BV23] by same author, _An arithmetic valuative criterion for proper maps of tame algebraic stacks_ , Manuscripta Math. (2023), https://link.springer.com/article/10.1007/s00229-023-01491-6.
  * [CQ05] Gabriel Cardona and Jordi Quer, _Field of moduli and field of definition for curves of genus 2_ , Computational aspects of algebraic curves, World Scientific, 2005, pp. 71–83.
  * [DD97] Pierre Dèbes and Jean-Claude Douai, _Algebraic covers: field of moduli versus field of definition_ , Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 3, 303–338.
  * [DE99] Pierre Dèbes and Michel Emsalem, _On fields of moduli of curves_ , J. Algebra 211 (1999), no. 1, 42–56.
  * [Hug07] Bonnie Huggins, _Fields of moduli of hyperelliptic curves_ , Math. Res. Lett. 14 (2007), no. 2, 249–262.
  * [Koi72] Shoji Koizumi, _The fields of moduli for polarized abelian varieties and for curves_ , Nagoya Mathematical Journal 48 (1972), 37–55.
  * [Kon09] Aristides Kontogeorgis, _Field of moduli versus field of definition for cyclic covers of the projective line_ , Journal de théorie des nombres de Bordeaux 21 (2009), no. 3, 679–693.
  * [Mar13] Andrea Marinatto, _The field of definition of point sets in $\mathbb{P}^{1}$_, J. Algebra 381 (2013), 176–199.
  * [Mat58] T. Matsusaka, _Polarized varieties, fields of moduli and generalized Kummer varieties of polarized abelian varieties_ , Amer. J. Math. 80 (1958), 45–82.
  * [Mur96] Naoki Murabayashi, _The field of moduli of abelian surfaces with complex multiplication_ , J. Reine Angew. Math. 470 (1996), 1–26.
  * [Rom05] Matthieu Romagny, _Group actions on stacks and applications_ , Michigan Math. J. 53 (2005), no. 1, 209–236.
  * [Shi59] Goro Shimura, _On the theory of automorphic functions_ , Ann. of Math. (2) 70 (1959), 101–144.
  * [Shi72] by same author, _On the field of rationality for an abelian variety_ , Nagoya Math. J. 45 (1972), 167–178.


# Dumbbell dynamics: a didactical approach

Benedetto Scoppola1 and Matteo Veglianti2

###### Abstract.

In this paper we propose a simplified model to describe the dissipative
effects of tides. We assume a spherical Earth with a dissipative coupling with
a mechanical dumbbell. The latter has a mass much smaller than the Earth’s,
and it models the presence of the tidal bulges. Using properly the scale
analysis, we will show that some of the consequences of tidal dissipation are
the circularization and the enlargement of orbit of the Moon and the slowing
down of the Earth’s rotation. We will also see that tidal dissipation plays a
fundamental role for the establishment of a regime of spin-orbit resonance in
the celestial systems. The mathematical tools used make our treatment
appropriate for senior high school students or college students.

1 Dipartimento di Matematica,
Università di Roma “Tor Vergata”
Via della Ricerca Scientifica - 00133 Roma, Italy
<EMAIL_ADDRESS>

2 Dipartimento di Fisica,
Università di Roma “Tor Vergata”
Via della Ricerca Scientifica - 00133 Roma, Italy
<EMAIL_ADDRESS>

##  1\. Introduction

All textbooks in introductory astronomy and many in physics and mechanics
mention the existence of oceanic tides as an interesting manifestation of
universal gravitation: indeed many teachers are interested in this topic. As
argued in [1], the most important aspects of the origin and properties of
tides are often treated inaccurately or even erroneously. Much of the
confusion over generating tides is related to the roles of the orbital motion
of the Moon and earth about their common center of mass and of the Earth’s
axial rotation. In discussing the physics behind this phenomenon, authors
usually explain (more or less successfully) why two tidal swells appear on the
opposite sides of the globe. However, it is difficult to find a plausible
explanation of the physical mechanism responsible for the phase shift between
the zenith of the moon and the moment of high tide, which at some places
approaches 90°. Misunderstandings also occur in discussions about the role of
tidal friction in the retardation of axial rotations and in the evolution of
orbital motions of the gravitationally coupled celestial bodies.
While the conservative aspects of the tides are masterfully treated using
elementary tools in [1], [2] and [3], the dissipative ones are only
qualitatively described. The scientific, non-pedagogical, works on tidal
dissipation are divided into two large areas according to the desired target:
rheological aspects (see [4] and [5]) or dynamic aspects (see [6] and [9]).
Our aim is therefore to propose a way to quantitatively deal with the
dissipative aspects of the tides and their consequences using high school
mathematics. To this end, we will use the "dumbbell model" developed in [7],
that consists in describing the planet in terms of a point $P$ of mass $M-\mu$
and a mechanical dumbbell centered in $P$, i.e., a system of two points, each
of which has mass $\mu/2$, constrained to be at fixed mutual distance $2r$,
having $P$ as center of mass.
The idea of a dumbbell model is not original, in fact is developed in [9] and
in many works (see, for instance [8]) where however it is used for other
purposes.
This model is useful to compute, using an elementary force’s approach, the
torque acting on the Earth’s ocean bulges due to the Moon and the torque
acting on the Moon due to the Earth’s ocean bulges. We perform the detailed
computation in section 2. In section 3 we present a way to describe the
evolution of the system imagining that the variation of the parameters does
not occur in a continuous way, but rather discretely. This allows us to avoid
a treatment through differential equations and replace them with finite
difference equations.
In this way, it will be possible to show how the tidal dissipation is
responsible for the circularization and the enlargement of the lunar orbit and
the slowing down of the earth’s rotation. We will see that the first two
events occur on very different time scales: the circularization of the lunar
orbit first is much faster than the enlargement. Thus the orbit will become
circular in shorter times than those of enlargement, and this is what we see
in many planet-satellite systems of our galaxy, particularly in the Earth-Moon
system. Finally, we will also see that tidal dissipation plays a fundamental
role for the establishment of a regime of spin-orbit resonance in the
celestial systems.
The mathematical tools used make our discussion appropriate for senior high
school students or college students. We believe that an appropriate and non-
sterile use of mathematics is useful to understand its functionality and to
make children passionate about studying this discipline. Furthermore, this
subject is very suitable for teaching and learning the scale analysis, that is
a very powerful tool to simplify complex equations by neglecting the suitable
small terms.

##  2\. Tidal torque

In this section we want to show how the dumbbell model is useful to compute
the torque acting on the Earth’s ocean bulges due to the Moon. The ocean
bulges are modeled by the aforementioned dumbbell: a pair of massive point
each of which with a mass of $\mu/2$ placed at the ends of a segment of length
$2r$, that we can assume, for simplicity, equal to the diameter $2R$ of the
Earth.111With a detailed computation using suitable integrals, one can show
that $r=\frac{3}{4}R$.
In figure 1 we imagine the Earth as a sphere with radius $R$ centered at the
origin $O$ of the reference frame and the Moon as a massive point, indicated
with $S$, with mass $m$, at a distance $a$ from the center of the Earth. Let
the $x$-axis be the line joining $O$ with $S$ and the $y$-axis perpendicular
to it. Moreover, for simplicity, we consider the rotation of the Earth
perpendicular to the Moon’s orbital plane.
On the Earth’s surface there are the two massive points $C_{1}$ and $C_{2}$,
that represent the ocean bulge’s center of gravity. Finally, in general, we
imagine that the dumbbell is inclined by an angle $\varepsilon$ with respect
to the $x$-axis.
Let $\vec{F_{1}}$ and $\vec{F_{2}}$ be the attractive forces acting on $C_{1}$
and $C_{2}$ due to the Moon and let $\vec{F_{1}^{\prime}}$ and
$\vec{F_{2}^{\prime}}$ the attractive forces acting on $S$ due to $C_{1}$ and
$C_{2}$ respectively. Clearly $\vec{F_{1}^{\prime}}=-\vec{F_{1}}$ and
$\vec{F_{2}^{\prime}}=-\vec{F_{2}}$.
Moreover, on $S$ also acts the attractive force $\vec{F}$, lying on the
$x$-axis, due to the rest of the Earth, deprived of the two masses of water.

Figure 1. Geometry of the system.

Let $G$ be the universal gravitational constant and let $\beta_{1}$ and
$\beta_{2}$ be the two angles $O{\hat{S}}C_{1}$ and $O{\hat{S}}C_{2}$
respectively; we have:

$F_{1}=-F_{1}^{\prime}=-\frac{Gm\frac{\mu}{2}}{SC_{1}^{2}}(-\hat{x}\cos\beta_{1}+\hat{y}\sin\beta_{1}),$
(1)
$F_{2}=-F_{2}^{\prime}=-\frac{Gm\frac{\mu}{2}}{SC_{2}^{2}}(-\hat{x}\cos\beta_{2}-\hat{y}\sin\beta_{2}),$
(2) $F=-\frac{Gm(M-\mu)}{a^{2}}\hat{x}.$ (3)

Moreover, from the geometry of the system we have that $\frac{r}{a}$ is a
dimensionless small parameter, so we can expand the forces up to the second
order in $\frac{r}{a}$.
To this end, we need the following geometric relation:

$\begin{matrix}SC_{1}^{2}=(a-r\cos\varepsilon)^{2}+(r\sin\varepsilon)^{2}=a^{2}\biggl{(}1-2\frac{r}{a}\cos\varepsilon+\frac{r^{2}}{a^{2}}\biggr{)}\\\
SC_{2}^{2}=(a+r\cos\varepsilon)^{2}+(r\sin\varepsilon)^{2}=a^{2}\biggl{(}1+2\frac{r}{a}\cos\varepsilon+\frac{r^{2}}{a^{2}}\biggr{)}\\\
\beta_{1}=\frac{r\sin\varepsilon}{a-r\cos\varepsilon}=\frac{r}{a}\sin\varepsilon\biggl{(}1+\frac{r}{a}\cos\varepsilon\biggr{)}\\\
\beta_{2}=\frac{r\sin\varepsilon}{a+r\cos\varepsilon}=\frac{r}{a}\sin\varepsilon\biggl{(}1-\frac{r}{a}\cos\varepsilon\biggr{)}\end{matrix}$
(4)

From (1) we have:

$F_{1}=-F_{1}^{\prime}=-\frac{Gm\frac{\mu}{2}}{a^{2}}\biggl{(}1+2\frac{r}{a}\cos\varepsilon-\frac{r^{2}}{a^{2}}+4\frac{r^{2}}{a^{2}}\cos^{2}\varepsilon\biggr{)}\biggl{[}-\hat{x}\biggl{(}1-\frac{r^{2}}{2a^{2}}\sin^{2}\varepsilon\biggr{)}+\hat{y}\biggl{(}\frac{r}{a}\sin\varepsilon\biggl{(}1+\frac{r}{a}\cos\varepsilon\biggr{)}\biggr{)}\biggr{]}=$
$=-\frac{Gm\frac{\mu}{2}}{a^{2}}\biggl{[}-\hat{x}\biggl{(}1+2\frac{r}{a}\cos\varepsilon-\frac{r^{2}}{a^{2}}\biggl{(}\frac{3}{2}-\frac{9}{2}\cos^{2}\varepsilon\biggr{)}\biggr{)}+\hat{y}\biggl{(}\frac{r}{a}\sin\varepsilon\biggl{(}1+3\frac{r}{a}\cos\varepsilon\biggr{)}\biggr{)}\biggr{]}.$
(5)

Similarly, from (2) we have:

$F_{2}=-F_{2}^{\prime}=-\frac{Gm\frac{\mu}{2}}{a^{2}}\biggl{[}-\hat{x}\biggl{(}1-2\frac{r}{a}\cos\varepsilon-\frac{r^{2}}{a^{2}}\biggl{(}\frac{3}{2}-\frac{9}{2}\cos^{2}\varepsilon\biggr{)}\biggr{)}-\hat{y}\biggl{(}\frac{r}{a}\sin\varepsilon\biggl{(}1-3\frac{r}{a}\cos\varepsilon\biggr{)}\biggr{)}\biggr{]}.$
(6)

We are now ready to calculate the torque acting on the dumbbell due to the
Moon and the torque acting on the Moon due to the dumbbell. We will show that
the two torques are exactly opposite, as it is expected from the conservation
of angular momentum for isolated systems.
For simplicity we impose that the Moon moves in a circular orbit around the
point G, the center of gravity of the Earth-Moon system. This is equivalent to
stating that the sum of the components along the $x$-axis of the forces acting
on the Moon is the centripetal force. Let $\omega$ be the angular velocity of
the revolution of the Moon and $r_{S}=SG$ the radius of the circular orbit
around the point $G$, so we have:

$m\omega^{2}r_{S}=F_{x}+F^{\prime}_{1x}+F^{\prime}_{2x}=$
$=\frac{Gm(M-\mu)}{a^{2}}+\frac{Gm\frac{\mu}{2}}{a^{2}}\biggl{(}1+2\frac{r}{a}\cos\varepsilon-\frac{r^{2}}{a^{2}}\biggl{(}\frac{3}{2}-\frac{9}{2}\cos^{2}\varepsilon\biggr{)}\biggr{)}+\frac{Gm\frac{\mu}{2}}{a^{2}}\biggl{(}1-2\frac{r}{a}\cos\varepsilon-\frac{r^{2}}{a^{2}}\biggl{(}\frac{3}{2}-\frac{9}{2}\cos^{2}\varepsilon\biggr{)}\biggr{)}=$
$=\frac{GmM}{a^{2}}-\frac{Gm\mu}{a^{2}}\frac{3}{2}\frac{r^{2}}{a^{2}}(1-3\cos^{2}\varepsilon)$

Hence:

$r_{S}=\frac{GM}{\omega^{2}a^{2}}\biggl{(}1+\frac{3}{2}\frac{\mu}{M}\frac{r^{2}}{a^{2}}(3\cos^{2}\varepsilon-1)\biggr{)}.$
(7)

Notice that, in the case of two point masses $m$ and $M$ placed at a distance
$a$, it turns out that $m$ makes a circular orbit around the common center of
gravity with a radius $r_{S}=\frac{GM}{\omega^{2}a^{2}}$.
Since in (7) both $\frac{\mu}{M}$ and $\frac{r^{2}}{a^{2}}$ are small
quantities, the correction to the Moon’s orbital radius due to the dumbbell is
completely negligible.

We can now compute both the torque acting on the dumbbell due to the Moon and
the torque acting on the Moon due to the dumbbell up to the smallest order in
$\frac{r}{a}$.
Let’s start with the latter. The magnitude of the torque acting on the Moon is
$\Gamma_{M}=a\lvert\vec{F_{M}}\rvert$, with $\vec{F_{M}}$ is the sum of the
forces acting on the Moon. Thanks to (7), $\vec{F_{M}}$ is parallel to the
$y$-axis, and the magnitude of the torque is:

$\Gamma_{M}=a(F^{\prime}_{1y}-F^{\prime}_{2y})=\frac{Gm\frac{\mu}{2}}{a^{2}}\biggr{[}\frac{r}{a}\sin\varepsilon\biggl{(}1+3\frac{r}{a}\cos\varepsilon\biggr{)}-\frac{r}{a}\sin\varepsilon\biggl{(}1-3\frac{r}{a}\cos\varepsilon\biggr{)}\biggl{]}=$
$=3\frac{r^{2}}{a^{3}}Gm\mu\sin\varepsilon\cos\varepsilon.$ (8)

On the other hand, the magnitude of the torque acting on the dumbbell due to
the Moon is:

$\Gamma_{D}=r\lvert\vec{F_{1}}\rvert\sin(\varepsilon+\beta_{1})-r\lvert\vec{F_{2}}\rvert\sin(\varepsilon-\beta_{2})=$
$=\frac{rGm\frac{\mu}{2}}{a^{2}}\biggr{(}1+2\frac{r}{a}\cos\varepsilon-\frac{r^{2}}{a^{2}}+4\frac{r^{2}}{a^{2}}\cos^{2}\varepsilon\biggl{)}\sin(\varepsilon+\beta_{1})-\frac{rGm\frac{\mu}{2}}{a^{2}}\biggr{(}1-2\frac{r}{a}\cos\varepsilon-\frac{r^{2}}{a^{2}}+4\frac{r^{2}}{a^{2}}\cos^{2}\varepsilon\biggl{)}\sin(\alpha-\beta_{2})=$
$=\frac{rGm\frac{\mu}{2}}{a^{2}}\biggr{(}1+2\frac{r}{a}\cos\varepsilon\biggl{)}(\sin\varepsilon\cos\beta_{1}+\cos\varepsilon\sin\beta_{1})-\frac{rGm\frac{\mu}{2}}{a^{2}}\biggr{(}1-2\frac{r}{a}\cos\varepsilon\biggr{)}(\sin\varepsilon\cos\beta_{2}-\cos\varepsilon\sin\beta_{2})=$
$=\frac{rGm\frac{\mu}{2}}{a^{2}}\biggr{(}1+2\frac{r}{a}\cos\varepsilon\biggl{)}(\sin\varepsilon+\frac{r}{a}\cos\varepsilon\sin\varepsilon)-\frac{rGm\frac{\mu}{2}}{a^{2}}\biggr{(}1-2\frac{r}{a}\cos\varepsilon\biggr{)}(\sin\varepsilon+\frac{r}{a}\cos\varepsilon\sin\varepsilon)=$
$=3\frac{r^{2}}{a^{3}}Gm\mu\sin\varepsilon\cos\varepsilon.$ (9)

As previously outlined, up to the smallest order in $\frac{r}{a}$ the two
torques are equal (and obviously in the opposite direction). This implies that
the angular momentum of the system is conserved, as we expected.

Moreover, $\mu$ represents the mass of the ocean bulge, that is the mass of
water whose shape is that of an ellipsoid (of semi-axis $R,R,R+h$) from which
it is subtracted a sphere of radius $R$ concentric to it, with $h$ represents
the tidal height of the ocean:

$h=\frac{3}{2}\frac{m}{M}\biggl{(}\frac{R}{a}\biggr{)}^{3}R.$ (10)

For a detailed computation of $h$ based on elementary mathematical tools see
[3]. So:

$\mu=\rho_{w}\frac{4}{3}\pi R^{2}h=\rho_{w}\frac{4}{3}\pi
R^{2}\frac{3}{2}\frac{m}{\rho_{E}\frac{4}{3}\pi
R^{3}}\biggl{(}\frac{R}{a}\biggr{)}^{3}R=\frac{\rho_{w}}{\rho_{E}}\frac{3}{2}\biggl{(}\frac{R}{a}\biggr{)}^{3}m.$
(11)

Finally, remembering that, according to our assumption, $r=R$ and using (11),
the torque can be written as:

$\Gamma=3\frac{R^{2}}{a^{3}}Gm\mu\sin\varepsilon\cos\varepsilon=\biggl{(}\frac{9}{2}\frac{\rho_{w}}{\rho_{E}}\biggr{)}G\frac{R^{5}}{a^{6}}m^{2}\sin(2\varepsilon)=kG\frac{R^{5}}{a^{6}}m^{2}\sin(2\varepsilon),$
(12)

where $k=\frac{9}{2}\frac{\rho_{w}}{\rho_{E}}$ is a dimensionless constant.
The formula (12) is known in literature as "MacDonald formula for body-tide
torques". We note that our dumbbell model allows us to derive this formula in
a simple way starting from reasonable physical considerations.

##  3\. Evolution of the system

In this section we want to study the evolution of the system avoiding advanced
mathematical tools. In order to do this, we imagine that the variations of
parameters is not continuous but discrete in time. This can be done because
the parameters vary on very large time scales, therefore at each revolution
they vary by very small quantities. For this reason the difference between a
discrete and a continuous evolution is irrelevant.
We start proving by this attitude the circularization of the orbit.
Since the results of the previous section hold in the case of circular orbit,
we can imagine that the real elliptical orbit of the Moon is the superposition
of two virtual semicircular orbits centered on the Earth and tangent to the
real trajectory in the perigee and in the apogee respectively, as shown in
figure 2. In this way we can apply the results obtained in the previous
section, but unfortunately we have introduced a discontinuity in the
trajectory of the Moon, which is very difficult to digest. However, imagining
that the evolution of the parameters occur in a discrete way at each semi-
revolution, the discontinuity in the virtual trajectory is irrelevant.

Figure 2. The real trajectory of the Moon around the Earth is the ellipse in
black, the point $P$ represents the focus occupied by the Earth. The virtual
trajectory is the dashed line in red, composed by the semi-circumference $ABC$
centered in $P$ and tangent to the ellipse in the perigee $B$ and the semi-
circumference $DEF$ centered in $P$ and tangent to the ellipse in the apogee
$E$.

So: $r_{a}$ represent the Earth-Moon distance in the apogee and $r_{p}$
represent the Earth-Moon distance in the perigee. Clearly: $r_{a}-r_{p}>0$.
Moreover:
$a=\frac{1}{2}(r_{a}+r_{p})$ represent the semi-major axis of the orbit;
$c=\frac{1}{2}|F_{1}-F_{2}|=\frac{1}{2}(r_{a}-r_{p})$ represent the half focal
distance;
$e=\frac{c}{a}$ represent the eccentricity of the orbit.
To determine the evolution of the system, the torque plays a crucial role. As
we calculated in the previous section, the torque acting on the Moon in the
perigee is:

$\Gamma_{M_{p}}=kG\frac{R^{5}}{r_{p}^{6}}m^{2}\sin(2\varepsilon).$ (13)

At the same time, the torque acting on the dumbbell is:

$\Gamma_{D_{p}}=-kG\frac{R^{5}}{r_{p}^{6}}m^{2}\sin(2\varepsilon).$ (14)

obviously the signs $\pm$ are arbitrary: in any case the torques have opposite
sign.
The torques in the apogee ($\Gamma_{M_{a}}$ and $\Gamma_{D_{a}}$) are similar:
just replace $r_{p}$ with $r_{a}$.
$\Gamma_{M}$ determines the variation of the orbital parameters (semi-major
axis, focal distance, eccentricity); $\Gamma_{D}$ determines the variation of
$\Omega$, the sidereal angular velocity of the Earth.
To determine the evolution of the parameters, we compare their values during
the $n$-th revolution ($a_{n};c_{n};e_{n};\Omega_{n}$) with their values
during the $(n+1)$-th revolution ($a_{n+1};c_{n+1};e_{n+1};\Omega_{n+1}$).
As we argued before, we imagine that the change of the parameters at the end
of $n$-th revolution consist of two contribution at the end of each virtual
semi-circumference.
Moreover: $r_{a,n}$ represents the Earth-Moon distance at the apogee during
the $n$-th revolution;
$r_{p,n}$ represents the Earth-Moon distance at the perigee during the $n$-th
revolution;
$a_{n}=\frac{1}{2}(r_{a,n}+r_{p,n})$ represents the semi-major axis during the
$n$-th revolution;
$c_{n}=r_{a,n}-r_{p,n}$ represents the half focal distance during the $n$-th
revolution;
$e_{n}=\frac{c_{n}}{a_{n}}$ represents the eccentricity during the $n$-th
revolution.
The same parameters during the $(n+1)$-th revolution are indicated with the
same notation, replacing the subscript $n$ with $(n+1)$.
Finally, the change of parameters between two successive revolutions are
indicated with $\Delta$:
$\Delta r_{p}=r_{p,n+1}-r_{p,n}$; $\Delta r_{a}=r_{a,n+1}-r_{a,n}$; $\Delta
a=a_{n+1}-a_{n}$; $\Delta c=c_{n+1}-c_{n}$; $\Delta e=e_{n+1}-e_{n}$.
We can start by considering the evolution of lunar orbital parameters. From
now on, we indicate generically with $r_{i}$ the Earth-Moon distance in the
position $i$: $i$ can be either $p$ (the perigee) or $a$ (the apogee).
Let $\Gamma_{Mn}=kG\frac{R^{5}}{r_{i}^{6}}m^{2}\sin(2\varepsilon)$ the torque
acting on the Moon in the $n$-th revolution. Then the variation of Moon’s
angular momentum $L=m\omega r_{i}^{2}$ between the $n$-th and the $(n+1)$-th
revolution is $\Delta L=L_{n+1}-L_{n}$ is:

$\Delta L=\Gamma_{Mn}T=\Gamma_{Mn}\frac{2\pi}{\omega}.$ (15)

From this equation we can derive the evolution of $a$, the semi-major axis of
the orbit. We remember that $\omega$ depends on $r_{i}$, indeed from the
Kepler’s third laws $\omega^{2}r_{i}^{3}=GM$. 222This result holds in the case
of a circular orbit. Thanks to the assumptions made above, this is our case.
Therefore we have:

$\omega=\frac{\sqrt{GM}}{r_{i}^{3/2}}$ (16)

Hence, from (15) and (16), we have:

$\Delta\left(m\sqrt{GMr_{i}}\right)=kG\frac{R^{5}}{r_{i}^{6}}m^{2}\sin(2\varepsilon)\frac{2\pi
r_{i}^{3/2}}{\sqrt{GM}},$ (17)

that we can rewrite as:

$\Delta\left(\sqrt{r_{i}}\right)=2\pi
k\frac{m}{M}\frac{R^{5}}{r_{i}^{9/2}}\sin(2\varepsilon).$ (18)

We can rewrite the l.h.s. of the previous equation as:

$\displaystyle\begin{aligned}
&\Delta\left(\sqrt{r_{i}}\right)=\sqrt{r_{i,n+1}}-\sqrt{r_{i,n}}=\sqrt{r_{i,n}+\Delta
r_{i}}-\sqrt{r_{i,n}}=\sqrt{r_{i,n}}\left(\sqrt{1+\frac{\Delta
r_{i}}{r_{i,n}}}-1\right)\\\
&\simeq\sqrt{r_{i,n}}\left(1+\frac{1}{2}\frac{\Delta
r_{i}}{r_{i,n}}-1\right)=\frac{\Delta r_{i}}{2\sqrt{r_{i,n}}}=\frac{\Delta
r_{i}}{2\sqrt{r_{i}}},\end{aligned}$ (19)

where we have used the approximation: $\sqrt{1+x}\simeq 1+\frac{1}{2}x$ (see
appendix), with $x=\frac{\Delta r_{i}}{r_{i,n}}<<1$.
Hence equation (18) becomes:

$\Delta r_{i}=4\pi
k\frac{m}{M}\frac{R^{5}}{r_{i}^{4}}\sin(2\varepsilon)=\frac{K}{r_{i}^{4}},$
(20)

with $K=4\pi k\frac{m}{M}R^{5}\sin(2\varepsilon)>0$ constant independent of
$r_{i}$.
Actually $K$ depends on $\varepsilon$ which depends on $r_{i}$, indeed
$\varepsilon$ is the difference between the angular position on the Moon (with
respect a certain reference axis), $\omega t$, and the sidereal angular
position of the Earth, $\Omega t$:

$\varepsilon=\Omega t-\omega t=\Omega t(1-\frac{\omega}{\Omega}).$ (21)

But if we suppose that $\omega<<\Omega$ (this assumption is currently true for
the Earth-Moon system, indeed:
$\frac{\omega}{\Omega}\simeq\frac{1\,\text{day}}{1\,\text{month}}\simeq\frac{1}{30}\simeq
0,03$), then $\varepsilon\simeq\Omega t$ is independent on $r_{i}$.
Let us now suppose, as we argued before, that the variation of $a$ consists of
two contributions: the variation when the Moon is at perigee ($\Delta a_{p}$)
and the variation when the Moon is at apogee ($\Delta a_{a}$):

$\begin{cases}\Delta r_{p}=\frac{K}{r_{p}^{4}}\\\ \Delta
r_{a}=\frac{K}{r_{a}^{4}}\end{cases}$ (22)

But $r_{a}>r_{p}$, then:

$\Delta r_{p}>\Delta r_{a}.$ (23)

From equation (23) we can derive two important results:
First, $\Delta a=\Delta r_{a}+\Delta r_{p}>0$, this implies

$a_{n+1}>a_{n},$ (24)

then the semi-major axis of the orbit increases.
Second, $\Delta
c=c_{n+1}-c_{n}=\frac{1}{2}\left(r_{a,n+1}-r_{p,n+1}\right)-\frac{1}{2}\left(r_{a,n}-r_{p,n}\right)=\frac{1}{2}\left(r_{a,n+1}-r_{a,n}\right)-\frac{1}{2}\left(r_{p,n+1}-r_{p,n}\right)=\frac{1}{2}(\Delta
r_{a}-\Delta r_{p})<0$, this implies

$c_{n+1}<c_{n},$ (25)

then the focal distance of the orbit decreases.
Finally, the two previous results implies that the eccentricity decreases and
so the orbit becomes circular. Indeed:

$\displaystyle\Delta e$
$\displaystyle=e_{n+1}-e_{n}=\frac{c_{n+1}}{a_{n+1}}-\frac{c_{n}}{a_{n}}=\frac{c_{n+1}a_{n}-c_{n}a_{n+1}}{a_{n+1}a_{n}}=\frac{c_{n+1}a_{n}-c_{n}a_{n}+c_{n}a_{n}-c_{n}a_{n+1}}{a_{n+1}a_{n}}$
$\displaystyle=\frac{a_{n}(c_{n+1}-c_{n})-c_{n}(a_{n+1}-a_{n})}{a_{n+1}a_{n}}=\frac{a_{n}\Delta
c_{n}-c_{n}\Delta a_{n}}{a_{n+1}a_{n}}<0,$

the last inequality follow from the fact that all the terms are positive,
except for $\Delta c$. This implies

$e_{n+1}<e_{n},$ (26)

then the eccentricity of the orbit decreases.
Moreover, we can determine the rate of decrease of $c_{n}$. Indeed:

$\displaystyle\Delta c$ $\displaystyle=\Delta(r_{a}-r_{p})=\Delta r_{a}-\Delta
r_{p}=\frac{K}{r_{a}^{4}}-\frac{K}{r_{p}^{4}}=\frac{K(r_{p}^{4}-r_{a}^{4})}{r_{a}^{4}r_{p}^{4}}$
$\displaystyle=\frac{K(r_{p}^{2}+r_{a}^{2})(r_{p}+r_{a})(r_{p}-r_{a})}{r_{a}^{4}r_{p}^{4}}=-c\frac{Ka(r_{p}^{2}+r_{a}^{2})}{r_{a}^{4}r_{p}^{4}}\simeq-c\frac{K}{a^{5}}=-\lambda
c,$

with $\lambda=\frac{K}{a^{5}}>0$.
Therefore $\Delta c_{n}$ decreases in a way directly proportional to $c_{n}$:
this kind of decrease is called "exponential decrease " where $\lambda$ is the
rate of decrease. Since $\lambda$ in positive, then $c_{n}$ decreases until it
becomes lesser than any prefixed positive quantity. When $c_{n}$ approaches
$0$, then $r_{a,n}\simeq r_{p,n}$ and hence, from (22), $\Delta
r_{a}\simeq\Delta r_{p}$. This implies $\Delta c\simeq 0$. Thus, when the
focal distance "becomes" zero, it no longer varies.
But $c_{n}\simeq 0$, $\Delta c\simeq 0$ imply that $e_{n}\simeq 0$, $\Delta
e\simeq 0$. Thus even the eccentricity of the orbit decreases until it becomes
lesser than any prefixed positive quantity and, "at the end" it no longer
varies. So the orbit becomes circular.
On the other hand, the semi-major axis $a$ increases indefinitely: even when
the orbit becomes circular, $a$ (that is its radius) continues to increase.
But the growth of $a$ and the decrease of $e$ occur on different time scales
and in different ways, indeed while the variation of $e$ is exponential, i.e.
very fast, that of $a$ is polynomial 333 Indeed,  $\Delta
a=\frac{1}{2}\Delta(r_{a}+r_{p})=\frac{1}{2}(\Delta r_{a}+\Delta
r_{p})=\frac{1}{2}\left(\frac{K}{r_{a}^{4}}+\frac{K}{r_{p}^{4}}\right)=\frac{K(r_{p}^{4}+r_{a}^{4})}{2r_{a}^{4}r_{p}^{4}}\simeq\frac{K}{2a^{4}}.$
, which is slower.
Let us now study the evolution of $\Omega$, the sidereal angular velocity of
the Earth: it’s variation is due to $\Gamma_{D}$, the torque acting on the
dumbbell. Indeed the dumbbell is pulled back from the Moon and thanks to the
friction between it and the underlying planet, it slows down the rotation of
the Earth.
The angular momentum of the Earth is $I_{E}\Omega$, where $I_{E}$ is the
moment of inertia. As we argued before, we suppose that it varies in each
lunar revolution. Thus, at the end of $n$-th revolution we have:

$\Delta\left(I_{E}\Omega\right)=\Gamma_{D,n}T=\Gamma_{D,n}\frac{2\pi}{\omega},$
(27)

and hence:

$\Delta\Omega=\frac{\Gamma_{D,n}}{I_{E}}\frac{2\pi}{\omega}=-2\pi
kG\frac{R^{5}}{r_{i}^{9/2}}\frac{m^{2}}{I_{E}}\sin(2\varepsilon)<0.$ (28)

So, as we expect, $\Omega$ decrease.
But if $\Omega$ decreases, also $\varepsilon$ decreases; indeed as argued
before (see equation (21)), $\varepsilon\simeq\Omega t$, then:

$\Delta\varepsilon=\varepsilon_{n+1}-\varepsilon_{n}=\Omega_{n+1}t-\Omega_{n}t=(\Omega_{n+1}-\Omega_{n})t=\Delta\Omega
t<0,$ (29)

this implies

$\varepsilon_{n+1}<\varepsilon_{n},$ (30)

then $\varepsilon$ decreases.
This dynamics has as a stationary situation that in which $\varepsilon=0$.
Indeed, when $\varepsilon$ approaches $0$, then $\Delta\Omega$ becomes $0$
(see equation (28)), and then $\Omega$ no longer varies. But, from equation
(21), when $\varepsilon=0$ then $\Omega-\omega=0$ that implies
$\Omega=\omega$.
So the stationary situation is the spin-orbit resonance regime: that is a
dynamics in which the Earth complete a rotation over a period of time equal to
a Moon revolution. In such a situation the dumbbell (that rotates with
velocity $\omega$) moves on the Earth at the same speed as that of the
underlying layer (that rotates with velocity $\Omega$) and thus friction
between the dumbbell and the underlying Earth ends: in such a situation tides
will no longer be observed on Earth and no more energy will be dissipated by
this mechanism.
This type of dynamics is often observed in the oldest planet-satellite systems
of the Solar System, indicating the fact that the equilibrium situation
predicted by our model is indeed attractive.

##  Appendix A Special approximation

In this section we want to show two ways to prove the special approximation we
used in equation (19):

$\sqrt{1+x}\simeq 1+\frac{1}{2}x,$ (31)

valid if $x<<1$.
The first way uses the linearization formula: given a function $f(x)$, if in
its point $x_{0}$ the function $f$ and its derivative $f^{\prime}$ are both
continuous, it is possible to approximate the function with the tangent line
in $x_{0}$:

$f(x_{0}+\delta)=f(x_{0})+f^{\prime}(x_{0})\delta+R(\delta),$ (32)

with $R(\delta)=o(\delta^{2})$.
So equation (31) follows from (32) if $f(x)=\sqrt{1+x}$, $x_{0}=0$ and
$\delta=x<<1$.
A second way to prove (31) does not use derivative, but only techniques of
Cartesian geometry. The issue consists to determine the tangent line to the
function $y=\sqrt{1+x}$ in its point $x_{0}=0$.
In the Cartesian plane, $y=\sqrt{1+x}$ represents the equation of a semi-
parabola. The tangent point is $(0,1)$ and then the equation of tangent line
is: $y=mx+1$. $m$ can be determined by imposing the tangent condition.
We can rewrite the equation of our semi-parabola as $y^{2}=1+x$ and we can
substitute $y$ with its expression $mx+1$, obtaining:

$m^{2}x^{2}+(2m-1)x=0.$ (33)

Finally we have to impose the tangent condition (uniqueness of the solution):
the discriminant of (33) is null. This implies: $m=1/2$.
Then, the equation of tangent line is $y=\frac{1}{2}x+1$.
So, the semi-parabola $y=\sqrt{1+x}$ can be approximated near the point
$x_{0}=0$ with its tangent line $y=\frac{1}{2}x+1$.

The authors have no conflicts to disclose.

## References

  * [1] Butikov, Eugene I., _A dynamical picture of the oceanic tides_ , Am. J. Phys. no. 70, pp. 1001-1010, 2002.
  * [2] Butikov, Eugene I., _Oceanic Tides: A Physical Explanation And Modeling_ , Computer tools in education, no. 5, pp. 12-34, 2017.
  * [3] Chiu-king Ng, _How tidal forces cause ocean tides in the equilibrium theory_ , Phys. Educ., no. 50, pp. 159-164, 2015.
  * [4] Efroimsky, Michael, _Bodily Tides near Spin-Orbit Resonances_ , Celestial Mech. Dynam. Astronom., vol. 112, no. 3, pp. 283-330, 2012.
  * [5] Efroimsky, Michael, _Tidal Evolution of Asteroidal Binaries. Ruled by Viscosity. Ignorant of Rigidity_ , Astronom. J., vol. 150, no. 4, Art. 98, 2015.
  * [6] Ferraz-Mello, S., Ragazzo, C. G., and dos Santos, L. R., _Dissipative Forces in Celestial Mechanics_ , 30° Coloquio Brasileiro de Matematica, Rio de Janeiro: IMPA, 2015.
  * [7] Scoppola, B., Troiani, A., Veglianti, M., _Tides and Dumbbell Dynamics_ , Regul. Chaot. Dyn. 27, 369–380 (2022). https://doi.org/10.1134/S1560354722030078
  * [8] Celletti, A. and Sidorenko, V., _Some Properties of the Dumbbell Satellite Attitude Dynamics_ , Celestial Mech. Dynam. Astronom., vol. 101, nos. 1–2, pp. 105–126, 2008
  * [9] Hut, P., _Tidal Evolution in Close Binary Systems_ , Astronomy and Astrophysics, 99, 126-140 (1981).


# Optimal Reeb graphs on two- and three-connected planar polygon

Oleksandr Pryshliak, Karolina Haieva

###### Abstract

To investigate the topological structure of planar polygon decomposition on
trapezoids, which is formed by height functions. We use the oriented Reeb
graph of the function with a marked vertex. We describe all possible optimal
Reeb graphs in the case of polygon in general position with one local minimum
and one local maximum. By optimal Reeb graph we mean Reeb graph, which cann’t
be obtaned from other Reeb graph by subdivision of an edge or a leaf
attaching. In this case polygon is a triangle with triangle holes. Constructed
Reeb graphs give topological structures of trapezoid maps on two-connected
polygon with six vertexes and three-connected polygon with nine vertexes.

2000 Mathematics Subject Classification. 58K05, 37D15.

Key words and phrases. Morse function, Morse-Smale flow, gradient flow, Reeb
graph.

## Introduction

In 1946, Reeb introduced [23] a graph, which describes the topological
properties of functions on two-dimensional manifolds. In the case of simple
Morse functions on closed oriented two-dimensional manifolds, the Reeb graph
is a complete topological invariant of fiberwise equivalence. It becomes a
complete topological invariant under topological equivalence, if a linear
order is specified on the set of vertices. In the case of a non-orientable
two-dimensional manifold [13], as well as manifolds with boundary [9, 10],
additional information is needed to construct a complete topological
invariant. Reeb graph on non-compact 2-manifolds wos investigated in [11, 19].
In this case it can be a non-Hausdorff space. If the function has by several
critical points on its level, then we need the information about the structure
of the function in the neighborhood of critical levels [2].

Another way to specify the topological structure of Morse functions is to use
Morse-Smale vector fields with given function values at singular points [13,
25]. Therefore, the topological classification of Morse–Smale vector fields is
closely related to the classification of functions.

Graphs as topological invariant of functions was used in [18, 10, 9, 19, 17,
13, 31, 32, 29, 1, 24], on surfaces with the boundary in [8, 10] and on closed
3-manifolds in [16, 26, 22].

By analogy to Reeb graph, complete invariants of flows ar also graph
(destiguioshed graphs). Its was constructed in [12, 15, 14, 12] on
classifications of flows on closed 2- manifolds and [21] on 2-manifolds with
the boundary and in [28, 30, 22, 20, 6] as Heegard diagrams of 3-manifolds.

We recoment [15, 5, 27, 33, 7, 4], where you can find the main invariants of
graphs and their embeddings in surfaces.

Effective computing of Reeb graphs was done in [3].

The main purpose of this paper is to find all possible Reeb graphs of height
function with minimal number of vertexes on two- and three-connected polygons.

The main results.

In Chapter 1, we describe the relationship between Reeb graphs and trapezoidal
maps of polygons.

Chapter 2 describes the properties of Reeb graphs on simple polygons and lists
all graphs with no more than 5 vertices.

Chapter 3 describes the properties of the optimal Reeb graph. In particular,
theorem 1 proves that such a graph has 3n vertices on an n-connected polygon,
and theorem 2 describes the number of vertices of each degree.

Chapter 4 lists all optimal graphs on biconnected polygons.

Chapter 5 describes all optimal graphs on three-connected polygons

##  1 Trapezoidal map and Reeb graph

Let $L$ be a polygon with a set of vertices $V=\\{v_{i}(x_{i},y_{i}),1\leq
i\leq m\\}$ and edges $E=\\{e_{j}=\\{v_{j},w_{j}\\},v_{j}\in V,w_{j}\in
V\\},1\leq j\leq n$. The polygon is in general position with the height
function $h(x,y)=y$, if $v_{i}\neq v_{j}$ implies $y_{i}\neq y_{j}$. In the
future, we will consider polygons that are in general position with the
function of height.

By trapezoid we will understand one of the following polygons: trapeze,
parallelogram, triangle.

Let’s choose a sufficiently large rectangle $S=[a,b]\times[c,d]$, that
$V\subset\text{Int}\ S$, (for example, $a=\min x_{i}-1,\ b=\max x_{i}+1,\
c=\min y_{i}-1,\ d=\max y_{i}+1$).

For each vertex $v_{i}$, we draw a horizontal ray to the left to the first
intersection with another edge of $L$ or the line $x=a$. Similarly, we draw
the ray to the right to the first point of intersection with the edges of $L$
or the line $x=b$. Consider those segments that lie inside the polygon $L$.
They split $L$ into trapezoids. The partition of $S$ into trapezoids
constructed in this way is called a trapezoidal map. We will denote it by
$T(L)$. By construction, a trapezoidal map is a flat rectilinear graph. Since
no more than 2 horizontal segments emerge from each vertex of the polygon $L$,
and none from the lowest and highest vertices, the total number of vertices of
the trapezoidal map does not exceed $3n-4$.

Figure 1: Example of the trapezoidal map (with red dashed bases) and Reeb
graph (black)

A component of the level line $h^{-1}(y)$ of the height function is called a
fiber.

Two functions are called fibre equivalent if there is a homeomorphism of the
polygon onto the polygon which maps fibers of one function to fibers of
another one.

The quotient space $M/\sim$ with orientation of edges according to the
direction of the increase of the function is called a Reeb graph, where
$h:M\to R$ is the height function, $x_{1}\sim x_{2}$ if $x_{1}$ and $x_{2}$
belongs to one fiber. Reeb graphs are considered to within the isomorphism of
oriented graphs. Examples of Reeb graphs are shown on Fig. 1.

By construction we have one to one correspondence trapezoids to edges of Reeb
graph. Thus, the Reeb graph define the structure of the trapezoidal map.

In addition, vertexes of degree 2 are divided on left and right. Vertex is
left if right horizontal ray intersect the polygon in its small neighbourhood.
Otherwise vertex is right.

Two simple height functions on the orientable surface are fiber equivalent if
and only if their Reeb graphs are isomorphic ([1], Theorem 2.4, p. 71). On the
non-orientable surfaces there are firer non-equivalent functions with the same
Reeb graphs. So additional invariants are nessersity to clasify height
functions. But in the case of planar polygon in general position Reeb graph is
complete invariant of height functions and structure of the trapezoidal map.

##  2 Reeb graph on simple polygon

By the definition of the Reeb graph, it is homotopically equivalent to a
polygon. So, if a polygon is simple, which means simply-connected, then the
Reeb graph has the same properties. This means that it is a tree. Since the
simple polygon with the smallest number of vertices is a triangle, the Reeb
graph has at least three vertices. This graph is shown in fig. 2.1.

12345678910

Figure 2: Reeb graphs on simple polygon with 3, 4 and 5 vertexes

Trees with 4 vertexes are either chains, two possible options are shown in
Fig. 2.2,3, or the tree is like the letter Y - 2.4. A tree with 5 vertices is
separated from a tree with 4 vertices by dividing one of the edges. All
possible options to divide Y are shown in Fig. 2.5–7. Chains with 5 vertexes
are shown in Fig. 2.8-10.

##  3 Optimal Reeb graph

The Reeb graph $G$ of a polygon is called optimal if there is no other Reeb
graph $H$ of a polygon such that $G$ and $H$ are homotopically equivalent and
the number of vertices of the graph $H$ is less than the number of vertices of
the graph $G$.

This means that the Reeb graph is not optimal, if it is possible to subtract
from another Reeb graph by the edge subdivision or add a leaf.

The polygon is called n-connected if it is connected, and its boundary has an
n component of connectivity.

###### Theorem 1

The optimal Reeb graph on an n-connected polygon has 3n vertices.

Proof. Each component of the boundary is a polygon which has at least 3
vertexes. Therefore, the number of vertices is no less than 3n. Let us show
that there is a height function that its Reeb graph has 3n vertices. For this
purpose, let us look at the rather large triangle A, the vertexes of which lie
at different heights. Consider $n-1$ triangle $A_{i}\subset\text{Int}A$, which
does not have intersection. The polygon $B=A\setminus\cup_{i=1}^{n-1}A_{i}$
contains 3n vertices, and the number of vertices of Reeb graph is same.

The completed theorem shows a method for generating all optimal Reeb graphs.

###### Theorem 2

The optimal Reeb graph of the height function on the n-connected polygon has
the following properties:

1) the graph is n-connected planar graph;

2) the graph has n vertexes of degree 2, two vertexes of degree 1, the other
vertices have degree 1 or 3;

3) only vertices of degree 1 are sinks and sources.

Proof. 1) It followed from the definition of Reeb graph.

2) Each triangle has a regular vertex of degree 2. The height function has a
maximum and a minimum. If there is other vertex of degree one or two, then the
graph is not optimal.

3) sinks and sources correspond to local extrema of the function, and vertices
of degree 2 and 3 correspond to saddle points.

##  4 Optimal Reeb graph on two-connected polygon

As it was meant above all, Reeb graph of two-connected polygon is optimal, if
it has 6 vertices, and the polygon has a triangle with a hole that looks like
a triangle.

1234

Figure 3: Optimal Reeb graph of two-connected polygon

The middle vertex of the triangle is called the vertex that is not the lowest
or the highest for the triangle. Since the axial symmetry of the vertical axis
is identical, we can take into account without limiting the formality that the
middle vertex on the external triangle is the rightmost (which is the largest
abscise). So, the axial symmetry of exactly horizontal straight lines leads to
equivalent graphs. Therefore, we can take into account that at least two
vertexes of the inner triangle lie lower than the middle vertex of the outer
triangle. There is two or three such vertexes. In addition, the inner triangle
can be left (middle apex of the left) or right (right). Therefore, there are 4
different possibilities for the development of the middle triangle and
according to the different Reeb graphs. Its is shown in Fig. 3.

Summarizing the results obtained above, we have the following

###### Theorem 3

There are 4 non-isomorphic optimal Reeb graphs on two-connected polygons.

##  5 Optimal Reeb graph of three-connected polygon

We denote by ABC the outer triangle, and by DEF and GHI the inner triangles,
such that A>B>C, D>E>F, G>H>I, D>G.

From now on, as before, we will be able to see only such triangles, in which B
is the right vertex of the outer triangle and no less than three vertices of
the inner triangles lie below it. We call the inner triangle DEF by upper and
we call GHI by the lower triangle. Let’s look at it first if F>G.

ABCDEFGHI12345678910111213

Figure 4: Optimal Reeb graph of three-connected polygon. Part 1

We suppose F>B>G. There are three options: 1) the inner triangles are right
Fig. 4.1, 2) one of them is left, and the other is right Fig. 4.2, 3) the
inner triangle are left Fig. 4.3. By moving the middle outer vertex up, a new
expansion of the triangle is removed. If we use it for Fig.4-1, we obtain Fig.
4.4, Fig. 4.5 and Fig. 4.10. If we use it for Fig. 4.2 we obtain Fig. 4,8 and
4.11. If we use it for horizontal symmetric of Fig. 4.2, we obtain Fig. 46,
7,13. Figure 4.3 gives Fig. 4.9,12.

12345678910111213

Figure 5: Optimal Reeb graph of three-connected polygon. Part 2

Next, we consider the cases when D>G>F. Fig. Morse4 shows all possible
arrangements of internal triangles. In the case of two solutions of a pair of
internal triangles, which are symmetrical about the vertical axis, we depict
only one of them. Therefore, the vertex B can be located both on the right and
on the left.

In Fig. 5.1 we have D>E>G>F>H>I. Possible positions of the vertex B on the
left - 4, and on the right - 6.

In Fig. 5.2 we have D>G>H>I>E>F. Possible positions of the vertex B on the
left - 3, and on the right - 7.

In Fig. 5.3 we have D>G>H>E>I>F. Possible positions of the vertex B on the
left - 3, and on the right - 6.

In Fig. 5.4, possible positions of the vertex B on the left - 4, and on the
right - 5.

In Fig. 5.5, possible positions of the vertex B on the left - 5, and on the
right - 4.

In Fig. 5.6, possible positions of the vertex B on the left - 5, and on the
right - 5.

In Fig. 5.7, possible positions of the vertex B on the left - 4, and on the
right - 6.

In Fig. 5.8, possible positions of the vertex B on the left - 5, and on the
right - 5.

In Fig. 5.9, possible positions of the vertex B on the left - 4, and on the
right - 5.

In Fig. 5.10, possible positions of the vertex B on the left - 3, and on the
right - 6.

In Fig. 5.11, possible positions of the vertex B on the left - 5, and on the
right - 4.

In Fig. 5.12, possible positions of the vertex B on the left - 4, and on the
right - 4.

In Fig. 5.13, possible positions of the vertex B on the left - 3, and on the
right - 5.

Summarizing the results obtained above, we have the following

###### Theorem 4

There are 133 non-isomorphic optimal Reeb graphs on three-connected polygons.

## Conclusion

In this paper, we have found all possible structures of optimal Reeb graphs on
two-connected (4 structures) and three-connected (133 structures) polygons.
Any other Reeb graph can be obtained from the optimal one by using edge
splitting and leaf addition operations. The resulting Reeb graphs describe the
structures of trapezoidal maps on these polygons.

## References

  * [1] S. Bilun and A. Prishlyak.  The closed morse 1-forms on closed surfaces.  Visn., Mat. Mekh., Kyv. Univ. Im. Tarasa Shevchenka, 2002(8):77–81, 2002.
  * [2] A.V. Bolsinov and A.T. Fomenko.  Integrable Hamiltonian systems. Geometry, Topology, Classification.  A CRC Press Company, Boca Raton London New York Washington, D.C., 2004\.  724 p.
  * [3] H. Doraiswamy, V. Natarajan  Computing Reeb Graphs as a Union of Contour Trees.  IEEE Trans Vis Comput Graph . 2013 Feb;19(2):249-62. doi: 10.1109/TVCG.2012.115.
  * [4] J. Gross and T. Tucker.  Topological graph theory.  Wiley, 1987.
  * [5] F. Harary.  Graph Theory.  Westview Press, 1969.
  * [6] C. Hatamian and A. Prishlyak.  Heegaard diagrams and optimal morse flows on non-orientable 3-manifolds of genus 1 and genus 2.  Proc. Int. Geom. Cent., 13(3):33–48, 2020. doi:10.15673/tmgc.v13i3.1779.
  * [7] P. Hilton and S. Wylie.  Homology theory: An introduction to algebraic topology.  Cambridge University Press, 1968.
  * [8] B. I. Hladysh and A. O. Pryshlyak.  Functions with nondegenerate critical points on the boundary of the surface.  Ukrainian Mathematical Journal, 68(1):29–40, jun 2016. doi:10.1007/s11253-016-1206-5.
  * [9] B.I. Hladysh and A.O. Prishlyak.  Topology of functions with isolated critical points on the boundary of a 2-dimensional manifold.  SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 13:050, 2017. doi:0.3842/SIGMA.2017.050.
  * [10] B.I. Hladysh and A.O. Prishlyak.  Simple morse functions on an oriented surface with boundary.  Журнал математической физики, анализа, геометрии, 15(3):354–368, 2019. doi:10.15407/mag15.03.354.
  * [11] A.S. Kronrod.  Functions of two variables.  Russian Mathematical Surveys, 5:24–134, 1950.
  * [12] Z. Kybalko, A. Prishlyak, and R. Shchurko.  Trajectory equivalence of optimal Morse flows on closed surfaces.  Proc. Int. Geom. Cent., 11(1):12–26, 2018. doi:10.15673/tmgc.v11i1.916.
  * [13] D.P. Lychak and A.O. Prishlyak.  Morse functions and flows on nonorientable surfaces.  Methods of Functional Analysis and Topology, 15(03):251–258, 2009\.
  * [14] A. Prishlyak, A. Prus, and S. Guraka.  Flows with collective dynamics on a sphere.  Proc. Int. Geom. Cent., 14(1):61–80, 2021. doi:10.15673/tmgc.v14i1.1902.
  * [15] A.O. Prishlyak.  On graphs embedded in a surface.  Russian Mathematical Surveys, 52(4):844, 1997. doi:10.1070/RM1997v052n04ABEH002074.
  * [16] A.O. Prishlyak.  Equivalence of morse function on 3-manifolds.  Methods of Func. Ann. and Topology, 5(3):49–53, 1999.
  * [17] A.O. Prishlyak.  Conjugacy of morse functions on surfaces with values on a straight line and circle.  Ukrainian Mathematical Journal, 52(10):1623–1627, 2000. doi:10.1023/A:1010461319703.
  * [18] A.O. Prishlyak.  Conjugacy of morse functions on 4-manifolds.  Russian Mathematical Surveys, 56(1):170–171, 2001. doi:10.1070/RM2001v056n01ABEH000370.
  * [19] A.O. Prishlyak.  Morse functions with finite number of singularities on a plane.  Meth. Funct. Anal. Topol, 8:75–78, 2002.
  * [20] A.O. Prishlyak.  Topological equivalence of smooth functions with isolated critical points on a closed surface.  Topology and its Applications, 119(3):257–267, 2002. doi:10.1016/S0166-8641(01)00077-3.
  * [21] A.O. Prishlyak.  On sum of indices of flow with isolated fixed points on a stratified sets.  Zhurnal Matematicheskoi Fiziki, Analiza, Geometrii [Journal of Mathematical Physics, Analysis, Geometry], 10(1):106–115, 2003.
  * [22] A.O. Prishlyak.  Regular functions on closed three-dimensional manifolds.  Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, 8:21–24, 2003.
  * [23] G. Reeb.  Sur les points singuliers d’une forme de pfaff complétement intégrable ou d’une fonction numérique.  C.R.A.S. Paris, 222:847—849, 1946.
  * [24] V.V. Sharko.  Functions on manifolds. Algebraic and topological aspects., volume 131 of Translations of Mathematical Monographs.  American Mathematical Society, Providence, RI, 1993.
  * [25] S. Smale.  On gradient dynamical systems.  Ann. of Math., 74:199–206, 1961.
  * [26] В.М. Кузаконь, В.Ф. Кириченко, and О.О. Пришляк.  Гладкi многовиди. Геометричнi та топологiчнi аспекти.  Працi Iнcтитуту математики НАН України.—2013.—97.—500 с, 2013.
  * [27] Л. С. Понтрягин.  Основы комбинаторной топологии.  Наука, 1986.
  * [28] А.О. Пришляк.  Векторные поля Морса–Смейла с конечным числом особых траекторий на трехмерных многообразиях.  Доповiдi НАН України, 6:43–47, 1998.
  * [29] А.О. Пришляк.  Сопряженность функций Морса.  Некоторые вопросы совр. математики. Институт математики АН Украины, Киев, 1998.
  * [30] А.О. Пришляк.  Топологическая эквивалентность функций и векторных полей Морса—Смейла на трёхмерных многообразиях.  Топология и геометрия. Труды Украинского мат. конгресса, pages 29–38, 2001.
  * [31] О.О. Пришляк.  Теорія Морса.  Київський університет, 2002.
  * [32] О.О. Пришляк.  Топологія многовидів.  Київський університет, 2015.
  * [33] У. Татт.  Теория графов.  Мир, 1988.

Taras Shevchenko National University of Kyiv

Oleksandr Pryshliak Email address<EMAIL_ADDRESS>Orcid ID:
0000-0002-7164-807X

Karolina Haieva Email address<EMAIL_ADDRESS>Orcid ID:
0009-0004-2613-4055