* [ Abelian relations ]. All the abelian relations of $\boldsymbol{\mathcal{B}}$ are polylogarithmic, of weight 1 or 2:
$-$ the subspace $\boldsymbol{AR}_{1}\big{(}\boldsymbol{\mathcal{B}}\big{)}$
of logarithmic ARs of $\boldsymbol{\mathcal{B}}$ has dimension 5;
$-$ the subspace $\boldsymbol{AR}_{2}\big{(}\boldsymbol{\mathcal{B}}\big{)}$
of dilogarithmic ARs of $\boldsymbol{\mathcal{B}}$ is spanned by
$\boldsymbol{\mathcal{A}b}$; and
$-$ there is a decomposition in direct sum
$\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{B}}\big{)}=\boldsymbol{AR}_{1}\big{(}\boldsymbol{\mathcal{B}}\big{)}\oplus\big{\langle}\boldsymbol{\mathcal{A}b}\big{\rangle}$
$(\star)$.

  * [ Rank ]. Bol’s web has maximal rank hence is exceptional.

  * [ Weyl group action ]. There is a natural action of $W(E_{4})=\mathfrak{S}_{5}$ on the space of abelian relations regarding which $(\star)$ is the decomposition into $\mathfrak{S}_{5}$-irreducible representations.

  * [ Hexagonality & Characterization ]. Bol’s web is hexagonal and is essentially characterized by this property since ‘for any $k\geq 3$, a hexagonal planar $k$-web either is linearizable and equivalent to a web formed by $k$ pencils of lines or $k=5$ and it is equivalent to $\boldsymbol{\mathcal{B}}$’.

  * [ Construction à la GM ]. $\boldsymbol{\mathcal{B}}$ can be constructed following the approach of Gelfand and MacPherson [GM]: it can be seen as the quotient, under the action of the Cartan torus $H_{4}\subset{\rm SL}_{5}(\mathbf{C})$, of a natural $H_{4}$-equivariant web defined on the grassmannian $G_{2}(\mathbf{C}^{5})$.

  * [ Cluster web ]. Bol’s web is of cluster type: up to equivalence, it can be defined by means of the $\mathscr{X}$-cluster variables of the finite type cluster algebra of type $A_{2}$.

#### 4.2.2.

Since $\boldsymbol{\mathcal{B}}$ is equivalent to
$\boldsymbol{\mathcal{W}}_{{\rm dP}_{5}}$ and because what can be considered
as its most important feature, namely the fact it carries the dilogarithmic
abelian relation $\boldsymbol{\mathcal{A}b}$, generalizes to
$\boldsymbol{\mathcal{W}}_{{\rm dP}_{4}}$, one can wonder whether the nice
properties listed above admit analogs for this 10-web. It turns out that it is
indeed the case since it can be proved that the following properties are
satisfied:

  * [ Non linearizability ]. $\boldsymbol{\mathcal{W}}_{{\rm dP}_{4}}$ is not linearizable hence not equivalent to an algebraic web.

  * [ Abelian relations ]. All the ARs of $\boldsymbol{\mathcal{W}}_{{\rm dP}_{4}}$ are hyperlogarithmic, of weight 1, 2 or 3:
$-$ the subspace $\boldsymbol{AR}_{1}\big{(}\boldsymbol{\mathcal{W}}_{{\rm
dP}_{4}}\big{)}$ of logarithmic ARs of $\boldsymbol{\mathcal{W}}_{{\rm
dP}_{4}}$ has dimension 20;
$-$ the subspace $\boldsymbol{AR}_{2}\big{(}\boldsymbol{\mathcal{W}}_{{\rm
dP}_{4}}\big{)}$ of weight 2 hyperlogarithmic ARs has dimension 15: it is the
$-$ direct sum of two subspaces, a first one $\boldsymbol{AR}_{2}^{\rm sym}$
of symmetric ARs, of dimension 5 and
$-$ another one $\boldsymbol{AR}_{2}^{\rm asym}$ of antisymmetric ARs, of
dimension 10;
$-$ the subspace $\boldsymbol{AR}_{3}\big{(}\boldsymbol{\mathcal{W}}_{{\rm
dP}_{4}}\big{)}$ of weight 3 hyperlogarithmic ARs of
$\boldsymbol{\mathcal{W}}_{{\rm dP}_{4}}$ is spanned by ${\bf Hlog^{3}}$;
$-$ there is a decomposition in direct sum
$\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{W}}_{{\rm
dP}_{4}}\big{)}=\boldsymbol{AR}_{1}\big{(}\boldsymbol{\mathcal{W}}_{{\rm
dP}_{4}}\big{)}\oplus\Big{(}\boldsymbol{AR}_{2}^{\rm
sym}\oplus\boldsymbol{AR}_{2}^{\rm asym}\Big{)}\oplus\big{\langle}\,{\bf
Hlog^{3}}\,\big{\rangle}$ $(\clubsuit)$

  * [ Rank ]. The web $\boldsymbol{\mathcal{W}}_{{\rm dP}_{4}}$ has maximal rank hence is exceptional.

  * [ Weyl group action ]. The Weyl group $W(D_{5})$ acts naturally on $\boldsymbol{AR}\big{(}\boldsymbol{\mathcal{W}}_{{\rm dP}_{4}}\big{)}$ and $(\clubsuit)$ actually is its decomposition into irreducible $W(D_{5})$-modules.

  * [ Hexagonality & Characterization ]. $\boldsymbol{\mathcal{W}}_{{\rm dP}_{4}}$ is characterized by its hexagonal 3-subwebs.313131See §5.7 further for a more precise statement.

  * [ Construction à la GM ]. $\boldsymbol{\mathcal{W}}_{{\rm dP}_{4}}$ can be seen as the quotient, under the action of the Cartan torus $H_{5}\subset{\rm S0}_{10}(\mathbf{C})$, of a natural $H_{5}$-equivariant web defined on the spinor variety $\mathbb{S}_{5}$.

  * [ Cluster web ]. $\boldsymbol{\mathcal{W}}_{{\rm dP}_{4}}$ is of cluster type: it can be obtained by means of some of the $\mathscr{X}$-cluster variables of the finite type cluster algebra of type $D_{4}$.

######  Remark 4.1.

1. Without additional explanations, the statements above may sound cryptic for non-specialists of web geometry. We will come back in detail to all of this in [Pi4].
2. Only very few of the statements above were known before:

  * $\boldsymbol{-}$

by means of explicit computations on a computer, we established in [Pi1,
§7.3.3] that the standard model of $\boldsymbol{\mathcal{W}}_{{\rm dP}_{4}}$
on the projective plane has maximal rank hence is exceptional. But nothing
else was known about its space of ARs at that time;

  * $\boldsymbol{-}$

in [Pe], Pereira introduced a way to associate a ‘resonnance web’
$\boldsymbol{\mathcal{W}}_{\mathcal{A}}$ to any arrangements of lines in the
plane $\mathcal{A}$. When specializing this constuction to a certain
arrangement ${\bf K}_{5}$, one recovers the planar model of
$\boldsymbol{\mathcal{W}}_{{\rm dP}_{4}}$. Then Pereira showed that the
‘hyperlogarithmic rank323232This is the immediate ‘hyperlogarithmic’
generalization of the notion of ‘polylogarithmic rank’ considered in [Pi2,
§1.4.3.2]. of $\boldsymbol{\mathcal{W}}_{{\bf K}_{5}}$ is $(20,15,1)$ which
shows in particular that this web carries a unique hyperlogarithmic AR of
weight 3. However, this AR was not explicited by Pereira.

One has to have in mind that considering that almost all (and possibly all cf.
§5.4 or §5.6 below) the remarkable properties of the pair Abel’s identity
($\boldsymbol{\big{(}\mathcal{A}b\big{)}}\simeq{\bf HLog}^{2}$) / (Bol’s web
$\boldsymbol{\mathcal{B}}\simeq\boldsymbol{\mathcal{W}}_{{\rm dP}_{5}}$) admit
formally very similar counterparts for the pair ${\bf HLog}^{3}$ /
$\boldsymbol{\mathcal{W}}_{{\rm dP}_{4}}$, then the latter may be considered
as the most natural weight 3 generalization of the former. The quest to find
natural generalizations to any weight of Abel’s dilogarithmic identity is a
long standing one which has received many great contributions by several
authors. But except until very recently and for very few cases, all of these
were sticking to the case of polylogarithms or more precisely to that of
iterated integrals on $\mathbf{P}^{1}\setminus\\{0,1,\infty\\}$ constructed
from words in the alphabet with $\omega_{0}=du/u$ and $\omega_{1}=du/(u-1)$ as
letters. The discovery of the identities ${\bf Hlog}^{r-2}$ for $r=4,\ldots,7$
as well as the striking fact that
$\boldsymbol{\mathcal{B}}\simeq\boldsymbol{\mathcal{W}}_{{\rm dP}_{5}}$ and
$\boldsymbol{\mathcal{W}}_{{\rm dP}_{4}}$ look so similar regading the
numerous remarkable properties they satisfy may be an indication that the
genuine natural generalizations of the five terms relation of the dilogarithm
have to be looked for in the more general setting of hyperlogarithms on
$\mathbf{P}^{1}$.

What makes the functional equations of polylogarithms so interesting is the
role these identities are playing in other areas of mathematics, such as in
hyperbolic geometry or in the K-theory of number fields (regulators and
Zagier’s conjecture). But for the moment, we are not aware of any occurence of
the identity ${\bf Hlog}^{3}$ within other fields of mathematics, which would
be necessary for it to be fully recognized as the genuine generalization in
weight 3 of Abel’s identity.

###  4.3. Webs by conics of singular del Pezzo’s surfaces

Del Pezzo surfaces have moduli and singular degenerations (see [Dem] for
instance). Moreover, the number of lines included in it as well as that of
conic fibrations on such a surface may vary depending on the kind of
singularities. One thus gets new examples of webs when considering the webs by
conics on singular del Pezzo’s which might carry interesting hyperlogarithmic
ARs obtained for instance as suitable degeneracies of the ARs associated to
the identities ${\bf HLog}^{r-2}$ ($r=4,\ldots,7$).

A rich but sufficiently interesting case to be considered is the one of del
Pezzo surfaces with finitely many singular points: these are classified as
well as the lines contained in each (see [Der] for a recent reference). We
will not discuss all the cases here but just consider one example which we
find interesting: the case of Cayley’s cubic.

Cayley’s surface $\mathscr{C}$ is the cubic surface in $\mathbf{P}^{3}$ with
four nodal points (it is unique up to projective equivalence). It is the
anticanonical image in $\mathbf{P}^{3}$ of the total space
${\mathscr{C}}^{\prime}$ of the blow-up
$\beta:{\mathscr{C}}^{\prime}\rightarrow\mathbf{P}^{2}$ of $\mathbf{P}^{2}$ at
the six vertices $v_{1},\ldots,v_{6}$ of the non Fano arrangement $A_{\rm
NF}$, pictured just below.

Figure 2. The non Fano arrangement $A_{\rm NF}$

Four of the seven lines of $A_{\rm NF}$ pass through three of the $v_{i}$’s.
The anticanonical image of the strict transform by $\beta$ of such a line is a
node of $\mathscr{C}$ and all its singular points are obtained in this way.

There are exactly nine lines contained in $\mathscr{C}$, namely the
anticanonical images of the six exceptional divisors $\beta^{-1}(v_{i})$ in
${\mathcal{C}}^{\prime}$ for $i=1,\ldots,6$ and the images of the strict
transforms of the three lines among the seven pictured in Figure 2 which
contain only two of the $v_{i}$’s. There are in total nine pencils of conics
on $\mathscr{C}$, which give rise to a 9-web on it, denoted by
$\boldsymbol{\mathcal{W}}_{\mathscr{C}}$, whose birational model on
$\mathbf{P}^{2}$ is easy to describe: it is the planar web formed by the six
pencils of lines passing through the vertices of $A_{\rm NF}$ and by the three
pencils of conics, one for each 4-tuple of vertices in general position among
the $v_{i}$’s. With such a description, it is then easy to recognize
$\boldsymbol{\mathcal{W}}_{\mathscr{C}}$ hence to know some of the remarkable
properties it satisfies:

> Up to equivalence, the web by conics
> $\boldsymbol{\mathcal{W}}_{\mathscr{C}}$ on Cayley’s cubic surface coincides
> with Spence-Kummer web $\boldsymbol{\mathcal{W}}_{\mathcal{S}\mathcal{K}}$
> associated to the identity with the same name satisfied by the trilogarithm.
> Consequently, the web $\boldsymbol{\mathcal{W}}_{\mathscr{C}}$
>
>   * •
>
> is not linearizable;
>
>   * •
>
> has maximal rank hence is exceptional;
>
>   * •
>
> has its space of ARs with a basis whose all elements are polylogarithmic,
> except one (which is however polylogarithmic but up to a finite covering).
> In particular, $\boldsymbol{\mathcal{W}}_{\mathscr{C}}$ carries two linearly
> independent polylogarithmic ARs of weight 3.
>
>

We see some significant differences between the properties
$\boldsymbol{\mathcal{W}}_{\mathscr{C}}$ satisfies and those enjoyed by the
27-web $\boldsymbol{\mathcal{W}}_{{\rm dP}_{3}}$ associated to ‘the’ smooth
cubic surface ${\rm dP}_{3}$: the latter carries a weight 4 hyperlogarithmic
AR (namely ${\bf HLog}^{4}$) but does not have maximal rank whereas the former
only carries trilogarithmic ARs and is exceptional. It is not clear whether
the properties of $\boldsymbol{\mathcal{W}}_{\mathscr{C}}$ can be deduced from
those satisfied by $\boldsymbol{\mathcal{W}}_{{\rm dP}_{3}}$ via a
degeneration process. This would be worth studying.

As is well-known, the number of lines on a singular cubic surface in
$\mathbf{P}^{3}$ with isolated singular points can be any one of the following
integers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 21, or 27 and any one
of these cases indeed occurs (see [BW, §3] for instance). We thus get as many
webs by conics on the corresponding cubic surfaces. We have already verified
that many of these webs have interesting web-theoretic properties.

In a future work, we plan to study systematically and thoroughly webs by
conics on del Pezzo surfaces with finitely many singular points.

##  5\. Questions and perspectives

We list below some questions or research projects that we find interesting and
that the discovery of the hyperlogarithmic identities ${\bf HLog}^{r-2}$
suggested to us.

###  5.1. The case of ${\rm dP}_{1}$

As mentioned above in §3.3.2, we do not know whether the element ${\bf
hlog}^{6}$ defined in (17) gives rise to a hyperlogarithmic abelian relation
for the 2160-webs $\boldsymbol{\mathcal{W}}_{{\rm dP}_{1}}$. We will prove
that this is indeed the case in a forthcoming joint work with A.-M. Castravet.

###  5.2. Behavior of the ${\mathcal{W}}_{{\bf dP}_{d}}$’s in families and
degeneracies

For $d\in\\{2,3,4\\}$ there is a $2(5-d)$-dimensional algebraic family
$d{\mathcal{P}}_{d}$ of degree $d$ del Pezzo surfaces giving rise to a family
of webs ${\mathscr{W}}_{{\rm dP}_{d}}\rightarrow d{\mathcal{P}}_{d}$ over it.
First, it is interesting to ask how the webs $\boldsymbol{\mathcal{W}}_{{\rm
dP}_{d}}$’s behave when ${\rm dP}_{d}$ varies in $d{\mathcal{P}}_{d}$.

Another kind of natural questions is how the $\boldsymbol{\mathcal{W}}_{{\rm
dP}_{d}}$’s may degenerate: there are modular compactifications
$\overline{d{\mathcal{P}}}_{d}$ of the moduli space $d{\mathcal{P}}_{d}$ and
it is natural to wonder first whether ${\mathscr{W}}_{{\rm dP}_{d}}\rightarrow
d{\mathcal{P}}_{d}$ can be extended over $\overline{d{\mathcal{P}}}_{d}$, then
if it is the case, what are the properties (in particular those concerning its
ARs) of a web $\boldsymbol{\mathcal{W}}_{{\rm dP}^{\prime}_{d}}$ on a
degenerate del Pezzo surface ${\rm
dP}^{\prime}_{d}\in\overline{d{\mathcal{P}}}_{d}\setminus d{\mathcal{P}}$.
Since the singular del Pezzo surfaces considered in §5.3 are such
degenerations, questions asked in the current subsection are generalizations
of those of raised in the above subsection.

In view of considering a very concrete example, let us consider the webs on
$\mathbf{P}^{2}$ defined by the 10 rational first integrals (4.1), denoted by
$\boldsymbol{\mathcal{W}}_{{\rm dP}_{4}(\gamma,\pi)}$. Assuming that (21) is
satisfied, each such web is a planar model of the 10-web by conics on a smooth
del Pezzo quartic surface which will be denoted by ${\rm dP}_{4}(\gamma,\pi)$.
Let us consider the naive specialization $\boldsymbol{\mathcal{W}}_{{\rm
dP}_{4},-1}=\lim_{\pi,\gamma\rightarrow-1}\boldsymbol{\mathcal{W}}_{{\rm
dP}_{4}(\gamma,\pi)}$ of the webs $\boldsymbol{\mathcal{W}}_{{\rm
dP}_{4}(\gamma,\pi)}$’s: it is the web whose first integrals are the limits of
all the $U_{i}$’s in (4.1) when both parameters $\pi$ and $\gamma$ go to -1.
It is no longer a 10-web but the following 8-web

$\boldsymbol{\mathcal{W}}_{{\rm
dP}_{4},-1}=\boldsymbol{\mathcal{W}}\left(\,x\,,\,\frac{1}{y}\,,\,\frac{y}{x}\,,\,\frac{x-y}{x-1}\,,\,\frac{x+1}{x-y}\,,\,\frac{x-y}{(x-1)(y+1)}\,,\,\frac{y(x+1)}{x(y+1)}\,,\,\frac{x(y-1)}{y(x-1)}\,\right)\,.$

One verifies that this web has maximal rank 21 with all its ARs being
generalized hyperlogarithms of weight 1 or 2.333333Here, the adjective phrase
‘generalized hyperlogarithmic’ refers to the notion of ‘generalized iterated
integrals’ considered in [Pi2, §1.5.3]. Denote by ${\bf
Hlog}^{3}_{\gamma,\pi}$ the weight 3 hyperlogarithmic functional identity
carried by $\boldsymbol{\mathcal{W}}_{{\rm dP}_{4}(\gamma,\pi)}$ for $\gamma$
and $\pi$ generic (that is, satisfying (21)). From the description of the ARs
of $\boldsymbol{\mathcal{W}}_{{\rm dP}_{4},-1}$ just mentioned, it comes that
${\bf Hlog}^{3}_{\gamma,\pi}$ somehow disapears under the specialization
$(\gamma,\pi)\rightarrow(-1,-1)$. It would be interesting to understand this
phenomenon better and more generally to see how the ARs of the generic web
$\boldsymbol{\mathcal{W}}_{{\rm dP}_{4}(\gamma,\pi)}$ and those of the
specialization $\boldsymbol{\mathcal{W}}_{{\rm dP}_{4},-1}$ are related.

###  5.3. Webs by conics on real del Pezzo surfaces

Let $S$ be a real del Pezzo surface with smooth complexification ${\rm
dP}_{d}\subset\mathbf{P}^{d}$ (for $d\in\\{2,3,4\\}$ say). Then it is known
that for some $S$, not all the lines contained in its complexification are
defined over $\mathbf{R}$. In such a case, it is natural to consider the web
$\boldsymbol{\mathcal{W}}_{S}$ by real pencils of conics on $S$, whose
complexification is a proper subweb of $\boldsymbol{\mathcal{W}}_{{\rm
dP}_{d}}$.

> When the number of real lines contained in $S$ is strictly less than
> $l_{r}=\lvert\mathcal{L}_{r}\lvert$ (with $r=9-d$), what can be said about
> the real web $\boldsymbol{\mathcal{W}}_{S}$? In particular, does this web
> carry interesting (hyperlogarithmic?) abelian relations?

When $d=3$, it is known that the number of real lines contained in $S$ can be
anyone of the following numbers 3, 7, 15 or 27 and that each of them indeed
occurs. This gives as many subcases of possibly interesting real webs by
conics which would be worth studying.

###  5.4. Global single-valued versions

An interesting well-known feature of Abel’s identity
$\boldsymbol{\big{(}\mathcal{A}b\big{)}}$ is that there is a global single-
valued but real analytic version of it. More precisely, let $D$ stand for the
‘Bloch-Wigner dilogarithm’, which is defined for any
$z\in\mathbf{P}^{1}\setminus\\{0,1,\infty\\}$ by

(25) $D(z)={\rm Im}\Big{(}{\bf L}{\rm i}_{2}(z)\Big{)}+{\rm
Arg}\big{(}1-z\big{)}\cdot{\rm Log}\,\lvert\,z\,\lvert\,$

where ${\rm Arg}:\mathbf{C}^{*}\rightarrow]-\pi,\pi]$ denotes the main branch
of the complex argument.

The function $D$ is real-analytic and extends continuously to the whole
Riemann sphere. Denoting again by $D$ this extension, one has
$D(0)=D(1)=D(\infty)=0$ and the most remarkable of its features is that it
satisfies the following global version of the identity
$\boldsymbol{\big{(}\mathcal{A}b\big{)}}$: one has

$D(x)-D(y)-D\bigg{(}\frac{x}{y}\bigg{)}-D\left(\frac{1-y}{1-x}\right)+D\left(\frac{x(1-y)}{y(1-x)}\right)=0\,$

for any $x,y\in\mathbf{P}^{1}$ such that none of the five arguments of $D$ in
this identity be indeterminate.

As we have considered them in this text, the hyperlogarithmic identities ${\bf
HLog}^{r-2}$ $(r=4,\ldots,7$) are holomorphic, because the functions
$AI^{r-2}_{i}$ involved in it are holomorphic. On the other hand ${\bf
HLog}^{r-2}$ is only locally satisfied, that because the $AI^{r-2}$’s extend
to global but multivalued functions on the whole projective line. Remark that
${\bf HLog}^{2}$ is a holomorphic version of Rogers dilogarithm $R$ which
admits $D$ as a single-valued global cousin. The point is that it has been
proved in [Br, §7] (see also [VZ, §2.5] or [CDG, §3]) that the most general
hyperlogarithm $I$ admits a single-valued global version $I^{\rm sing}$.
Specializing this to the case of our antisymmetric hyperlogaritms
$AI^{r-2}_{i}$ involved in ${\bf HLog}^{r-2}$ gives global single-valued
functions $\widetilde{AI}^{\,r-2}_{i}$’s (for $i=1,\ldots,\kappa_{r}$)
regarding which the following questions immediately arise for each
$r=4,\ldots,7$:

  * 1\. can one give an explicit formula for $\widetilde{AI}^{r-2}_{i}$?

  * 2\. do the $\widetilde{AI}^{\,r-2}_{i}$’s satisfy the global univalued version of ${\bf HLog}^{r-2}$, i.e. using the same notations
than in Theorem 1.1, do we have
$\sum_{i=1}^{\kappa_{r}}\epsilon_{i}\,\widetilde{AI}^{r-2}_{i}\big{(}U_{i}\big{)}=0$
identically on the whole $X_{r}$ ?

We believe that for any $r\in\\{4,\ldots,7\\}$, the answer to 2. is positive,
possibly up to considering suitable single-valued versions of the
antisymmetric hyperlogarithms $AI_{i}^{r-2}$. An approach for answering the
questions might be to use Theorem 1.1 of [CDG, §3] but this would require
first to build a ‘motivic lift’ of the identity ${\bf HLog}^{r-2}$.

###  5.5. Motivic lifts

In [CDG], the authors discuss motivic avatars of hyperlogarithms, some single-
valued (still motivic) versions of these, classical realisations of them, as
well as a motivic approach of the functional identities satisfied by the
hyperlogarithms. Thanks to its algebraic nature (cf. §3.3 where it is
identified to a generator of the signature subrepresentation of the
$W(E_{r})$-module
$\oplus_{\mathfrak{c}\in{\mathcal{K}}_{r}}\wedge^{r-2}\mathcal{U}_{\mathfrak{c}}$),
we expect that ${\bf HLog}^{r-2}$ indeed can be lifted into the ‘motivic
world’. It would be interesting to investigate this more rigourously.

###  5.6. An approach à la Gelfand and MacPherson

From our point of view, the representation-theoretic proof we give in sub-
section §3.3 that identity ${\bf HLog}^{r-2}$ holds true is somehow conceptual
and uniform in $r$ but actually does not explain much about why this identity
exists.

A remarkably nice construction of (the real form) of Abel’s identity
$\boldsymbol{\big{(}\mathcal{A}b\big{)}}$ has been given by Gelfand and
MacPherson in their wonderful paper [GM]. In their approach:

  * $(i).$

Bol’s web is obtained as the quotient of a $H_{5}$-equivariant web on the
grassmannian $G_{2}(\mathbf{R}^{5})$ induced by five natural rational maps
$\tilde{f}_{i}:G_{2}(\mathbf{R}^{5})\dashrightarrow G_{2}(\mathbf{R}^{4})$ of
linear origin, where $H_{5}$ stands for the (diagonal) Cartan torus of ${\rm
SL}_{5}(\mathbf{R})$ (see [GM, §1.1.3 and §2.2.3]);

  * $(ii).$

Abel’s identity follows from several geometrical or homological facts:

    1. 1.

the closure of a generic $H_{5}$-orbit in $G_{2}(\mathbf{R}^{5})$ can be
identified to a certain polytope, the so-called hypersimplex $\Delta^{2}_{3}$
(see [GM, Thm. 2.3.4]);

    2. 2.

a certain homological relation is satisfied between the class of the boundary
of $\Delta^{2}_{3}$ and the homological classes of its facets when these are
intrinsically considered (see [GM, Prop. 2.1.3]);

    3. 3.

the homological relation referred to above plus the version of Stokes’ theorem
for fibers integration give rise to a differential relation between the push-
forwards under the action of the suitable Cartan tori, of the invariant
4-forms on $G_{2}(\mathbf{R}^{5})$ and on $G_{2}(\mathbf{R}^{4})$ representing
the first Pontryagin class of the tautological bundles of these two
grassmannian varieties (see [GM, Thm. 1.2.2]); ;

    4. 4.

in the case under scrutiny, one of the members of the differential identity
mentioned in 3. vanishes (cf. [GM, Thm. 1.2.3]), which gives us after
integration and normalization that the sum of the five dilogarithmic terms in
the left hand-side of $\boldsymbol{\big{(}\mathcal{A}b\big{)}}$ vanishes
identically.

In [Pi4], we show that all of the ingredients used by Gelfand and MacPherson
to describe Bol’s web
$\boldsymbol{\mathcal{B}}\simeq\boldsymbol{\mathcal{W}}_{{\rm dP}_{5}}$ as a
torus quotient of a $H_{5}$-equivariant web admit counterparts for the web
$\boldsymbol{\mathcal{W}}_{{\rm dP}_{4}}$ (case $r=5$). It is then natural to
wonder whether the construction of
$\boldsymbol{\big{(}\mathcal{A}b\big{)}}\simeq{\bf HLog}^{2}$ mentioned in
$(ii).$ just above can be generalized to the case of ${\bf HLog}^{3}$ or not:

  * $\bullet$ does the identity ${\bf HLog}^{3}$, or a real version of it, can be constructed by means of the
Gelfand-MacPherson theory of ‘generalized dilogarithmic forms’ developed in
[GM]?

We believe that this question can be answered by the affirmative. If indeed
true, we expect that this would furnish a conceptual explanation of why ${\bf
HLog}^{3}$ holds true: in the case of ${\bf HLog}^{2}$, a key result is the
homological identity about the hypersimplex $\Delta_{m}^{n}$ referred to in
$(ii).2$. As mentioned in [GM, §2.1.3], this formula ‘may be interpreted as
computing the connecting homomorphism for the triple
$(\Delta_{m}^{n},\partial\Delta_{m}^{n},c)$ where $c$ is the codimension two
skeleton of $\Delta_{m}^{n}$’. We believe that something similar holds true
for identity ${\bf HLog}^{3}$, but in this case with respect to the polytope
involved in the description à la Gelfand and MacPherson of
$\boldsymbol{\mathcal{W}}_{{\rm dP}_{4}}$, which is the $D_{5}$-moment
polytope $\Delta_{D_{5}}$343434This convex polytope is known by many other
names: the weight polytope of type $D_{5}$, the 5-dimensional demihypercube or
half measure polytope, Gosset’s polytope $1_{21}$ or even the demipenteract.:
we expect that in its algebraic (tensorial) form (cf. §3.3), ${\bf HLog}^{3}$
may be interpreted as a kind of dual of a homological identity computing the
connecting homomorphism for the triple
$(\Delta_{D_{5}},\partial\Delta_{D_{5}},c_{D_{5}})$ where $c_{D_{5}}$ is the
codimension two skeleton of $\Delta_{D_{5}}$. If this is indeed true, this
possibly could be extended to the other identities ${\bf HLog}^{r-2}$ for
$r=4,\ldots,7$ as well. We hope to clarify this in a future work.

###  5.7. Combinatorial characterization

In §4.2 we discussed some nice properties of the webs
$\boldsymbol{\mathcal{W}}_{{\rm dP}_{5}}$ and $\boldsymbol{\mathcal{W}}_{{\rm
dP}_{4}}$, in particular the fact that both are characterized by their
hexagonal 3-subwebs. But we did not explain the precise meaning of this. In
this subsection, we consider this more generally and more rigorously and ask
whether the characterization just mentioned extends to the other webs
$\boldsymbol{\mathcal{W}}_{{\rm dP}_{d}}$’s.

Let us introduce general notions which we believe are relevant for studying
webs. Let $\boldsymbol{\mathcal{W}}$ be a planar $d$-web, formed by $d$
distinct foliations $\mathcal{F}_{1},\ldots,\mathcal{F}_{d}$ on its definition
domain in $\mathbf{C}^{2}$.

For any $k=3,\ldots,d$, one sets/defines

  * •

${\rm SubW}_{k}(\boldsymbol{\mathcal{W}})$ is the set of $k$-subwebs of
$\boldsymbol{\mathcal{W}}$;

  * •

${\rm SubW}(\boldsymbol{\mathcal{W}})$ is the set of subwebs of
$\boldsymbol{\mathcal{W}}$ (i.e. ${\rm
SubW}(\boldsymbol{\mathcal{W}})=\cup_{k=3}^{d}{\rm
SubW}_{k}(\boldsymbol{\mathcal{W}})$);

  * •

$\boldsymbol{AR}_{3}(\boldsymbol{\mathcal{W}})$ denotes the subspace of
$\boldsymbol{AR}(\boldsymbol{\mathcal{W}})$ spanned by the ARs of the
3-subwebs of $\boldsymbol{\mathcal{W}}$;

  * •

${r}_{3}(\boldsymbol{\mathcal{W}})$ stands for the dimension of
$\boldsymbol{AR}_{3}(\boldsymbol{\mathcal{W}})$;

  * •

$r_{\boldsymbol{\mathcal{W}}}:{\rm
SubW}(\boldsymbol{\mathcal{W}})\rightarrow\mathbf{N}$ is the function
associating $r_{3}(\boldsymbol{W})$ to any subweb $\boldsymbol{W}$ of
$\boldsymbol{\mathcal{W}}$;

  * •

$r_{\boldsymbol{\mathcal{W}},k}:{\rm
SubW}_{k}(\boldsymbol{\mathcal{W}})\rightarrow\mathbf{N}$ is the restriction
of $r_{\boldsymbol{\mathcal{W}}}$ to ${\rm
SubW}_{k}(\boldsymbol{\mathcal{W}})$.

The functions $r_{\boldsymbol{\mathcal{W}},k}$’s are combinatorial objects
invariantly attached to $\boldsymbol{\mathcal{W}}$. They allow to state in a
nice and concise way some properties of webs. For instance, let us consider
the interesting case of Bol’s web
$\boldsymbol{\mathcal{B}}\simeq\boldsymbol{\mathcal{W}}_{{\rm dP}_{5}}$: since
it is hexagonal, one has identically $r_{\boldsymbol{\mathcal{B}},3}=1$ and
$r_{\boldsymbol{\mathcal{B}},4}=3$ and one verifies that
$r_{\boldsymbol{\mathcal{B}},5}\equiv 5$. On the other hand, if
$\boldsymbol{\mathcal{L}}$ is a planar web formed by 5 pencils of lines, then
$r_{\boldsymbol{\mathcal{L}},3}\equiv 1$ (hexagonality),
$r_{\boldsymbol{\mathcal{L}},4}\equiv 3$ and
$r_{\boldsymbol{\mathcal{L}},5}\equiv 6$. Hence Bol’s characterization of
$\boldsymbol{\mathcal{B}}$ can be stated nicely as follows:

Theorem. (Bol [Bo]) Bol’s web is characterized by
$r_{\boldsymbol{\mathcal{B}}}$: any 5-web $\boldsymbol{\mathcal{W}}$ such that
$r_{\boldsymbol{\mathcal{W}},3}\equiv 1$ and
$r_{\boldsymbol{\mathcal{W}},5}\leq 5$ is equivalent to Bol’s web.

This result shows that the $r_{\boldsymbol{\mathcal{W}},k}$’s can be useful to
study webs up to equivalence. Given two planar $d$-webs
$\boldsymbol{\mathcal{W}}^{\prime}$ and
$\boldsymbol{\mathcal{W}}^{\prime\prime}$ as above, let us say that the
functions $r_{\boldsymbol{\mathcal{W}}^{\prime}}$ and
$r_{\boldsymbol{\mathcal{W}}^{\prime\prime}}$ (or
$r_{\boldsymbol{\mathcal{W}}^{\prime},3}$ and
$r_{\boldsymbol{\mathcal{W}}^{\prime\prime},3}$) are equivalent, denoted by
$r_{\boldsymbol{\mathcal{W}}^{\prime}}\sim
r_{\boldsymbol{\mathcal{W}}^{\prime\prime}}$ (and similarly for
$r_{\boldsymbol{\mathcal{W}}^{\prime},3}$ and
$r_{\boldsymbol{\mathcal{W}}^{\prime\prime},3}$) if both coincide, possibly up
to a relabeling of the foliations composing one of them.

In [Bu1], Burau first determines $r_{\boldsymbol{\mathcal{W}}_{{\rm
dP}_{3}},3}$ and then proves the following characterization result:

Theorem. (Burau [Bu1]) The class of 27-webs $\boldsymbol{\mathcal{W}}_{{\rm
dP}_{3}}$’s is characterized by $r_{\boldsymbol{\mathcal{W}}_{{\rm
dP}_{3}},3}$: any 27-web $\boldsymbol{\mathcal{W}}$ such that
$r_{\boldsymbol{\mathcal{W}},3}\sim r_{\boldsymbol{\mathcal{W}}_{{\rm
dP}_{3}},3}$ is equivalent to the 27-web in conics on a cubic surface in
$\mathbf{P}^{3}$.

Considering the two theorems above, one is naturally led to wonder if such
results would not hold for all del Pezzo webs $\boldsymbol{\mathcal{W}}_{{\rm
dP}_{d}}$. Here are a few questions regarding this that we find interesting:

  1. 1.

A 3-subweb of $\boldsymbol{\mathcal{W}}_{{\rm dP}_{d}}$ corresponds to a
triple $\boldsymbol{\mathfrak{c}}$ of pairwise distinct elements of
$\boldsymbol{\mathcal{K}}_{r}$. Does the hexagonality of the associated 3-web
$\boldsymbol{\mathcal{W}}(\boldsymbol{\mathfrak{c}})$ only depend on the
$W_{r}$-orbit of $\boldsymbol{\mathfrak{c}}$ in
$\big{(}\boldsymbol{\mathcal{K}}_{r}\big{)}^{3}$?

  2. 2.

Determine the functions $r_{\boldsymbol{\mathcal{W}}_{{\rm dP}_{d}},k}$ for
$k$ small. When there are moduli for the considered del Pezzo surfaces (i.e.
when $d\leq 4$), show that all of the $r_{\boldsymbol{\mathcal{W}}_{{\rm
dP}_{d}},k}$’s do not depend on ${\rm dP}_{d}$.

  3. 3.

Does $r_{\boldsymbol{\mathcal{W}}_{{\rm dP}_{d}},3}$ determine the class of
webs $\boldsymbol{\mathcal{W}}_{{\rm dP}_{d}}$’s, i.e. is the following
statement true: ‘any $\kappa_{r}$-web $\boldsymbol{\mathcal{W}}$ such that
$r_{\boldsymbol{\mathcal{W}},3}\sim r_{\boldsymbol{\mathcal{W}}_{{\rm
dP}_{d}},3}$ is necessarily equivalent to a web
$\boldsymbol{\mathcal{W}}_{{\rm dP}_{d}}$’?

The answers to these questions are known when $d=5$ (not to mention the
trivial case $d=6$). Burau’s theorem above gives an answer to 3. in the case
when $d=3$. The statement [ Hexagonality & Characterization ] mentioned in
§4.2.2 means that we have verified that the answer to the third question for
$\boldsymbol{\mathcal{W}}_{{\rm dP}_{4}}$ is affirmative as well. More
generally, we expect that the answers to 1. and 3. are affirmative for any
$d$. For the moment, the considered cases have been solved by ad hoc
arguments. It would be interesting to have a uniform approach (independent on
$d$).

###  5.8. Other surfaces

The setting we are working with in this paper can be very roughly described as
follows: we deal with some surfaces (del Pezzo’s in this case), each one
carrying a finite number of fibrations of a fixed type, with the property that
any component (a line in the case under scrutiny) of a non-irreducible fiber
of one of these fibrations is also a component of a fiber of the same kind of
another of the fibrations considered. Stating things in that way makes us
realise that such situations can be encountered for other kinds of algebraic
surfaces than del Pezzo’s. We will discuss in a rather vague way the case of
some elliptic projective surfaces.

Thanks to Kodaira, there is a classification of the analytic type of the
possible singular fibers of an elliptic fibration on a projective surface $S$.
In particular, there is only a finite number of possible singular fibers and
all of these have rational irreducible components. Then given a web
$\boldsymbol{\mathcal{W}}_{S}$ formed by a finite number of elliptic
fibrations on $S$, all over $\mathbf{P}^{1}$ say, the components of their
singular fibers form a finite set $\mathcal{L}_{\boldsymbol{\mathcal{W}}_{S}}$
of rational curves in $S$ and if $L_{\boldsymbol{\mathcal{W}}_{S}}$ stands for
the reduced effective divisor in $S$ whose components are the elements of
$L_{\boldsymbol{\mathcal{W}}_{S}}$, it follows that
$\boldsymbol{\mathcal{W}}_{S}$ is singularity free on the complement of
$L_{\boldsymbol{\mathcal{W}}_{S}}$ in $S$. Given an elliptic fibration
$\pi:S\rightarrow\mathbf{P}^{1}$ in $\boldsymbol{\mathcal{W}}_{S}$, let
$\sigma_{\pi}$ be the set of its singular values (i.e.
$\sigma_{\pi}=\\{\,\lambda\in\mathbf{P}^{1}\,\lvert\,\pi^{-1}(\lambda)\mbox{
is singular}\,\\}$). We denote by $H_{\pi}$ the space of global logarithmic
1-forms on $\mathbf{P}^{1}$ with poles in $\sigma_{\pi}$ and we define
$H_{\boldsymbol{\mathcal{W}}_{S}}$ as the space of rational 1-forms on $S$
spanned by the pull-backs $\pi^{*}(H_{\pi})$ for all $\pi$ in
$\boldsymbol{\mathcal{W}}_{S}$. We can play now the same game as the one
played successfully in this paper with del Pezzo’s surfaces: for a given
weight $w>0$, we are interested in the kernel of the linear map
$\oplus_{\pi\in\boldsymbol{\mathcal{W}}_{S}}\pi^{*}\big{(}(H_{\pi})^{\otimes
w}\big{)}\rightarrow(H_{\boldsymbol{\mathcal{W}}_{S}})^{\otimes w}$ since any
element in it will give rise to a weight $w$ hyperlogarithmic AR for
$\boldsymbol{\mathcal{W}}_{S}$.

An interesting class of surfaces to which the approach just sketched could
apply is that of projective $K3$ surfaces with a finite automorphism group.
Surfaces of this kind have been classified by different authors (see [Rou] and
the reference therein) and any of them admits only a finite number of elliptic
fibrations (by [St, Cor. 2.7]). Given such a $K3$ surface $X$, the finite set
of elliptic fibrations on it (up to $\mathcal{J}_{0}(X)$-equivalence, see
[BKW]) gives rise to the ‘elliptic web’ of $X$, denoted by
$\boldsymbol{\mathcal{E}\mathcal{W}}_{X}$ which may carry interesting
hyperlogarithmic abelian relations.

Note that K3 surfaces share with del Pezzo surfaces similar features
relatively to their Picard lattice: in both cases, there is a notion of
‘root’, of ‘root system’, with an associated Weyl group, etc. (e.g. see [BL,
§4]). All this suggests that elliptic webs on K3 surfaces, or possibly just
some of them, may carry interesting ARs. We are not aware of any results in
this direction for now, everything remains to be done.

###  5.9. The hyperlogarithms $AI^{r-2}$ and antisymmetization in the realm of
polylogarithmic functions.

As far as we know, the totally antisymmetrized hyperlogarithms $AI^{r-2}$
(defined for any $r\geq 4$) have already been considered in the mathematical
literature only in the case when $r=4$, in which case one recovers (a
holomorphic version of) Rogers’ dilogarithm (see §2.1.4 above). Indeed,
$AI^{2}$ is nothing but a holomorphic version (localized at the chosen base
point) of Rogers’ dilogarithm $R$ hence it is polylogarithmic in the most
basic way. It follows from (2.1.4) that $AI^{3}$ and $AI^{4}$ can be expressed
in terms of (holomorphic) polylogarithmic functions (see also (7) for
$AI^{3}$). These remarks lead to wonder about the other cases:

> $\bullet$ for $r\geq 5$, can $AI^{r-2}$ be expressed by means of classical
> polylogarithms?

We are aware of only a few examples where antisymmetrizations do occur in the
realm of (multivariable) polylogarithms:353535Antisymmetrizations also occur
in the realm of Nielsen polylogarithms, see [CGR].

  * $-$

first in [G2] (for $n=3$) then in [G3] for any $n\geq 3$, Goncharov defines
what he calls the $n$-th grassmannian polylogarithm $L_{n}^{G}$ by means of
antisymmetrizations (see Definition 4.1 of [G3]). For any $n\geq 3$, the
function $L_{n}^{G}$ is a holomorphic but multi-valued function defined on the
moduli space ${\rm Conf}_{2n}^{0}(\mathbf{P}^{n-1})$ of generic configurations
of $2n$ points in $\mathbf{P}^{n-1}$. For suitable choices of the branches of
$L_{n}^{G}$ involved, the following identity is (locally) satisfied for any
$\boldsymbol{y}=[y_{1},\ldots,y_{n}]\in{\rm
Conf}_{2n+1}^{0}(\mathbf{P}^{n-1})$:

$\sum_{i=1}^{2n+1}(-1)^{i}L_{n}^{G}\big{(}\boldsymbol{y}_{\hat{\imath}}\big{)}=0$

where $\boldsymbol{y}_{\hat{\imath}}$ stands for
$[y_{1},\ldots,\widehat{y_{i}},\ldots,y_{2n}]\in{\rm
Conf}^{0}_{2n}(\mathbf{P}^{n-1})$ for any $i=1,\ldots,2n+1$;

  * $-$

in the recent preprints [Ru] and [MR], the authors define what they call
‘quadrangular polylogarithms’ ${\rm QLi}_{n}$ and ‘cluster grasmmannian
polylogarithms’ ${\rm GLi}_{n}$ respectively, again using some
antisymmetrizations, but partial ones now (see formulas (5.1) and (4.12) in
[Ru] and [MR] respectively). From the point of view we are interested in in
this text, the cluster grassmannian polylogarithms are most interesting since
for any $n\geq 2$, ${\rm GLi}_{n}$ satisfies an interesting functional
equation (see [MR, Theorem 1.5.(4)]).

These results suggest (at least to the author of these lines) the following
question:

> $\bullet$ for $n\geq 2$, are Goncharov’s grassmannian polylogarithm
> $L_{n}^{G}$, Matveiakin-
>  Rudenko cluster grassmannian polylogarithm ${\rm GLi}_{n}$ and the single-
> variable
>  hyperlogarithm $AI^{n}$ considered in this text, related in any way?

The answer to this question being obvious (and affirmative) for $n=2$, the
first case to be considered is the one when $n=3$. For instance, one can ask
whether $AI^{3}$ can be expressed by means of $L_{3}^{G}$ or of ${\rm
GLi}_{3}$ or not.

###  5.10. Curvilinear webs on Fano blow-ups of projective spaces.

The planar webs considered in this paper arise quite naturally when
considering blow-ups of the projective plane which are Fano. If one is
interested by possible higher dimensional generalizations of these webs, it is
natural, in a first step at least, to consider the blow-ups of projective
spaces of dimension bigger than 2 which are Fano manifolds again.

In this subsection, we would like to discuss two cases, one which is classical
and has been studied recently, for which most of things are clear, a second
one which looks very similar but which has not been considered beyond the two
dimensional case yet. The former case is that of the blow-ups of
$\mathbf{P}^{n}$ in $n+2$ points, the latter in $n+3$ points, this for any
$n\geq 2$.

We fix $n\geq 2$ and a subset $P\subset\mathbf{P}^{n}$ of $N$ points
$p_{1},\ldots,p_{N}$ in general position, with $N=n+2$ or $n+3$. We denote by
$\boldsymbol{\mathcal{W}}_{P}$ the web by rational curves on $\mathbf{P}^{n}$
formed by the following foliations:

  1. $-$

the linear radial foliations $\mathcal{L}_{p_{i}}$ of lines passing through
$p_{i}$, for $i=1,\ldots,N$;

  2. $-$

the congruences $\mathcal{C}_{J}$ of degree $n$ rational normal curves (RNCs)
passing through the $p_{j}$’s for $j\in J$, for any $J\subset\\{1,\ldots,N\\}$
of cardinality $n+2$.

When $N=n+2$, then

$\boldsymbol{\mathcal{W}}_{P}=\boldsymbol{\mathcal{W}}\Big{(}\mathcal{L}_{p_{1}},\ldots,\mathcal{L}_{p_{n+2}},\mathcal{C}\Big{)}$

where $\mathcal{C}$ stands here for the congruence of degree $n$ RNCs passing
through all the $p_{i}$’s. In particular, $\boldsymbol{\mathcal{W}}_{P}$ is a
$(n+3)$-web for $n$ arbitrary and it is equivalent to
$\boldsymbol{\mathcal{W}}_{{\rm dP}_{5}}\simeq\boldsymbol{\mathcal{B}}$ when
$n=2$.

In case $N=n+3$, let $\mathcal{C}_{\widehat{\imath}}$ stand for the congruence
associated to $\\{1,\dots,n+3\\}\setminus\\{i\\}$ for $i=1,\ldots,n+3$. Then
one has

$\boldsymbol{\mathcal{W}}_{P}=\boldsymbol{\mathcal{W}}\Big{(}\,\mathcal{L}_{p_{1}},\ldots,\mathcal{L}_{p_{n+2}},\mathcal{C}_{\widehat{1}},\ldots,\mathcal{C}_{\widehat{n+3}}\,\Big{)}$

which thus is a $2(n+3)$-web. In particular when $n=2$, one recovers (a
birational model of) $\boldsymbol{\mathcal{W}}_{{\rm dP}_{4}}$.

####  5.10.1. The curvilinear webs $\boldsymbol{\mathcal{W}}_{P}$ when $N=n+2$

When $N=n+2$ (for any $n\geq 2$), then the web $\boldsymbol{\mathcal{W}}_{P}$
is a birational model of the web
$\boldsymbol{\mathcal{W}}_{\mathcal{M}_{0,n+3}}$ defined on the moduli space
$\mathcal{M}_{0,n+3}$ by the $n+3$ maps
$f_{i}:\mathcal{M}_{0,n+3}\rightarrow\mathcal{M}_{0,n+2}$ induced by
forgetting the $i$-th point (for $i=1,\ldots,n+3$).

These webs have been studied by a few classical authors and more recently by
Damiano and in the paper [Pi3] (see the references therein). Building on
Gelfand-MacPherson theory of ‘generalized dilogarithmic forms’, Damiano
investigated the $(n-1)$-abelian relations of
$\boldsymbol{\mathcal{W}}_{\mathcal{M}_{0,n+3}}$ for any $n\geq 2$, and
claimed that the structure of the corresponding space of ARs
$\boldsymbol{AR}^{(n-1)}\big{(}\boldsymbol{\mathcal{W}}_{\mathcal{M}_{0,n+3}}\big{)}$
was for $n$ arbitrary the most straightforward generalization one can think of
that of
$\boldsymbol{\mathcal{W}}_{\mathcal{M}_{0,5}}\simeq\boldsymbol{\mathcal{B}}$
(cf. §4.2.1 above). In [Pi3] it is proved that Damiano’s arguments and claims
are flawed when $n$ is odd and a correct way to handle this case is given.

The situation for $n\geq 2$ arbitrary is similar by many aspects to that of
del Pezzo’s surfaces considered in this paper: if $X_{P}$ denotes the blow-up
of $\mathbf{P}^{n}$ at the $p_{i}$’s, then ${\bf
Pic}(X_{P})\simeq\mathbf{Z}^{n+3}$ and one can define ‘roots’ which are
‘$(-1)$-divisors’ and which form a root system of Dynkin type $A_{n+2}$ in the
orthogonal $(K_{X_{P}})^{\perp}\subset{\bf Pic}(X_{P})$ (cf. Section 2 of
[CT]). The associated Weyl group $W_{P}\simeq W(A_{n+2})$ in this case is
naturally isomorphic to ${\rm
Aut}\big{(}{\mathcal{M}_{0,n+3}}\big{)}\simeq\mathfrak{S}_{n+3}$ and acts
naturally on
$\boldsymbol{AR}^{(n-1)}\big{(}\boldsymbol{\mathcal{W}}_{\mathcal{M}_{0,n+3}}\big{)}$.
And one of the main result about the latter space concerns its structure as a
$\mathfrak{S}_{n+3}$-module.363636See [Pi3] for more details. One denotes by
$\boldsymbol{AR}^{(n-1)}_{C}\big{(}\boldsymbol{\mathcal{W}}_{\mathcal{M}_{0,n+3}}\big{)}$
the subspace generated by the $(n-1)$-ARs of the $(n+1)$-subwebs of
$\boldsymbol{\mathcal{W}}_{\mathcal{M}_{0,n+3}}$. It is clearly a
subrepesentation of the whole space
$\boldsymbol{AR}^{(n-1)}\big{(}\boldsymbol{\mathcal{W}}_{\mathcal{M}_{0,n+3}}\big{)}$.
Using Gelfand-MacPherson theory and following Damiano, one constructs a
special abelian relation
$\boldsymbol{\mathcal{E}}_{n}\in\boldsymbol{AR}^{(n-1)}\big{(}\boldsymbol{\mathcal{W}}_{\mathcal{M}_{0,n+3}}\big{)}$
from the ($n$-th wedge power of the) ${\rm SO}_{n+3}(\mathbf{R})$-invariant
representative of the Euler class of the tautological bundle on the oriented
grassmannian $G_{2}^{or}\big{(}\mathbf{R}^{n+3}\big{)}$. Then the following
dichotomy holds true, according to the parity of $n$:

  * $\bullet$

when $n$ is odd, one has
$\boldsymbol{\mathcal{E}}_{n}\in\boldsymbol{AR}^{(n-1)}\big{(}\boldsymbol{\mathcal{W}}_{\mathcal{M}_{0,n+3}}\big{)}=\boldsymbol{AR}^{(n-1)}_{C}\big{(}\boldsymbol{\mathcal{W}}_{\mathcal{M}_{0,n+3}}\big{)}$
and the latter space is an irreducible $\mathfrak{S}_{n+3}$-representation,
with Young symbol $\big{[}31^{n}\big{]}$;

  * $\bullet$

when $n$ is even

    * –

there is a decomposition in direct sum of $\mathfrak{S}_{n+3}$-irreps

$\boldsymbol{AR}^{(n-1)}\big{(}\boldsymbol{\mathcal{W}}_{\mathcal{M}_{0,n+3}}\big{)}=\boldsymbol{AR}^{(n-1)}_{C}\big{(}\boldsymbol{\mathcal{W}}_{\mathcal{M}_{0,n+3}}\big{)}\oplus\big{\langle}\boldsymbol{\mathcal{E}}_{n}\big{\rangle}$

with corresponding Young symbols $\big{[}221^{n-1}\big{]}$ and
$\big{[}1^{n+3}\big{]}$;

    * –

at least for $n\leq 12$ (but conjecturally for any $n$ even) there is an
explicit formula for the component of $\boldsymbol{\mathcal{E}}_{n}$,
involving only rational expressions and logarithms of rational expressions
(cf. [Pi3, Prop. 5.28]). In short: $\boldsymbol{\mathcal{E}}_{n}$ is a
polygarithmic AR of weight 1.373737 When $n=2$, the weight 2 polylogarithmic
identity ${\bf HLog}^{2}\simeq\boldsymbol{\big{(}\mathcal{A}b\big{)}}$ and the
weight 1 abelian relation $\boldsymbol{\mathcal{E}}_{2}$ are related as
follows: the latter corresponds to the differential identity obtained by
taking the total derivative of the former functional equation.

  * $\bullet$

in both cases (i.e. regardless of the parity of $n$), the $(n-1)$-rank of
$\boldsymbol{\mathcal{W}}_{P}\simeq\boldsymbol{\mathcal{W}}_{\mathcal{M}_{0,n+3}}$
is equal to $(n+1)(n+2)/2$ hence this web has maximal rank.

One of the key ingredients to obtain the results above is the action of the
Weyl group $W(A_{n+2})$ on the space of ARs of the web under scrutiny. All
this makes of the $\boldsymbol{\mathcal{W}}_{\mathcal{M}_{0,n+3}}$’s natural
and meaningful generalisations of the del Pezzo conical web
$\boldsymbol{\mathcal{W}}_{{\rm dP}_{5}}$, especially when $n$ is even.

####  5.10.2. The curvilinear webs $\boldsymbol{\mathcal{W}}_{P}$ when $N=n+3$

Considering the previous discussion about the case when $N=n+2$, one can
wonder about the case when $N+3$, and more precisely whether the
$\boldsymbol{\mathcal{W}}_{P}$’s would not satisfy natural generalizations of
(some of ) the nice properties enjoyed by $\boldsymbol{\mathcal{W}}_{{\rm
dP}_{4}}$ listed in §4.2.2.

The setting for any $n\geq 2$ appears as a natural generalisation of that of
${\rm dP}_{4}$ (which corresponds to the case when $n=2$). As in the previous
case, setting $X_{P}={\bf Bl}_{P}(\mathbf{P}^{n})$, the $(-1)$-divisors give
rise to a root system, now of Dynkin type $D_{n+3}$, in the orthogonal of the
canonical class in ${\rm Pic}(X_{P})\simeq\mathbf{Z}^{n+4}$. Moreover, there
is a natural action of the Weyl group $W(D_{n+3})$ on the Picard lattice of
$X_{P}$ (see [CT] and more specifically [SV]). It is then natural to ask the
following questions:

  * 1.

Does the Weyl group $W(D_{n+3})$ act naturally on the space
$\boldsymbol{AR}^{(n-1)}\big{(}\boldsymbol{\mathcal{W}}_{P}\big{)}$ of
$(n-1)$-ARs of $\boldsymbol{\mathcal{W}}_{P}$ (or otherwise on a certain
subspace of it) ?

  * 2.

If the answer to the first question is affirmative, what is the decomposition
of $\boldsymbol{AR}^{(n-1)}\big{(}\boldsymbol{\mathcal{W}}_{P}\big{)}$ into
irreducible $W(D_{n+3})$-representations?

  * 3.

What is the dimension of
$\boldsymbol{AR}^{(n-1)}\big{(}\boldsymbol{\mathcal{W}}_{P}\big{)}$ ? Does
this web has maximal $(n-1)$-th rank ?

We do not have answers to these questions for $n\geq 3$ yet.

In this case (namely $N=n+3$) as well, there might exist a dichotomy depending
on the parity of $n$. This is suggested by the very nice results of [AC],
which concern the case when $n$ is even. In this case, we set $m=n/2\geq 1$
and consider a projective parametrization
$\Lambda:\mathbf{P}^{1}\rightarrow\mathbf{P}^{n}$ of the unique degree $n$
rational normal curve passing through $p_{1},\ldots,p_{n+3}$. Let
$\mathbf{C}=\mathbf{P}^{1}\setminus\\{\infty\\}$ be an affine chart such that
$P\subset\Lambda(\mathbf{C})$ and set
$\lambda_{i}=\Lambda^{-1}(p_{i})\in\mathbf{C}$ for $i=1,\ldots,n+3$. Let $Z$
be the complete intersection in $\mathbf{P}^{n+2}$ of the two quadrics cut out
by the equations $\sum_{i=1}^{n+3}x_{i}^{2}=0$ and
$\sum_{i=1}^{n+3}\lambda_{i}\,x_{i}^{2}=0$ and consider the variety
$G=F_{m-1}(Z)\subset G_{m-1}\big{(}\mathbf{P}^{n+2}\big{)}$ of $(m-1)$-planes
contained in $Z$.

In [AC], the following statements have been proved to be true:

  1. $(i).$

there are pseudo-isomorphisms $f:G\dashrightarrow X_{P}$, that is birational
maps inducing isomorphisms in codimension 1 (and there exist precisely
$2^{n+2}$ such pseudo-isomorphisms);

  2. $(ii).$

$G$ contains $2^{n+2}$ subspaces of dimension $m$ which are linear (that is
give rise to $m$-planes in the image) relatively to the Plücker embedding
$G\hookrightarrow\mathbf{P}\big{(}\wedge^{m}\mathbf{C}^{n+3}\big{)}$;

  3. $(iii).$

given a pseudo-isomorphism $f:G\dashrightarrow X_{P}$, the pull-back
$f^{*}\big{(}\boldsymbol{\mathcal{W}}_{P}\big{)}$ admits as first integrals
$2(n+3)$ morphisms onto rational varieties $\varphi_{i}:G\rightarrow
T_{\varphi_{i}}$ and $\psi_{i}:G\rightarrow T_{\psi_{i}}$ (with
$i=1,\ldots,n+3$) which all satisfy the following properties:

     * $-$

any smooth fiber is isomorphic to $\mathbf{P}^{1}$;

     * $-$

there are exactly $2^{n}$ singular fibers, each of which has two irreducible
components, namely two $m$-planes in $G$ (cf. $(ii)$ above) intersecting
transversely in one point.

All the properties above together are generalizations in higher but even
dimension of the properties of ${\rm dP}_{4}$ which were useful regarding the
identity ${\bf HLog}^{3}$. As the identity ${\bf
HLog}^{2}\simeq\boldsymbol{\big{(}\mathcal{A}b\big{)}}$ admits a
generalization in any even dimension (the so-called Euler’s abelian relation
$(\boldsymbol{\mathcal{E}}_{n})$, see [Pi3] and footnote 37 above), we find
natural to ask the following:

  * 4.

Can ${\bf HLog}^{3}$ be generalized to the case of
$\boldsymbol{\mathcal{W}}_{P}$ when $N=n+3$ is odd? More precisely, in this
case, does this web admit a $(n-1)$-AR whose components are weight 2
hyperlogarithms? If yes, does this AR corresponds to a symbolic tensorial
identity which is stable up to sign under the action of $W(D_{n+3})$ (hence
spans the $W(D_{n+3})$-signature representation)?

###  5.11. Curvilinear webs on Mori dream spaces.

The considerations above lead us to notice that any Mori dream space (ab. MDS)
naturally carries a curvilinear web (by rational curves?). One can ask about
the properties of such webs.

Among the simplest examples of non Fano Mori dream spaces obtained by blowing-
up projective spaces, one can mention the following ones:

  * •

the blow-up of $\mathbf{P}^{2}$ at nine points lying on a smooth conic;

  * •

the blow-up of $\mathbf{P}^{3}$ at seven points in general position;

  * •

the Losev-Manin space obtained by two successive blow-ups: first four points
in general position in $\mathbf{P}^{3}$, then the strict transforms of the six
lines passing through two of the considered points.

As references about these examples, see [CT] for the first two and [C, §8] for
the third.

Let $X$ be one of the three MDS above and let $\boldsymbol{\mathcal{W}}_{X}$
be a natural web by rational curves on it.383838We are quite vague about what
has to be considered as ‘natural’ here. These are the webs on $X$ the
foliations of which are families of (necessarily rational?) curves
corresponding to generators of extremal classes in the cone of movable classes
on $X$. We hope to come back to this with more details in a future work. For
any $k=0,\ldots,\dim(X)-1$, one can ask the following question:

  * $\bullet$

What can be said about the $k$-abelian relations of
$\boldsymbol{\mathcal{W}}_{X}$? Does this web carry hyperlogarithmic or even
better, polylogarithmic, $k$-ARs?

Curvilinear webs on Fano manifolds or more generally on MDS may be very
interesting webs. It is our opinion that they would deserve being investigated
more in depth.

## Appendix: the structure of $\mathbf{C}^{{\mathcal{L}}_{r}}$ as a
$W(E_{r})$-module

In this appendix we briefly explain the computations we used to establish some
representation-theoretic facts about the $W(E_{r})$-module
$\mathbf{C}^{{\mathcal{L}}_{r}}$. These computations were mainly performed
using the software GAP3 [S&al] which has the advantage of being available
online in CoCalc.393939One can run GAP3 by typing the command /ext/bin/gap3 in
a CoCalc terminal window. For the sake of clarity and conciseness, we will
essentially discuss the case $r=5$, as all the other ones can be treated in a
completely analogous way.

### The case $r=5$

Our aim here is to explain how, using GAP, it is easy to determine the
character $\chi_{5}$ of the $W(D_{5})$-module $\mathbf{C}^{{\mathcal{L}}_{5}}$
explicitly and then to perform the computations used in Section §3.

First, in order to define $W(D_{5})$ and its character table in GAP, we run
the following commands:

⬇

gap> WD5:=CoxeterGroup("D",5);;

gap> TableWD5:=CharTable(WD5);

gap> Display(TableWD5);

From the outcome of the last command, one can extract the character 4 (in
which the dot $\cdot$ is used instead of $0$ to make its reading easier).

| $\boldsymbol{\big{(}1^{5}.\big{)}}$ | $\boldsymbol{(1^{3}.1^{2})}$ | $\boldsymbol{\big{(}1.1^{4}\big{)}}$ |  $\boldsymbol{\big{(}21^{3}.\big{)}}$ |  $\boldsymbol{\big{(}1^{2}.21\big{)}}$ |  $\boldsymbol{\big{(}21.1^{2}\big{)}}$ |  $\boldsymbol{\big{(}.21^{3}\big{)}}$ |  $\boldsymbol{\big{(}221.\big{)}}$ |  $\boldsymbol{\big{(}1.22\big{)}}$ |  $\boldsymbol{\big{(}2.21\big{)}}$ |  $\boldsymbol{\big{(}311.\big{)}}$ |  $\boldsymbol{\big{(}1.31\big{)}}$ |  $\boldsymbol{\big{(}3.11\big{)}}$ |  $\boldsymbol{\big{(}32.\big{)}}$ |  $\boldsymbol{\big{(}.32\big{)}}$ |  $\boldsymbol{\big{(}41.\big{)}}$ |  $\boldsymbol{\big{(}.41\big{)}}$ |  $\boldsymbol{\big{(}5.\big{)}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\boldsymbol{[1^{2}.1^{3}]}$ | 10 | -2 | 2 | -4 | 2 | . | -2 | 2 | -2 | . | 1 | -1 | 1 | -1 | 1 | . | . | .
$\boldsymbol{[1.1^{4}]}$ | 5 | 1 | -3 | -3 | -1 | 1 | 3 | 1 | 1 | -1 | 2 | . | -2 | . | . | -1 | 1 | .
$\boldsymbol{[.1^{5}]}$ | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1
$\boldsymbol{[1^{3}.2]}$ | 10 | -2 | 2 | -2 | . | 2 | -4 | -2 | 2 | . | 1 | -1 | 1 | 1 | -1 | . | . | .
$\boldsymbol{[1^{2}.21]}$ | 20 | -4 | 4 | -2 | 2 | -2 | 2 | . | . | . | -1 | 1 | -1 | 1 | -1 | . | . | .
$\boldsymbol{[1.21^{2}]}$ | 15 | 3 | -9 | -3 | -1 | 1 | 3 | -1 | -1 | 1 | . | . | . | . | . | 1 | -1 | .
$\boldsymbol{\big{[}.21^{3}\big{]}}$ | 4 | 4 | 4 | -2 | -2 | -2 | -2 | . | . | . | 1 | 1 | 1 | 1 | 1 | . | . | -1
$\boldsymbol{\big{[}1.2^{2}\big{]}}$ | 10 | 2 | -6 | . | . | . | . | 2 | 2 | -2 | -2 | . | 2 | . | . | . | . | .
$\boldsymbol{\big{[}2.21\big{]}}$ | 20 | -4 | 4 | 2 | -2 | 2 | -2 | . | . | . | -1 | 1 | -1 | -1 | 1 | . | . | .
$\boldsymbol{\big{[}.2^{2}1\big{]}}$ | 5 | 5 | 5 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | .
$\boldsymbol{\big{[}1^{2}.3\big{]}}$ | 10 | -2 | 2 | 2 | . | -2 | 4 | -2 | 2 | . | 1 | -1 | 1 | -1 | 1 | . | . | .
$\boldsymbol{\big{[}1.31\big{]}}$ | 15 | 3 | -9 | 3 | 1 | -1 | -3 | -1 | -1 | 1 | . | . | . | . | . | -1 | 1 | .
$\boldsymbol{\big{[}.31^{2}\big{]}}$ | 6 | 6 | 6 | . | . | . | . | -2 | -2 | -2 | . | . | . | . | . | . | . | 1
$\boldsymbol{\big{[}2.3\big{]}}$ | 10 | -2 | 2 | 4 | -2 | . | 2 | 2 | -2 | . | 1 | -1 | 1 | 1 | -1 | . | . | .
$\boldsymbol{\big{[}.32\big{]}}$ | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | .
$\boldsymbol{\big{[}1.4\big{]}}$ | 5 | 1 | -3 | 3 | 1 | -1 | -3 | 1 | 1 | -1 | 2 | . | -2 | . | . | 1 | -1 | .
$\boldsymbol{\big{[}.41\big{]}}$ | 4 | 4 | 4 | 2 | 2 | 2 | 2 | . | . | . | 1 | 1 | 1 | -1 | -1 | . | . | -1
$\boldsymbol{\big{[}.5\big{]}}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1

Table 4. Character table of the Weyl group of type $D_{5}$.

A few explanation are in order regarding this table. As is well-known (e.g.
see [GP, §5.6]), for any odd integer $n\geq 4$,404040The case when $n$ is even
is a bit more involved but fully understood as well. the characters as well as
the conjugacy classes of the Weyl group of type $D_{n}$ can be labelled by
means of the ‘bipartitions of $n$’, that is pairs $(\lambda,\lambda^{\prime})$
of partitions such that $\lvert\lambda\lvert+\lvert\lambda^{\prime}\lvert=n$.
In Table 4, we use $[\lambda.\lambda^{\prime}]$ for labelling the line
associated to the corresponding character while $(\lambda.\lambda^{\prime})$
serves to label the column associated to the corresponding conjugacy class.
When $\lambda$ is the empty partition, we just write $[.\lambda^{\prime}]$ and
$(.\lambda^{\prime})$ respectively and similarly when
$\lambda^{\prime}=\emptyset$.

In view of determining $\chi_{5}$ as explained in the proof of Proposition
3.2, we need to construct explicit representatives of the conjugacy classes in
$W(D_{5})$. This can be achieved in GAP3 by means of the following command:

⬇

gap> List(ConjugacyClasses(WD5),x->CoxeterWord(WD5,Representative(x)));

which returns the following string of 18 sequences:

⬇

[ [ ], [ 1, 2 ], [ 1, 2, 3, 1, 2, 3, 4, 3, 1, 2, 3, 4 ], [ 1 ], [ 1, 2, 3 ],

[ 1, 2, 4 ], [ 1, 2, 3, 1, 2, 3, 4, 3, 1, 2, 3, 4, 5 ], [ 1, 4 ],

[ 1, 3, 1, 2, 3, 4 ], [ 1, 2, 3, 5 ], [ 1, 3 ], [ 1, 2, 3, 4 ],

[ 1, 2, 4, 5 ], [ 1, 3, 5 ], [ 1, 3, 1, 2, 3, 4, 5 ], [ 1, 4, 3 ],

[ 1, 2, 3, 4, 5 ], [ 1, 4, 3, 5 ] ]

The $k$-th element of the above list corresponds to an explicit representative
of the $W(D_{5})$-conjugacy class associated to the $k$-th column of Table 4
in terms of the generators of WD5 the software GAP is dealing with. More
precisely, if this string is $[i_{1},\ldots,i_{m}]$, the corresponding
representative is $\zeta_{i_{1}}\cdots\zeta_{i_{m}}$, where
$\zeta_{1},\ldots,\zeta_{5}$ stand for the generators of WD5 implicitly
defined when the latter group was defined (by using the GAP command
CoxeterGroup("D",5).

In view of relating GAP generators $\zeta_{i}$’s with the $s_{i}$’s considered
in the proof of Proposition 3.2, one makes explicit the labelling implemented
in GAP of the vertices of the Dynkin diagram $D_{5}$ via the command
PrintDiagram(CoxeterGroup("D",5)) which returns the diagram

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{5\,.}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Comparing it with the diagram for $D_{5}$ in Figure 1, one can easily express
GAP generators of WD5 in terms of those we considered in Section
§3:414141Several (two in case of $D_{5}$) equivalent choices are possible for
expressing the $\zeta_{i}$’s in terms of the $s_{i}$’s; all give rise to the
same results of course. one has

$\zeta_{1}=s_{4}\,,\qquad\zeta_{2}=s_{5}\,,\qquad\zeta_{3}=s_{3}\,,\qquad\zeta_{4}=s_{2}\qquad\mbox{
and }\qquad\zeta_{5}=s_{1}\,.$

We denote by $\sigma$ the permutation such that $\zeta_{i}=s_{\sigma(i)}$ for
$i=1,\ldots,5$.

At this point, it is straightforward to compute the value of the character
$\chi_{5}$ evaluated on any conjugacy class. If $[i_{1},\ldots,i_{m}]$ stands
for the string given by GAP which encodes a representative of a conjugacy
class $\Gamma\subset W(D_{5})$, then one has

$\chi_{5}(\Gamma)={\rm Tr}\Big{(}M_{\sigma(i_{1})}\cdots
M_{\sigma(i_{m})}\Big{)}$

where the $M_{i}$’s $(i=1,\ldots,5$) stand for the $16\times 16$ matrices
defined in (14). Since these matrices can be computed explicitly, its is
straighforward to compute the trace above hence to build the 18-tuple of
values corresponding to the character $\chi_{5}$ (for the same labelling as
the one used to label the column of the character table above). Identifying
$\chi_{5}$ with this 18-tuple, one has

$\chi_{5}=\big{(}16,0,0,8,0,0,0,4,0,0,4,0,0,2,0,2,0,1\big{)}.$

Elementary linear algebra computations give us that

$\chi_{5}=\boldsymbol{\big{[}.5\big{]}}+\boldsymbol{\big{[}1.4\big{]}}+\boldsymbol{\big{[}2.3\big{]}}$

where we use their labels (of the form
$\boldsymbol{\big{[}\lambda.\lambda^{\prime}\big{]}}$) for denoting the
correspond line in the character table. Since
$\boldsymbol{\big{[}.5\big{]}}={\bf 1}$, this gives us the decomposition for
$\mathbf{C}^{{\mathcal{L}}_{5}}$ in (3.2).

Once $\chi_{5}$ is explicitly known, it is not difficult to compute the
character of $\wedge^{3}\mathbf{C}^{{\mathcal{L}}_{5}}$, denoted by
$\wedge^{3}\chi_{5}$, thanks to the following formula which holds true for any
element $g$ of $W(D_{5})$:

(26)
$\wedge^{3}\chi_{5}\big{(}g\big{)}=\frac{1}{3!}\,\bigg{(}\,\chi_{5}\big{(}g\big{)}^{3}-3\,\chi_{5}\Big{(}g^{2}\Big{)}\,\chi_{5}\big{(}g\big{)}+2\,\chi\Big{(}g^{3}\Big{)}\,\bigg{)}\,.$

Using the same notations and conventions as above, we then get that

$\wedge^{3}\chi_{5}=\big{(}560,0,0,24,0,0,0,-20,0,0,8,0,0,0,0,-2,0,0\Big{)}\,.$

Denoting now by $\chi^{s}$ the character of $W(D_{5})$ corresponding to the
$s$-th line of Table 4, we have
$\wedge^{3}\chi_{5}=\sum_{s=1}^{18}w_{s}\,\,\chi^{s}$ with

$\big{(}w_{s}\big{)}_{s=1}^{18}=\big{(}1,1,0,4,5,4,1,1,6,0,5,6,3,3,1,2,2,0\big{)}\,.$

In particular, since $w^{3}=0$ we get that
$\chi^{3}=\boldsymbol{[.1^{5}]}={\bf sign}$ does not appear in the
decomposition of $\wedge^{3}\mathbf{C}^{{\mathcal{L}}_{5}}$ into
$W(D_{5})$-irreducibles, which is the content of (19) in the case under
scrutiny.

All the computations above can be performed quite easily within GAP. First one
constructs $\chi_{5}$ and $\wedge^{3}\chi_{5}$ as characters of $W(D_{5})$ by
means of the following two commands:

⬇

gap>chi5:=TableWD5.irreducibles[14]+TableWD5.irreducibles[16]

+TableWD5.irreducibles[18];;

gap>Wedge3chi5:=AntiSymmetricParts(TableWD5,[chi5],3);

And one computes the multiplicity of the signature representation as an
irreducible component of $\wedge^{3}\mathbf{C}^{{\mathcal{L}}_{5}}$ by
computing the scalar product

⬇

gap>ScalarProduct(TableWD5,Wedge3Chi5[1],TableWD5.irreducibles[3]);

which turns out to be 0 as instantly returned by GAP.

### A few words about the other cases

All that has been discussed above can be adapted quite straightforwardly for
any other $r\in\\{4,\ldots,8\\}$. Once $\chi_{r}$ has been determined in terms
of the character table of the associated Weyl group $W(E_{r})$, one computes
its $(r-2)$-th wedge product either using the general determinental formula
expressing $\wedge^{r-2}\chi_{r}$ in terms of $\chi_{r}$424242This classical
formula can found in completely explicit form in [GR, Exercise 2.9.13.(a)]. or
the GAP command AntiSymmetricParts. Then the multiplicity $m_{r}^{\bf sign}$
of ${\bf sign}$ in the decomposition of $\wedge^{r-2}\chi_{r}$ into
irreducibles is given by their scalar product which we compute in GAP using
ScalarProduct.

We obtain that ${\bf sign}$ does not appears in the decomposition of
$\wedge^{r-2}\chi_{r}$ for $r=4,\ldots,7$, while its multiplicity $m_{8}^{\bf
sign}$ as a component of $\wedge^{6}\chi_{8}$ can be computed to be 5.

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Luc Pirio<EMAIL_ADDRESS>
Laboratoire de Mathématiques de Versailles
Université Paris-Saclay, UVSQ & CNRS (UMR 8100)