# Monodromy of the Radon transform

Meirav Amram111M. Amram: Mathematics Department, Shamoon College of
Engineering, 77245 Ashdod, Israel<EMAIL_ADDRESS>
Michael Dettweiler222M. Dettweiler: Department of Mathematics, University of
Bayreuth, 95440 Bayreuth, Germany<EMAIL_ADDRESS>
Stefan Reiter333S. Reiter: Department of Mathematics, University of Bayreuth,
95440 Bayreuth, Germany<EMAIL_ADDRESS>
###### Abstract

We algorithmically determine the monodromy of the local system on the smooth
part of the Radon transformation of a generic simple perverse sheaf on the
projective plane.

###### Contents

  1. Introduction
  2. 1 Definition and first properties of the Radon transformation
  3. 2 Encoding local systems on plane curve complements
  4. 3 Deformations of local systems and cocycles using the braid group
  5. 4 The monodromy of the Radon transform
  6. 5 Examples: an elliptic fibration and the classical Zariski pair
  7. 6 Appendix - the Radon program

## Introduction

The classical Radon transform is an integral transformation which sends a
suitably integrable function $f$ on the projective plane
${\mathbb{P}}^{2}(\mathbb{R})$ to the function

$\mathcal{R}(f)\,:\,\widetilde{{\mathbb{P}}}^{2}(\mathbb{R})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{R},\,\,\,\widetilde{x}\mapsto\int_{L_{\widetilde{x}}}f\,,$

on the dual plane $\widetilde{{\mathbb{P}}}^{2}(\mathbb{R}),$ where
$L_{\widetilde{x}}\subseteq{\mathbb{P}}^{2}(\mathbb{R})$ is the line
corresponding to an element $\widetilde{x}$ in the dual plane [14].

The algebraic-geometric version of the classical Radon transformation was
introduced by Brylinsky [6]. Following Grothendieck’s fonctions-faisceaux
philosophy (cf. [6], [12]), the basic idea is to replace functions by objects
in the derived category $D^{b}_{c}({\mathbb{P}}^{2})$ and integration of
functions by direct image functors between derived categories. This leads to a
functor (the Radon transform)

$\mathcal{R}:D^{b}_{c}({\mathbb{P}}^{2})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}D^{b}_{c}(\widetilde{{\mathbb{P}}}^{2}),\,\,K\mapsto
Rp_{2*}p_{1}^{*}(K)[1],$

where $p_{1},p_{2}$ are the two canonical projections of the incidence
subvariety $H\subseteq{\mathbb{P}}^{2}\times\widetilde{{\mathbb{P}}}^{2}$ (see
[6] and Sec. 1 for the formal definition).

Recall that one has the abelian category of perverse sheaves ${\rm
Perv}({\mathbb{P}}^{2})$ inside $D^{b}_{c}({\mathbb{P}}^{2})$ and that, using
perverse cohomology, any object in $D^{b}_{c}({\mathbb{P}}^{2})$ can be
described by perverse sheaves ([4], [11]). By Radon inversion (see Section 1
and [6], [11]), Brylinsky’s Radon transform sends simple perverse sheaves to
simple perverse sheaves, up to constant sheaves in
$D^{b}_{c}(\widetilde{{\mathbb{P}}}^{2}).$ It is this result which makes the
Radon transform useful in many applications.

Generically, a simple perverse sheaf $K$ on ${\mathbb{P}}^{2}$ is an
intermediate extension $j_{!*}{\mathcal{L}}[2]$ of an irreducible non-constant
local system ${\mathcal{L}}$ (shifted to cohomological degree $-2$) along an
open inclusion $j:Y\hookrightarrow{\mathbb{P}}^{2}$ with $Y$ a plane curve
complement $Y={\mathbb{P}}^{2}\smallsetminus C.$ Let
$\widetilde{Y}=\widetilde{{\mathbb{P}}}^{2}\smallsetminus\widetilde{C}$ be the
complement of the union $\widetilde{C}$ of the dual curve of $C$ and the lines
in the dual plane, which are determined by the singularities of $C.$ Then the
Radon transform $\mathcal{R}(K)$ is locally constant on $\widetilde{Y}$ (Prop.
4.1). We call the local system
$\mathcal{R}({\mathcal{L}}):=\mathcal{R}(K)[-2]|_{\widetilde{Y}}$ the Radon
transform of ${\mathcal{L}}.$ Under some additional assumptions which are
often fulfilled in the applications, the Radon transform $\mathcal{R}(K)$ is
the intermediate extension

$\mathcal{R}(K)=\widetilde{j}_{!*}\left(\mathcal{R}({\mathcal{L}})[2]\right)$

of $\mathcal{R}({\mathcal{L}})[2]$ along
$\widetilde{j}:\widetilde{Y}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{{\mathbb{P}}}^{2}.$
Hence $\mathcal{R}({\mathcal{L}})$ determines $\mathcal{R}(K)$ in these cases
by the uniqueness of the intermediate extension.

The local systems ${\mathcal{L}}$ and $\mathcal{R}({\mathcal{L}})$ are
determined in turn by their monodromy representations

$\rho_{\mathcal{L}}:\pi_{1}({\mathbb{P}}^{2}\smallsetminus
C)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{GL}(V)\quad\textrm{and}\quad\rho_{\mathcal{R}({\mathcal{L}})}:\pi_{1}(\widetilde{{\mathbb{P}}}^{2}\smallsetminus\widetilde{C})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{GL}(\widetilde{V}).$

It is the aim of this article, to _determine the monodromy representation of
the Radon transform ${\mathcal{R}({\mathcal{L}})}$ in an algorithmic way_ from
the fundamental data (= monodromy tuple of ${\mathcal{L}}$ together with the
braid monodromy of $C$), used to describe the local system ${\mathcal{L}},$
see Thm. 4.6.

The main idea is to interprete the Radon transform
$\mathcal{R}({\mathcal{L}})$ as a variation of parabolic cohomology groups
over $\widetilde{Y}$ in the sense of [7] and to use the algorithmic approach
for such variations which is introduced in [7].

We remark that the étale version of our results on the Radon transform leads
to new instances of the regular inverse Galois problem, details will appear
elsewhere.

## 1\. Definition and first properties of the Radon transformation

In the following we recall the definition of the Radon transform on the
projective plane, following Brylinsky [6]. For results on the étale version of
the Radon transform we refer to [11]. We remark that most of the stated
results also hold for higher dimensional projective spaces over more general
bases (cf. loc.cit.).

We consider projective spaces and varieties over the complex numbers. We
assume additionally in Section 1 that the coefficient ring is a field $k$ of
characteristic zero. In the other sections, the coefficient ring is allowed to
be any field $k,$ unless otherwise stated.

Let ${\mathbb{P}}^{2}$ be the usual projective plane and let
$\widetilde{{\mathbb{P}}}^{2}$ be the dual projective plane which parametrizes
the lines in ${\mathbb{P}}^{2}.$ Then $\widetilde{{\mathbb{P}}}^{2}$ is also a
projective plane and the line in ${\mathbb{P}}^{2}$ corresponding to
$[\widetilde{x}_{0},\widetilde{x}_{1},\widetilde{x}_{2}]\in\widetilde{{\mathbb{P}}}^{2}$
has the defining equation

$x_{0}\cdot\widetilde{x}_{0}+x_{1}\cdot\widetilde{x}_{1}+x_{2}\cdot\widetilde{x}_{2}=0,$

also proving that ${\mathbb{P}}^{2}$ is the dual of
$\widetilde{{\mathbb{P}}}^{2}.$

Notation: a point $\widetilde{x}\in\widetilde{{\mathbb{P}}}^{2}$ will
correspond to a line $L_{\widetilde{x}}\subseteq{\mathbb{P}}^{2}.$ By duality,
an element $x\in$ ${\mathbb{P}}^{2}$ can be identified with a line
$\widetilde{L}_{x}\subseteq\widetilde{{\mathbb{P}}}^{2}$ in the dual plane. By
construction, the line $\widetilde{L}_{x}$ parametrizes the pencil of lines
${\bf L}_{x}$ in ${\mathbb{P}}^{2}$ containing the point $x.$

Let

$H:=\\{({x},\widetilde{{x}})\in{\mathbb{P}}^{2}\times\widetilde{{\mathbb{P}}}^{2}\mid{x}\in
L_{\widetilde{{x}}}\\}\subset{\mathbb{P}}^{2}\times\widetilde{{\mathbb{P}}}^{2}$

denote the incidence relation which is equipped with the obvious projections
$p_{1}:H\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathbb{P}}^{2}$
and
$p_{2}:H\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{{\mathbb{P}}}^{2}.$

Then one has the _Radon transformation_

$\mathcal{R}:D^{b}_{c}({\mathbb{P}}^{2})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}D^{b}_{c}(\widetilde{{\mathbb{P}}}^{2}),\,K\mapsto
Rp_{2*}p_{1}^{*}(K)[1].$

The same construction applies to the dual space leading to the _dual Radon
transform_

$\widetilde{\mathcal{R}}:D^{b}_{c}(\widetilde{{\mathbb{P}}}^{2})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}D^{b}_{c}({\mathbb{P}}^{2}),\,\widetilde{K}\mapsto
Rp_{1*}p_{2}^{*}(\widetilde{K})[1]\,.$

For an object $K\in D^{b}_{c}({\mathbb{P}}^{2}),$ let
$\mathcal{R}^{n}(K):={}^{p}H^{n}\mathcal{R}(K)$ denote the $n$-th perverse
cohomology of $\mathcal{R}(K)$ and let ${\mathcal{K}}:=Rp_{*}(K),$ where $p$
denotes the structure morphism to the spectrum of $\mathbb{C}.$

###### Remark 1.1

The definition of the Radon transform makes sense also for more general
coefficient rings and also for more general fields of definition (cf.[6],
[13]). Then results whose proof depends on the decomposition theorem (see
[4]), like the proof of Radon inversion in [6], [11], become more involved.
This is the reason, why we assume that the coefficient ring is a field of
characteristic $0$ in this section. We remark that by [13], Thm. 1.1 and Rem.
1.2(3), Radon inversion still holds in our context for coefficient rings like
$\mathbb{Z}/\ell^{n}\mathbb{Z}$ and algebraically closed fields of definition
of arbitrary characteristic.

Let ${\rm Perv}^{\mathrm{smooth}}({\mathbb{P}}^{2})$ denote the category of
smooth(=constant) perverse sheaves on ${\mathbb{P}}^{2}$ and let
$M({\mathbb{P}}^{2})$ be the Verdier quotient ${\rm
Perv}({\mathbb{P}}^{2})/{\rm Perv}^{\mathrm{smooth}}({\mathbb{P}}^{2}).$ Let
$M(\widetilde{{\mathbb{P}}}^{2})$ be analogously defined. Within these
categories one has _Radon inversion_ (cf. [6], Thm. 5.5, [11], Lemma 1.4):

(1.1) $\widetilde{\mathcal{R}}\circ\mathcal{R}\equiv\operatorname{Id}.$

Especially, simple objects in $M({\mathbb{P}}^{2})$ correspond to simple
objects in $M(\widetilde{{\mathbb{P}}}^{2})$ under Radon transform. Note that
any object $K$ in $M({\mathbb{P}}^{2})$ is represented uniquely by a _reduced_
perverse sheaf (i.e., a perverse sheaf which does not admit any nontrivial
constant subobject/quotient), see [11], convention on Page 211. Sometimes it
is useful to work directly within the categories ${\rm
Perv}({\mathbb{P}}^{2})$ and ${\rm Perv}(\widetilde{{\mathbb{P}}}^{2})$ and
not in their associated quotients. This leads to the following definition:

###### Definition 1.2

An object $K$ in ${\rm Perv}({\mathbb{P}}^{2})$ has the _property $P$_ if it
has no nontrivial cohomology outside degree $0,$ i.e., ${\mathcal{K}}^{i}=0$
for $i\neq 0,$ where ${\mathcal{K}}$ is as above. Let ${\mathcal{P}}$ denote
the category of semisimple perverse sheaves having the property $P.$ Let
$\widetilde{{\mathcal{P}}}$ be the subcategory of ${\rm
Perv}(\widetilde{{\mathbb{P}}}^{2})$ defined analogously.

###### Remark 1.3

The property $P$ appears often as follows: let $K=j_{!*}({\mathcal{L}})[2]$ be
an intermediate extension of an irreducible local system ${\mathcal{L}}$ on a
dense open subset $U$ of ${\mathbb{P}}^{2}$ such that ${\mathcal{L}}$ has
local monodromy with a trivial $1$-eigenspace along a projective line
$L\subset{\mathbb{P}}^{2}\smallsetminus U,$ then $K$ is in ${\mathcal{P}}.$

To see this we argue as follows: the inclusion
$j:U\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathbb{P}}^{2}$
is the composition of
$j^{\prime}:U\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{A}^{2}={\mathbb{P}}^{2}\smallsetminus
L$ with
$j^{{}^{\prime\prime}}:\mathbb{A}^{2}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathbb{P}}^{2}.$
The long exact sequence of cohomology with compact supports with respect to
the inclusion
$j^{{}^{\prime\prime}}:\mathbb{A}^{2}\hookrightarrow{\mathbb{P}}^{2}$ applied
to the constructible sheaf $j_{!*}({\mathcal{L}})=K[-2]$ reads

$\cdots\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H^{1}_{c}(\mathbb{A}^{2},j^{\prime}_{!*}({\mathcal{L}}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H^{1}({\mathbb{P}}^{2},K[-2])\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H^{1}(L,K[-2]|_{L})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H^{2}_{c}(\mathbb{A}^{2},j^{\prime}_{!*}({\mathcal{L}}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\cdots\;.$

By Artin’s vanishing theorem one has
$H^{1}_{c}(\mathbb{A}^{2},j^{\prime}_{!*}({\mathcal{L}}))=0,$ see [9], Thm.
4.1.26 and Cor. 5.2.19. Since $K[-2]|_{L}$ has at most punctual support by the
assumption on the local monodromy at $L,$ the cohomology group
$H^{1}(L,K[-2]|_{L})$ also vanishes, implying that
$H^{1}({\mathbb{P}}^{2},K[-2])$ and hence
$H^{-1}({\mathbb{P}}^{2},K)={\mathcal{K}}^{-1}$ is zero. The other cohomology
groups ${\mathcal{K}}^{i},\,i\neq 0,$ also vanish by the irreducibility
assumption, resp. by duality.

We refer to [9], Chapter 6, for other vanishing theorems in this direction.

###### Proposition 1.4

Let $K$ be an object in ${\rm Perv}({\mathbb{P}}^{2}).$ Then the following
holds:

  1. (1)

The Radon transform $\mathcal{R}(K)$ is perverse if $K$ is in ${\mathcal{P}}$
or if $K$ is a simple and non-constant perverse sheaf situated in degree $-2.$

  2. (2)

If $K$ is a simple object in ${\mathcal{P}},$ then $\mathcal{R}(K)$ is simple
as an object in ${\rm Perv}(\widetilde{{\mathbb{P}}}^{2}).$

###### Proof.

By [11], Lemma IV.2.2, (cf. [6], Thm. 5.5) one has

$\mathcal{R}^{n}(K)\simeq\widetilde{p}^{*}[2]{\mathcal{K}}^{n-1}$

for $n<0$ and

$\mathcal{R}^{n}(K)\simeq\widetilde{p}^{*}[2]{\mathcal{K}}^{n+1}$

for $n>0.$ Hence the perverse sheaves $\mathcal{R}^{n}(K)$ vanish for $n\neq
0,$ using that ${\mathcal{K}}^{i}=0$ for $i\neq 0,$ proving the first claim of
$(1).$ If $K$ is a simple and non-constant perverse sheaf situated in degree
$-2,$ then it is the intermediate extension of a non-constant irreducible
local system on a dense open subset of ${\mathbb{P}}^{2}.$ This implies that
${\mathcal{K}}^{-2}$ is trivial (and hence also ${\mathcal{K}}^{2}$ by
duality), implying that also in this case, one has
$\mathcal{R}(K)=\mathcal{R}^{0}(K)$ implying that $\mathcal{R}(K)$ is
perverse. It has to be non-constant because of Radon inversion.

Let $K$ be a simple object in ${\mathcal{P}}.$ Since the morphism
$p_{2}:H\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{{\mathbb{P}}}^{2}$
is projective, the Radon transform $\mathcal{R}(K)$ is semisimple. The
Lefschetz sequence given on page 211 of [11] implies that
$\widetilde{p}^{*}[2]{\mathcal{K}}^{-1}$ is isomorphic to the maximal constant
perverse subsheaf of $\mathcal{R}^{0}(K)=\mathcal{R}(K)$ (the latter equality
follows as in the proof of $(1)$ using
${\mathcal{K}}^{-2}={\mathcal{K}}^{2}=0$). Since ${\mathcal{K}}^{-1}=0$ by
property $P$ and since $\mathcal{R}({K})$ is semisimple, the Radon transform
$\mathcal{R}(K)$ is reduced. It follows now from Radon inversion in the form
stated in (1.1) that $\mathcal{R}(K)$ is again simple. $\Box$

## 2\. Encoding local systems on plane curve complements

Let $C\subseteq{\mathbb{P}}^{2}$ be a reduced plane algebraic curve of degree
$r.$ In this section we collect the data which describe a local system
${\mathcal{L}}$ over an arbitrary field $k$ of rank $n\in\mathbb{N}$ on
${\mathbb{P}}^{2}\smallsetminus C,$ suitable for the Radon transformation. In
the following, we will summarize the main results of [7], adapted to our
situation:

We choose a generic point ${p}_{0}\in{\mathbb{P}}^{2}\smallsetminus C.$ Using
a projective coordinate change on ${\mathbb{P}}^{2}$ we can assume that
$p_{0}=[1:0:0].$ Using the conventions from the beginning of Section 1, we
write $\widetilde{L}_{{p}_{0}}\subseteq\widetilde{{\mathbb{P}}}^{2}$ for the
corresponding line in the dual plane. Hence

$\widetilde{L}_{p_{0}}=\\{[0:\widetilde{x}_{1}:\widetilde{x}_{2}]\mid[\widetilde{x}_{1}:\widetilde{x}_{2}]\in{\mathbb{P}}^{1}\\}.$

We also choose an element
$\widetilde{p}_{0}\in\widetilde{L}_{p_{0}}\subseteq\widetilde{{\mathbb{P}}}^{2}$
such that $L_{\widetilde{p}_{0}}\subseteq{{\mathbb{P}}}^{2}$ intersects $C$
transversally, where $L_{\widetilde{p}_{0}}$ is the line in ${\mathbb{P}}^{2}$
determined by $\widetilde{p}_{0}.$ By possibly using a further coordinate
change fixing $p_{0}$ we can assume that
$\widetilde{p}_{0}=[0:1:0]\in\widetilde{{\mathbb{P}}}^{2}.$

As in the previous section, the incidence relation is defined as

$H:=\\{([x_{0}:x_{1}:x_{2}],[\widetilde{x}_{0}:\widetilde{x}_{1}:\widetilde{x}_{2}])\in{\mathbb{P}}^{2}\times\widetilde{{\mathbb{P}}}^{2}\mid
x_{0}\cdot\widetilde{x}_{0}+x_{1}\cdot\widetilde{x}_{1}+x_{2}\cdot\widetilde{x}_{2}=0\\}\subset{\mathbb{P}}^{2}\times\widetilde{{\mathbb{P}}}^{2},$

coming with its projections
$p_{1}:H\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathbb{P}}^{2}$
and
$p_{2}:H\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{{\mathbb{P}}}^{2}.$
We can view the pencil ${\bf L}_{p_{0}}$ as a subvariety of $H:$

${\bf
L}_{p_{0}}=p_{2}^{-1}(\widetilde{L}_{p_{0}})=\\{([x_{0}:x_{1}:x_{2}],[0:\widetilde{x}_{1}:\widetilde{x}_{2}])\in
H\\}.$

Let ${\widetilde{y}_{1}},\ldots,{\widetilde{y}_{s}}\in\widetilde{L}_{p_{0}}$
denote the points which are _exceptional_ with respect to $C,$ meaning that
each $L_{\widetilde{y}_{i}}$ is either tangent to $C$ or that
$L_{\widetilde{y}_{i}}$ contains a singularity of $C.$ By possibly using
another projective coordinate change (which fixes $p_{0}$ and fixes
$L_{\widetilde{p}_{0}}$ setwise), we can assume that
$\widetilde{y}_{s}=[0:0:1].$

One has an isomorphism of ${\mathbb{P}}^{1}$-bundles

(2.1) $\Lambda:{\bf L}_{p_{0}}\smallsetminus
p_{2}^{-1}({\widetilde{y}_{s}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathbb{P}}^{1}_{\mathbb{A}^{1}}={\mathbb{P}}^{1}\times\mathbb{A}^{1}$

given by

$([x_{0}:x_{1}:x_{2}],[0:1:\widetilde{x}_{2}])\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}([x_{0}:x_{1}],\widetilde{x}_{2}),$

where the bundle maps are the respective second projections.

Let

$S:=\widetilde{L}_{p_{0}}\smallsetminus\\{{\widetilde{y}_{1}},\ldots,{\widetilde{y}_{s}}\\},$

let $X:=p_{2}^{-1}(S)$ and $X_{0}:=p_{2}^{-1}(\widetilde{p}_{0}).$ Then
$D:=p_{1}^{-1}(C)\cap X$ is a divisor of relative degree $r$ over $S$ (denoted
$r$-configuration in [7], Section 2.1). Let $U:=X\smallsetminus D,$ write
$\pi:U\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}S$
for the restriction of $p_{2}$ to $U,$ and let
$U_{0}:=p_{2}^{-1}({\widetilde{p}_{0}})\cap U.$ Then
$\pi:U\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}S$ is
a topological fibration with typical fibre $U_{0}$ and hence yields a short
exact sequence as follows (cf. [7], Formula (6)):

(2.2)
$1\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\pi_{1}(U_{0},(p_{0},\widetilde{p}_{0}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\pi_{1}(U,(p_{0},\widetilde{p}_{0}))\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\pi_{1}(S,\widetilde{p}_{0})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}1\,.$

By restricting the above ${\mathbb{P}}^{1}$-bundle isomorphism $\Lambda:{\bf
L}_{p_{0}}\smallsetminus
p_{2}^{-1}({\widetilde{y}_{s}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathbb{P}}^{1}_{\mathbb{A}^{1}}$
to $X$ we obtain a trivializing isomorphism
$\lambda:X\simeq{\mathbb{P}}^{1}_{S}$ of ${\mathbb{P}}^{1}$-bundles which
identifies the map
$p_{2}:X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}S$
with the structural map
${\mathbb{P}}^{1}_{S}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}S.$
One has

$\lambda((p_{0},[0,1,\widetilde{x}_{2}]))=(\infty,\widetilde{x}_{2}),$

where $\infty=[1:0]$ is the point at infinity in ${\mathbb{P}}^{1}.$ Moreover,
since $p_{0}\notin C$ as $p_{0}$ is generic w.r. to $C,$ the section
$\infty\times S=\lambda(p_{1}^{-1}(p_{0}))$ is disjoint from $\lambda(D).$

As in [7], Section 1.2, we fix a homeomorphism (called a _marking_)
$\kappa:X_{0}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathbb{P}}^{1}(\mathbb{C}),$
mapping $P_{0}:=X_{0}\smallsetminus D$ to a subset $\\{x_{1},\ldots,x_{r}\\}$
of the real line inside the Riemann sphere ${\mathbb{P}}^{1}(\mathbb{C})$ and
$p_{0}$ to an element $x_{0}$ in the upper half plane. Let us, just for the
moment, identify $U_{0}=X_{0}\smallsetminus P_{0}$ with
${\mathbb{P}}^{1}(\mathbb{C})\smallsetminus\\{x_{1},\ldots,x_{r}\\}$ via
$\kappa.$ As in [7] now choose counterclockwise simple loops
$\gamma_{1},\ldots,\gamma_{r}$ around $x_{1},\ldots,x_{r}$ which are based at
$x_{0}$ as follows: Each $\gamma_{i},\,i=1,\ldots,r$ is a simple closed loop
which intersects the real line exactly twice, first the open interval
$(x_{i-1},x_{i}),$ then the interval $(x_{i},x_{i+1})$ (where for $i=1,$ the
interval $(x_{i-1},x_{i})$ is iterpreted as $(-\infty,x_{1})$ and for $i=r$
the interval $(x_{i},x_{i+1})$ is interpreted as $(x_{i+1},+\infty)$).

By taking inverse images under $\kappa,$ this leads to a standard presentation
of the fundamental group

$\pi_{1}(U_{0},(p_{0},\widetilde{p}_{0}))=\langle\gamma_{1},\ldots,\gamma_{r}\mid\gamma_{1}\cdots\gamma_{r}=1\rangle.$

We remark that (up to a Möbius transform) the construction of the
$\gamma_{1},\ldots,\gamma_{r}$ is the same as the construction of a geometric
basis of meridians in [2].

By the above presentation of $\pi_{1}(U_{0}),$ the restriction
${\mathcal{L}}_{0}:={\mathcal{L}}|_{U_{0}}$ is determined by the tuple of
matrices

${\bf g}_{\mathcal{L}}:=(g_{1},\ldots,g_{r})\in\operatorname{GL}(V)^{r},\quad
g_{1}\cdots g_{r}=1,$

where $\gamma_{i}$ is sent to $g_{i}$ under the monodromy representation
$\rho_{{\mathcal{L}}}$ of ${\mathcal{L}}$ (here $V\simeq k^{n}$). We call
${\bf g}$ the _monodromy tuple_ of ${\mathcal{L}}_{0}$ (resp. of
${\mathcal{L}}$) with respect to $\gamma_{1},\ldots,\gamma_{r}.$

###### Remark 2.1

By Lefschetz’ hyperplane theorem (see [10], [8]), the fundamental group
$\pi_{1}(U_{0})$ maps surjectively onto
$\pi_{1}({\mathbb{P}}^{2}\smallsetminus C).$ Hence the monodromy tuple ${\bf
g}$ of ${\mathcal{L}}_{0}$ also determines the local system ${\mathcal{L}}$ up
to isomorphism.

Let

$\mathcal{O}_{r}:=\\{P\subseteq\mathbb{A}^{1}\mid\\#P=r\\}$

denote the configuration space parametrizing subsets of $\mathbb{C}$ of
cardinality $r$ and let

$\mathcal{O}_{r,1}:=\\{(q,P)\in{\mathbb{P}}^{1}\times\mathcal{O}_{r}\mid
q\notin P\\}.$

As explained in [7], Section 2.2, our choice of a marking defines generators
$\beta_{1},\ldots,\beta_{r-1}$ of the braid group
${\mathcal{B}}_{r}:=\pi_{1}(\mathcal{O}_{r},P_{0})$ as follows: under
$\kappa,$ a representative of $\beta_{i}$ fixes all points different from
$\\{x_{i},x_{i+1}\\}$ and the image under $\kappa$ of $x_{i}$ moves to
$x_{i+1}$ in the lower half plane and (the image under $\kappa$ of) $x_{i+1}$
travels to $x_{i}$ in the upper half plane. This leads to the standard
presentation of ${\mathcal{B}}_{r}:$

$\langle\beta_{1},\ldots,\beta_{r-1}\mid\beta_{i}\beta_{i+1}\beta_{i}=\beta_{i+1}\beta_{i}\beta_{i+1}\textrm{
for }i=1,\ldots,r-1\textrm{ and }\beta_{i}\beta_{j}=\beta_{j}\beta_{i}\textrm{
for }|i-j|>1\rangle.$

Let ${\mathcal{B}}_{r,1}:=\pi_{1}(\mathcal{O}_{r,1},(\infty,P_{0})).$ The
natural projection
$\mathcal{O}_{r,1}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{O}_{r}$
is a topological fibration with typical fibre $U_{0}$, and admits a section
$P\mapsto(\infty,P).$ It yields a split exact sequence of fundamental groups

(2.3)
$1\;\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\;\pi_{1}(U_{0},{\infty})\;\mathchoice{\xrightarrow{\textstyle\kern
4.0pt\;\kern
3.0pt}}{\stackrel{{\scriptstyle\;}}{{\longrightarrow}}}{}{}\pi_{1}(\mathcal{O}_{r,1},(\infty,P_{0}))\;\mathchoice{\xrightarrow{\textstyle\kern
4.0pt\;\kern
3.0pt}}{\stackrel{{\scriptstyle\;}}{{\longrightarrow}}}{}{}{\mathcal{B}}_{r}\;\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\;1.$

We may identify ${\mathcal{B}}_{r}$ with its image in
$\pi_{1}(\mathcal{O}_{r,1})$ under the splitting induced from the above
section. Then the conjugation operation of ${\mathcal{B}}_{r}$ on
$\pi_{1}({\mathbb{P}}^{1}\smallsetminus P_{0},{\infty})$ is as follows ([7],
Formula 10):

(2.4)
$\beta_{i}^{-1}\,\gamma_{j}\beta_{i}\;=\quad\begin{cases}\quad\gamma_{i}\gamma_{i+1}\gamma_{i}^{-1},&\quad\text{\rm
for $j=i$,}\\\ \quad\gamma_{i},&\quad\text{\rm for $j=i+1$,}\\\
\quad\gamma_{j},&\quad\text{\rm otherwise.}\end{cases}$

Since $\lambda(D)$ is disjoint from $\infty\times S$ we obtain a map
$p:S\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathcal{O}_{r}$
by sending $s$ to the fibre of $\lambda(D)$ over $s.$ Let
$\varphi:\pi_{1}(S,\widetilde{p}_{0})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{B}}_{r}$
be the homomorphism induced by $p.$ Then the above short exact sequence (2.2)
can be viewed as the pullback of the short exact sequence (2.3) along
$\varphi$ (cf. [7], Section 2.3, for these results). Using the splitting of
(2.2) coming from the $\infty$-section, one can view $\pi_{1}(S)$ as a
subgroup of $\pi_{1}(U).$ By construction, the resulting conjugation action of
$\pi(S)$ on $\pi_{1}(U_{0})$ factors through the map $\varphi$ and is
determined by Formula (2.4).

###### Definition 2.2

Let $\widetilde{\gamma}_{1},\ldots,\widetilde{\gamma}_{s}$ be simple loops in
$S$ satisfying the product rule
$\widetilde{\gamma}_{1}\cdots\widetilde{\gamma}_{s}=1,$ constructed using a
marking as above. Then their images under $\varphi$ are words in the braids
$\beta_{1},\ldots,\beta_{r-1}:$

$\varphi(\widetilde{\gamma}_{i})=\omega_{i}(\beta_{1},\ldots,\beta_{r-1})=\omega_{i}\in{\mathcal{B}}_{r}.$

These words constitute the _braid monodromy_ of $C.$

###### Definition 2.3

The _fundamental data_ of a local system ${\mathcal{L}}$ on
${\mathbb{P}}^{2}\smallsetminus C$ consist of the following:

  * $\scriptscriptstyle\bullet$

The monodromy tuple $(g_{1},\ldots,g_{r})\in\operatorname{GL}(V)^{r}\,(V\simeq
k^{n})$ of ${\mathcal{L}}$ with respect to fixed generators
$\gamma_{1},\ldots,\gamma_{r}$ of $\pi_{1}(U_{0}).$

  * $\scriptscriptstyle\bullet$

The braid monodromy $(\omega_{1},\ldots,\omega_{s})\in{\mathcal{B}}_{r}^{s}$
of $C$ with respect to fixed generators
$\widetilde{\gamma}_{1},\ldots,\widetilde{\gamma}_{s}$ of $\pi_{1}(S).$

###### Remark 2.4

  1. (1)

As mentioned above, the trivializing morphism
$\lambda:X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathbb{P}}^{1}_{S}$
has the property that $\lambda(D)$ and $\infty\times S$ are disjoint. It
follows that the restriction of $\lambda$ to
$U\smallsetminus\lambda^{-1}(\infty\times S)$ constitutes an isomorphism

$U\smallsetminus\lambda^{-1}(\infty\times
S)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{A}^{2}\smallsetminus(C\cup
L_{\widetilde{y}_{1}}\cup\ldots\cup L_{\widetilde{y}_{s-1}})$

under which the map $U\smallsetminus\lambda^{-1}(\infty\times
S)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}S$ is
transformed into the second projection

${{\rm pr}}_{2}:\mathbb{A}^{2}\smallsetminus(C\cup
L_{\widetilde{y}_{1}}\cup\ldots\cup
L_{\widetilde{y}_{s-1}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{A}^{1}_{y}\smallsetminus\\{-{y_{1}},\ldots,-{y_{s-1}}\\},\,(x,y)\mapsto
y,$

composed by the map

$-{\bf
1}:\mathbb{A}^{1}_{y}\smallsetminus\\{-{y_{1}},\ldots,-{y_{s-1}}\\}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathbb{A}^{1}_{y}\smallsetminus\\{{y_{1}},\ldots,{y_{s-1}}\\},\,y\mapsto-y.$

Hence the braid monodromy in the above sense coincides, up to a twist by the
morphism $-{\bf 1},$ indeed with the classical notion of braid monodromy (cf.
[8]).

  2. (2)

In practise, we first choose generators
$-\widetilde{\gamma}_{1},\ldots,-\widetilde{\gamma}_{s}$ of
$\pi_{1}(\mathbb{A}^{1}_{y}\smallsetminus\\{-{y_{1}},\ldots,-{y_{s-1}}\\})$
satisfying the product rule
$(-\widetilde{\gamma}_{1})\cdots(-\widetilde{\gamma}_{s})=1$ and define
$\widetilde{\gamma}_{1},\ldots,\widetilde{\gamma}_{s}$ to be their images
under $\lambda^{-1}\circ-{\bf 1}.$ Then the classical braid monodromy of the
plane curve complement yields the braid monodromy defined above.

  3. (3)

If $\widetilde{\gamma}_{1},\ldots,\widetilde{\gamma}_{s}$ are constructed as
in (2), by van Kampens theorem, the fundamental group
$\pi_{1}({\mathbb{P}}^{2}\smallsetminus C)$ has a presentation as follows
(where the conjugation operation of $\omega_{i}$ on $\gamma_{i}$ is as in
(2.4)):

$\langle\gamma_{1},\ldots,\gamma_{r}\mid\gamma_{1}\cdots\gamma_{r}=1,\,{\omega_{i}}^{-1}\gamma_{j}\omega_{i}=\gamma_{j}\textrm{
for }i\in\\{1,\ldots,s\\},j\in\\{1,\ldots,r\\}\rangle.$

Hence the monodromy tuple $(g_{1},\ldots,g_{r})\in\operatorname{GL}(V)^{r}$ of
${\mathcal{L}}$ also satisfies these relations.

## 3\. Deformations of local systems and cocycles using the braid group

We continue by summarizing the results of [7] concerned with deformations of
local system along braids: Using the notions of the previous section, define

$\mathcal{E}_{r}\;:=\;\\{\;{\bf g}=(g_{1},\ldots,g_{r})\mid
g_{i}\in\operatorname{GL}(V),\;\;\prod_{i}g_{i}=1\;\\}.$

As described in the previous Section, an element ${\bf g}\in\mathcal{E}_{r}$
corresponds to a representation
$\rho_{0}:\pi_{1}(U_{0})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{GL}(V)$
(set $\rho_{0}(\gamma_{i}):=g_{i}$) and hence to a local system on $U_{0}$.

As described in [7], Section 2.2, the natural deformation operation of the
braid group on local systems (by composing $\rho_{0}$ with the operation of
$\beta\in{\mathcal{B}}_{r}$ on $\pi_{1}(U_{0})$) amounts to a right-action of
${\mathcal{B}}_{r}$ on $\mathcal{E}_{r}$ as follows:

(3.1) ${\bf
g}^{\beta_{i}}\;=\;(g_{1},\ldots,g_{i+1},g_{i+1}^{-1}g_{i}g_{i+1},\ldots,g_{r}).$

This deformation operation enables the computation of the braiding operation
on cocycles which will be considered in the next section: given ${\bf
g}\in\mathcal{E}_{r}$, we have the vector space

$H_{\bf g}\;=\;\\{\;(v_{1},\ldots,v_{r})\mid v_{i}\in{\rm
Im}(g_{i}-1),\;v_{1}\cdot g_{2}\cdots g_{r}+v_{2}\cdot g_{3}\cdots
g_{r}+\cdots+v_{r}=0\;\\}\subset V^{r}.$

We remark that we have a natural isomorphism between $H_{\bf g}/E_{\bf g}$
(where $E_{\bf g}\leqslant H_{\bf g}$ is as defined below) and the parabolic
cohomology group $H^{1}({\mathbb{P}}^{1},j_{*}{\mathcal{L}})$ (where $j$
denotes the inclusion of $U_{0}$ to ${\mathbb{P}}^{1},$ cf. [7], Section 2).

In loc.cit., Section 3, for ${\bf g}\in\mathcal{E}_{r}$ and
$\beta\in{\mathcal{B}}_{r}$ a linear map

$\Phi({\bf g},\beta):\,H_{{\bf g}}\;\mathchoice{\xrightarrow{\textstyle\kern
4.0pt\;\kern
3.0pt}}{\stackrel{{\scriptstyle\;}}{{\longrightarrow}}}{}{}H_{{\bf
g}^{\beta}}.$

is defined as follows (recall our convention of linear maps acting from the
right): for a standard generator $\beta_{i}\in{\mathcal{B}}_{r}$ one has

(3.2) $\qquad(v_{1},\ldots,v_{r}){\Phi({\bf g},\beta_{i})}\\\
\;=\;(v_{1},\ldots,v_{i+1},\,\underbrace{v_{i+1}(1-g_{i+1}^{-1}g_{i}g_{i+1})+v_{i}g_{i+1}}_{\text{\rm$(i+1)$th
entry}},\,\ldots,v_{r}).$

Note that both vector spaces $H_{{\bf g}}$ and $H_{{\bf g}^{\beta}}$ are
subspaces of $V^{r}.$ Hence ${\Phi({\bf g},\beta_{i})}$ is given by
restriction of the following endomorphism $M({\bf g},\beta_{i})\in{\rm
End}(V^{r})$ in matrix form to $H_{{\bf g}}:$

(3.3) $M({\bf g},\beta_{i})=\left(\begin{array}[]{llllllllll}1&&&\\\
&&\ddots&&&\\\ &&&\quad 1&&\\\ &&&&\,\,\,0&\quad g_{i+1}&\\\
&&&&\,\,\,1&\quad(1-g_{i+1}^{-1}g_{i}g_{i+1})&&\\\ &&&&&&&1&&\\\
&&&&&&&&\quad\ddots&\\\ &&&&&&&&&\quad\quad 1\end{array}\right)\,.$

Moreover, one has the ‘cocycle rule’ which allows one to compute $\Phi({\bf
g},\beta)$ for a general element $\beta\in{\mathcal{B}}_{r}$ in terms of the
$\Phi({\bf g},\beta_{i}):$

(3.4) $\Phi({\bf g},\beta^{\prime}\beta^{\prime\prime})\;=\;\Phi({\bf
g},\beta^{\prime})\cdot\Phi({\bf
g}^{\beta^{\prime}},\beta^{\prime\prime})\quad(\beta^{\prime},\beta^{\prime\prime}\in{\mathcal{B}}_{r}).$

The product on the right hand side of (3.4) is defined as the linear map from
$H_{{\bf g}}$ to $H_{{\bf g}^{\beta^{\prime}\beta^{\prime\prime}}},$ acting
from the right, obtained by first applying $\Phi({\bf g},\beta^{\prime})$ and
then $\Phi({\bf g}^{\beta^{\prime}},\beta^{\prime\prime})$.

The cocycle rule especially implies that

${\Phi({\bf g},\beta_{i}^{-1})}=\Phi({\bf g}^{\beta_{i}^{-1}},\beta_{i})^{-1}$

is given by restriction of

$M({\bf g}^{\beta_{i}^{-1}},\beta_{i})^{-1}\in{\rm End}(V^{r})$

to $H_{{\bf g}}.$ The latter endomorphism is denoted by $M({\bf
g},\beta_{i}^{-1})$ for convenience. By the definition of $M({\bf
g}^{\beta_{i}^{-1}},\beta_{i})$ and inversion one derives the following
matrix-expression:

(3.5) $M({\bf g},\beta_{i}^{-1})=\left(\begin{array}[]{llllllllll}1&&&\\\
&&\ddots&&&\\\ &&&\quad 1&&\\\ &&&&\,\,\,(g_{i+1}-1)g_{i}^{-1}&\quad 1&\\\
&&&&\,\,\,\quad\quad g_{i}^{-1}&\quad 0&&\\\ &&&&&&&1&&\\\
&&&&&&&&\quad\ddots&\\\ &&&&&&&&&\quad\quad 1\end{array}\right)\,.$

###### Proposition 3.1

  1. (1)

Let
$\omega=\beta_{i_{1}}^{\varepsilon_{1}}\cdots\beta_{i_{k}}^{\varepsilon_{k}}\in{\mathcal{B}}_{r}$
be a word in the standard generators $\beta_{1},\ldots,\beta_{r-1}$ of
${\mathcal{B}}_{r}$ (with $\varepsilon_{1},\ldots,\varepsilon_{k}\in\\{\pm
1\\}$) and let ${\bf g}\in\mathcal{E}_{r}.$ Then the isomorphism ${\Phi({\bf
g},\omega)}:H_{{\bf
g}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}H_{{\bf
g}^{\omega}}$ is given by the restriction of the following endomorphism
$M({\bf g},\omega)\in{\rm End}(V^{r})$ to $H_{{\bf g}}:$

$M({\bf g},\omega)=M({\bf g},\beta_{i_{1}}^{\varepsilon_{1}})\cdot M({\bf
g}^{\beta_{i_{1}}^{\varepsilon_{1}}},\beta_{i_{2}}^{\varepsilon_{2}})\cdots
M({\bf
g}^{\beta_{i_{1}}^{\varepsilon_{1}}\cdots\beta_{i_{k-1}}^{\varepsilon_{k-1}}},\beta_{i_{k}}^{\varepsilon_{k}}).$

  2. (2)

Let

$E_{{\bf g}}\;:=\;\\{\;(v\cdot(g_{1}-1),\ldots,v\cdot(g_{r}-1))\;\mid\;v\in
V\;\\}\,.$

Then $\Phi({\bf g},\omega)$ induces an isomorphism

$\overline{\Phi}({\bf g},\omega):\,W_{{\bf g}}:=H_{{\bf g}}/E_{{\bf
g}}\;\;\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\;\;W_{{\bf
g}^{\omega}}.$

###### Proof.

The first claim follows from the cocycle rule (3.4). The second claim follows
from the fact that $\Phi({\bf g},\omega)$ maps $E_{\bf g}$ to $E_{{\bf
g}^{\omega}},$ cf. [7], Section 2.2. $\Box$

## 4\. The monodromy of the Radon transform

We assume in the following, that $K=j_{!*}{\mathcal{L}}[2]$ with
${\mathcal{L}}$ an irreducible and non-constant local system of rank $n$ on
$Y={\mathbb{P}}^{2}\smallsetminus C,$ where $C$ is a reduced plane curve of
degree $r.$

Using a suitable coordinate change, we can assume that we are in a situation
as described in Section 2. We will freely use the notation of the previous
sections. Especially, let

$\left((g_{1},\ldots,g_{r}),(\omega_{1},\ldots,\omega_{s})\right)\in\operatorname{GL}(V)^{r}\times{\mathcal{B}}_{r}^{s}$

be the fundamental data of ${\mathcal{L}}$ with respect to a fixed choice of
generators

$\gamma_{1},\ldots,\gamma_{r}\in\pi_{1}(U_{0},(p_{0},\widetilde{p}_{0}))\,\,\textrm{
and
}\,\,\widetilde{\gamma}_{1},\ldots,\widetilde{\gamma}_{s}\in\pi_{1}(S,\widetilde{p}_{0})$

as in the previous sections.

Let $\widetilde{C}\subseteq\widetilde{{\mathbb{P}}}^{2}$ be the union of the
dual curve of $C$ and the lines in $\widetilde{{\mathbb{P}}}^{2}$ which
correspond to pencils of lines in ${\mathbb{P}}^{2},$ passing through the
singularities of $C.$ Moreover, let
$\widetilde{Y}:=\widetilde{{\mathbb{P}}}^{2}\smallsetminus\widetilde{C}.$ For
$\widetilde{x}\in\widetilde{{\mathbb{P}}}^{2}$ let
$j_{\widetilde{x}}:L_{\widetilde{x}}\smallsetminus C\hookrightarrow
L_{\widetilde{x}}$ be the natural inclusion and let
${\mathcal{L}}_{\widetilde{x}}:={\mathcal{L}}|_{L_{\widetilde{x}}\smallsetminus
C}.$

###### Proposition 4.1

For $\widetilde{x}\in\widetilde{Y}$ let $\mathcal{R}(K)_{\widetilde{x}}$ be
the stalk of the Radon transform at $\widetilde{x}$ in the derived sense.
Consider the _parabolic cohomology group_

$H^{1}_{par}({\mathcal{L}}_{\widetilde{x}}):=H^{1}(L_{\widetilde{x}},j_{\widetilde{x}*}({\mathcal{L}}_{\widetilde{x}})).$

Then there is an isomorphism

$\mathcal{R}(K)_{\widetilde{x}}[-2]\simeq
H^{1}_{par}({\mathcal{L}}_{\widetilde{x}}).$

Moreover, the dimension of $H^{1}_{par}({\mathcal{L}}_{\widetilde{x}})$ is

(4.1) $n(r-2)-\sum_{i=1}^{r}\dim(V^{g_{i}}).$

Especially, the restriction of $\mathcal{R}(K)$ to $\widetilde{Y}$ is a local
system whose rank is given by (4.1), placed in cohomological degree $-2.$

###### Proof.

If $\widetilde{x}\in\widetilde{Y},$ then
$\pi_{1}(L_{\widetilde{x}}\smallsetminus C)$ maps surjectively onto
$\pi_{1}(U)$ (see [8]), implying that
${\mathcal{L}}_{\widetilde{x}}={\mathcal{L}}|_{L_{\widetilde{x}}\smallsetminus
C}$ is an irreducible and non-constant local system on
$L_{\widetilde{x}}\smallsetminus C.$ Hence
$H^{i}(L_{\widetilde{x}},j_{\widetilde{x}*}({\mathcal{L}}_{\widetilde{x}}))=0$
for $i\neq 1.$ By proper basechange and the irreducibility assumption, the
stalk $\mathcal{R}(K)_{\widetilde{x}}$ of the Radon transform

$\mathcal{R}(K)=Rp_{2*}p_{1}^{*}(j_{!*}{\mathcal{L}}[2])$

is the cohomology group
$H^{1}(p_{2}^{-1}(\widetilde{x}),p_{1}^{*}(j_{!*}({\mathcal{L}}))|_{p_{2}^{-1}(\widetilde{x})}),$
placed in degree $-2.$ The first projection induces an isomorphism
$p_{1}:p_{2}^{-1}(\widetilde{x})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}L_{\widetilde{x}}$
with

$p_{1}^{*}(j_{\widetilde{x}*}({\mathcal{L}}_{\widetilde{x}}))\simeq\left(p_{1}^{*}(j_{!*}({\mathcal{L}}))\right)|_{p_{2}^{-1}(\widetilde{x})}.$

Hence

$\mathcal{R}(K)_{\widetilde{x}}[-2]\simeq
H^{1}(L_{\widetilde{x}},j_{\widetilde{x}*}({\mathcal{L}}_{\widetilde{x}})).$

It is a consequence of the Ogg-Shafarevich Formula (cf. [7]), that for all
$\widetilde{x}\in\widetilde{Y}$ the dimension of
$H^{1}_{par}({\mathcal{L}}_{\widetilde{x}})$ is as in Formula (4.1), since the
local monodromy is constant along the components of $C$ and since each line
$L_{\widetilde{x}}$ intersects $C$ transversally. This implies that the
restriction of $\mathcal{R}(K)$ to $\widetilde{Y}$ is a local system whose
degree is given by Eq. (4.1), placed in cohomological degree $-2.$ $\Box$

###### Definition 4.2

Let $K$ and ${\mathcal{L}}$ be as above. The _Radon transformation_ of
${\mathcal{L}}$ is the local system

$\mathcal{R}({\mathcal{L}}):=(\mathcal{R}(K)|_{\widetilde{Y}})[-2]$

on $\widetilde{Y}$.

###### Remark 4.3

It follows from Prop. 4.1 that $\mathcal{R}({\mathcal{L}})$ is the zero-local
system if

$n(r-2)-\sum_{i=1}^{r}\dim(V^{g_{i}})=0.$

However, if

$n(r-2)-\sum_{i=1}^{r}\dim(V^{g_{i}})\neq 0,$

then usually $\mathcal{R}({\mathcal{L}})$ is non-constant and hence determines
$\mathcal{R}(K)$ uniquely.

An immediate consequence of Lefschetz’ hyperplane theorem is the following:

###### Lemma 4.4

Let
$S=\widetilde{L}_{p_{0}}\smallsetminus\\{\widetilde{y}_{1},\ldots,\widetilde{y_{s}}\\}\subseteq\widetilde{Y}.$
Then the Radon transform $\mathcal{R}({\mathcal{L}})$ is determined by its
restriction to $S.$ Then $\mathcal{R}({\mathcal{L}})$ is determined up to
isomorphism by the monodromy tuple

$\widetilde{{\bf
g}}=(\widetilde{g}_{1},\ldots,\widetilde{g}_{s})\in\operatorname{GL}(\mathcal{R}({\mathcal{L}})_{\widetilde{p}_{0}})^{s}$

of $\mathcal{R}({\mathcal{L}})$ with respect to
$\widetilde{\gamma}_{1},\ldots,\widetilde{\gamma}_{s}.$ $\Box$

Let $H$ be the incidence relation used in the definition of the Radon
transform, equipped with its projections
$p_{1}:H\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathbb{P}}^{2}$
and
$p_{2}:H\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{{\mathbb{P}}}^{2}.$
Let ${\mathcal{L}}_{0}={\mathcal{L}}_{\widetilde{p}_{0}}$ be the restriction
of $p_{1}^{*}{\mathcal{L}}$ to $U_{0}\subseteq U,$ with $U_{0}\subseteq
U\subseteq X\subseteq H$ as in Section 2. Then $p_{1}^{*}{\mathcal{L}}$ is a
variation of ${\mathcal{L}}_{0}$ over $\widetilde{Y}$ with
$\mathcal{R}({\mathcal{L}})$ the parabolic cohomology of this variation (in
the sense of [7], Def. 2.1).

Recall from [7] that the parabolic cohomology group
$H^{1}_{par}({\mathcal{L}}_{0})=\mathcal{R}({\mathcal{L}})_{\widetilde{p}_{0}}$
is isomorphic to a subspace of the first group cohomology
$H^{1}(\pi_{1}(\mathbb{A}^{1}\smallsetminus
P_{0}),V)\,(P_{0}=X_{0}\smallsetminus U_{0})$ where $V\simeq k^{n}$ is the
stalk of ${\mathcal{L}}_{0}$ at $(p_{0},\widetilde{p}_{0})\in U_{0}$ and $V$
is a $\pi_{1}(\mathbb{A}^{1}\smallsetminus P_{0})$-module via the monodromy
representation $\rho_{{\mathcal{L}}_{0}}$ of ${\mathcal{L}}_{0}.$ The group
cohomology $H^{1}(\pi_{1}(\mathbb{A}^{1}\smallsetminus P_{0}),V)$ is formed,
up to equivalence induced by boundaries, by maps (called cocycles)

$\delta\colon\pi_{1}(\mathbb{A}^{1}\smallsetminus
P_{0})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}V$

satisfying the cocycle rule

(4.2)
$\delta(\gamma\gamma^{\prime})=\delta(\gamma)\rho_{{\mathcal{L}}_{0}}(\gamma^{\prime})+\delta(\gamma^{\prime}).$

###### Lemma 4.5

Let ${\bf g}:=(g_{1},\ldots,g_{r})\in\operatorname{GL}(V)^{r}$ be the
monodromy tuple of ${\mathcal{L}}_{0}.$ Let $W_{{\bf g}}:=H_{{\bf g}}/E_{{\bf
g}}$ be as in Prop. 3.1(ii). Then the evaluation map

$\mathcal{R}({\mathcal{L}})_{\widetilde{p}_{0}}=H^{1}_{par}({\mathcal{L}}_{0})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}W_{{\bf
g}},\quad[\delta]\mapsto(\delta(\gamma_{1}),\ldots,\delta(\gamma_{r}))+E_{{\bf
g}}$

is an isomorphism.

###### Proof.

Follows from [7], Section 1.3. $\Box$

###### Theorem 4.6

Let ${\mathcal{L}}$ be an irreducible and non-constant local system on a plane
curve complement ${\mathbb{P}}^{2}\smallsetminus C$ with fundamental data

$\left({\bf
g}=(g_{1},\ldots,g_{r}),(\omega_{1},\ldots,\omega_{s})\right)\in\operatorname{GL}(V)^{r}\times{\mathcal{B}}_{r}^{s}$

with respect to a fixed choice of generators

$\gamma_{1},\ldots,\gamma_{r}\in\pi_{1}(U_{0},p_{0})\,\,\textrm{ and
}\,\,\widetilde{\gamma}_{1},\ldots,\widetilde{\gamma}_{s}\in\pi_{1}(S,\widetilde{p}_{0}).$

Then the monodromy tuple

$\widetilde{{\bf
g}}=(\widetilde{g}_{1},\ldots,\widetilde{g}_{s})\in\operatorname{GL}(W_{{\bf
g}})^{s}$

of the Radon transform $\mathcal{R}({\mathcal{L}})$ with respect to
$\widetilde{\gamma}_{1},\ldots,\widetilde{\gamma}_{s}$ is as follows: For
$i\in\\{1,\ldots,s\\},$ write

$\omega_{i}=\beta_{i_{1}}^{\varepsilon_{1}}\cdots\beta_{i_{k}}^{\varepsilon_{k}}\in{\mathcal{B}}_{r}$

as a word in the standard generators $\beta_{1},\ldots,\beta_{r-1}$ of
${\mathcal{B}}_{r}$ with $\varepsilon_{1},\ldots,\varepsilon_{k}\in\\{\pm
1\\}.$ Let

$M({\bf g},\omega_{i})=M({\bf g},\beta_{i_{1}}^{\varepsilon_{1}})\cdot M({\bf
g}^{\beta_{i_{1}}^{\varepsilon_{1}}},\beta_{i_{2}}^{\varepsilon_{2}})\cdots
M({\bf
g}^{\beta_{i_{1}}^{\varepsilon_{1}}\cdots\beta_{i_{k-1}}^{\varepsilon_{k-1}}},\beta_{i_{k}}^{\varepsilon_{k}})\in{\rm
End}(V^{r})$

be as in Prop. 3.1. Then $\widetilde{g}_{i}$ is induced by the operation of
$M({\bf g},\omega_{i})\in{\rm End}(V^{r})$ on the subquotient $W_{\bf
g}=H_{\bf g}/E_{\bf g}\leqslant V^{r}/E_{\bf g}.$

###### Proof.

Consider the second projection
$p_{2}:H\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\widetilde{{\mathbb{P}}}^{2},$
where $H$ is the incidence relation, and let
$S=\widetilde{L}_{p_{0}}\smallsetminus\\{\widetilde{y}_{1},\ldots,\widetilde{y}_{s}\\}\subseteq\widetilde{{\mathbb{P}}}^{2}.$
Let $X=p_{2}^{-1}(S)$ and let $D=p_{1}^{-1}(C)\cap X$ be as in the previous
section. Then the pair $(X,D)$ is an _$r$ -configuration_ in the sense of [7],
Section 2.1 (i.e., for $s\in S,$ the fibres $X_{s}$ are Riemann surfaces of
genus $0$ and $\\#D\cap X_{s}=r$). Recall that $U$ was defined to be
$X\smallsetminus D.$

There exists an isomorphism
$\lambda:X\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathbb{P}}^{1}_{S}$
such that $\lambda(D)$ is disjoint from $\infty\times S.$ Therefore we can
embed $S$ into $U\subseteq X$ via $\lambda^{-1}(\infty\times S).$ This induces
a section
$s:\pi_{1}(S)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\pi_{1}(U)$
to the homomorphism
$\pi_{1}(U)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\pi_{1}(S)$
from the short exact sequence in (2.2). Write
$\chi:\pi_{1}(S)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{GL}(V)$
for the composition $\rho_{p_{1}^{*}({\mathcal{L}})}\circ s.$

For $h\in\operatorname{GL}(V)$ and ${\bf g}^{h}\in\operatorname{GL}(V)^{r}$
the tuple obtained by simultaneously conjugating the components of ${\bf g}$
by $h,$ one gets an isomorphism

$\overline{\Psi}(g,h):W_{{\bf
g}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}W_{{\bf
g}^{h}}$

from the diagonal operation of $h$ on $V^{r}.$ Since $p_{0}$ is a smooth point
of ${\mathcal{L}}$ and since

$p_{1}^{-1}(p_{0})=\lambda^{-1}(\infty\times S),$

the homomorphism $\chi$ is trivial, implying that
$\overline{\Psi}(g,\chi(\widetilde{\gamma}_{i}))$ is trivial.

As already mentioned, the divisor $\lambda(D)$ is disjoint from $\infty\times
S$ (hence $\lambda$ is an _affine frame_ as defined in loc.cit., Section 2.4).
Hence, by [7], Thm. 2.5, under the identification of $W_{\bf
g}=\mathcal{R}({\mathcal{L}})_{\widetilde{p}_{0}},$ the monodromy
representation
$\rho_{\mathcal{R}({\mathcal{L}})}:\pi_{1}(S)\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\operatorname{GL}(W_{\bf
g})$ is as follows:

$\rho_{\mathcal{R}({\mathcal{L}})}(\widetilde{\gamma}_{i})=\overline{\Phi}({\bf
g},\omega_{i})\cdot\overline{\Psi}(g,\chi(\widetilde{\gamma}_{i}))=\overline{\Phi}({\bf
g},\omega_{i})\in\operatorname{GL}(W_{\bf g}),$

where $\overline{\Phi}({\bf g},\omega_{i})$ is as in Prop. 3.1. $\Box$

###### Remark 4.7

Let $K$ be a simple and non-constant perverse sheaf whose coefficient field
$k$ has characteristic $0.$ It is a consequence of Radon inversion that the
Radon transform $\mathcal{R}(K)$ has a unique simple non-constant subfactor
$S$ (cf. the discussion in Section 1). Let us additionally assume that the
following generic condition holds: the Radon transform
$\mathcal{R}({\mathcal{L}})$ is a non-constant local system on
$\widetilde{Y}.$ Then $j_{!*}(\mathcal{R}({\mathcal{L}}))[-2]$ contains this
unique subfactor $S.$ If moreover $K$ satisfies the condition $P$ then Prop.
1.4 implies that

$j_{!*}(\mathcal{R}({\mathcal{L}}))[-2]=\mathcal{R}(K).$

The local system $\mathcal{R}({\mathcal{L}})$ in turn is determined up to
isomorphism by the monodromy tuple (with respect to
$\widetilde{\gamma}_{1},\ldots,\widetilde{\gamma}_{s}$), which is
algorithmically determined by Thm. 4.6.

## 5\. Examples: an elliptic fibration and the classical Zariski pair

Let $k$ be a field of characteristic $\neq 2.$ Let

$\widetilde{x}_{1}:=[2:-4:3],\,\widetilde{x}_{2}:=[1,-1,1],\,\widetilde{x}_{3}:=[1:0:0],\,\widetilde{x}_{4}:=[1:0:-1]\,\in\widetilde{{\mathbb{P}}}^{2}$

and let ${\mathcal{L}}$ be the rank-one local system on

${\mathbb{P}}^{2}\smallsetminus
C\quad(C:=\mathop{\textstyle\bigcup}\displaylimits_{i=1}^{4}L_{\widetilde{x}_{i}})$

having quadratic local monodromy $-1\in k^{*}$ along
$L_{\widetilde{x}_{i}}\,(i=1,2,3,4).$ Hence the monodromy tuple of
${\mathcal{L}}$ (defined as in Section 2) is ${\bf
g}=(-1,-1,-1,-1)\in(k^{*})^{4}.$

We remark that $C$ represents a generic quadruple of projective lines in
${\mathbb{P}}^{2}.$

The singularities of $C$ are

$x_{1}:=[-\frac{1}{2}:\frac{1}{2}:1],\,x_{2}:=[0:\frac{3}{4}:1],\,x_{3}:=[0:1:1],$
$x_{4}:=[1:\frac{5}{4}:1],\,x_{5}:=[1:2:1],\,x_{6}:=[0:1:0].$

Hence the Radon transform $\mathcal{R}({\mathcal{L}})$ of ${\mathcal{L}}$ is a
$k$-local system on $\widetilde{{\mathbb{P}}}^{2}\smallsetminus\widetilde{C},$
where $\widetilde{C}$ is the union of
$\widetilde{L}_{x_{1}},\ldots,\widetilde{L}_{x_{6}}.$

###### Proposition 5.1

The Radon transform $\mathcal{R}({\mathcal{L}})$ is an irreducible $k$-local
system of rank $2$ on
$\widetilde{{\mathbb{P}}}^{2}\smallsetminus\widetilde{C},$ having monodromy
tuple

$\left(\left(\begin{array}[]{rr}1&-2\\\
0&1\end{array}\right),\left(\begin{array}[]{rr}1&0\\\
-2&1\end{array}\right),\left(\begin{array}[]{rr}-1&-2\\\
2&3\end{array}\right),\left(\begin{array}[]{rr}-1&-2\\\
2&3\end{array}\right),\left(\begin{array}[]{rr}-3&-8\\\
2&5\end{array}\right),\left(\begin{array}[]{rr}1&-2\\\
0&1\end{array}\right)\right).$

###### Proof.

By the Ogg-Shafarevich Formula (4.1), the rank of $\mathcal{R}({\mathcal{L}})$
is $2$ since $n=1,$ $r=4,$ and since $-1\in k^{*}$ has no nontrivial
invariants acting on the one-dimensional vector space $k.$ Using the theory of
wiring diagrams (encoding the intersection pattern of the underlying real line
arrangement, cf. [1], [2]) one derives the braid monodromy of $U$ in a
standard way as follows:

$\omega_{1}=\beta_{1}^{2},\,\omega_{2}={}(\beta_{2}^{2})^{\beta_{1}},\,\omega_{3}={}(\beta_{1}^{2})^{\beta_{2}\beta_{1}},\,\omega_{4}={}(\beta_{3}^{2})^{\beta_{1}\beta_{2}\beta_{1}},\,\omega_{5}={}(\beta_{2}^{2})^{\beta_{3}\beta_{1}\beta_{2}\beta_{1}}$

with $\omega_{6}$ being the inverse of the product
$\omega_{1}\cdots\omega_{5}.$

By a computation using Thm. 4.6, the monodromy tuple of
$\mathcal{R}({\mathcal{L}})$ is as stated. Note that this computation is
considerably simplified by the obvious observation that ${\bf
g}^{\omega_{i}}={\bf g}$ for $i=1,\ldots,6.$ Hence Prop. 3.1 in combination
with Thm. 4.6 implies that for the above expression of $\omega_{i}$ as a word
in the standard generators

$\omega_{i}=\beta_{i_{1}}^{\varepsilon_{1}}\cdots\beta_{i_{k}}^{\varepsilon_{k}}\quad(\varepsilon_{j}\in\\{\pm
1\\}),$

the matrix inducing
$\rho_{\mathcal{R}({\mathcal{L}})}(\widetilde{\gamma}_{i})$ is the
corresponding matrix product

$M({\bf g},\omega_{i})=M({\bf g},\beta_{i_{1}})^{\varepsilon_{1}}\cdots M({\bf
g},\beta_{i_{k}})^{\varepsilon_{k}}.$

The latter product can easily be evaluated using

$M({\bf g},\beta_{1})=\left(\begin{array}[]{lrlll}0&\quad-1&\\\
1&\quad\,2&&\\\ &&&1&\\\ &&&&1\end{array}\right),\,\,M({\bf
g},\beta_{2})=\left(\begin{array}[]{llrll}1&&&&\\\ &0&\quad-1&\\\
&1&\quad\,2&&\\\ &&&&1\\\ \end{array}\right),\,\,M({\bf
g},\beta_{3})=\left(\begin{array}[]{lllrl}1&&&&\\\ &1&&&\\\ &&0&\quad-1\\\
&&1&\quad\,2\\\ \end{array}\right)\,$

and passing to the subquotient $H_{{\bf g}}/E_{{\bf g}}.$ $\Box$

###### Remark 5.2

  1. (1)

Again, by standard arguments, the braid monodromy of $\widetilde{C}$ is
generated by the following braids:

(5.1)
$(\beta_{4}\beta_{5}\beta_{4})^{2},\quad\beta_{3}^{2},\quad\beta_{3}^{-1}(\beta_{1}\beta_{2}\beta_{1})^{2}\beta_{3}.$

It can be checked that the local system $\mathcal{R}({\mathcal{L}})$ satisfies
indeed the relations coming from Eq. (5.1).

  2. (2)

Using the braid monodromy in Eq. (5.1) the dual Radon transform applied to
$\mathcal{R}({\mathcal{L}})$ has rank $2$ and decomposes into a trivial local
system and the restriction of ${\mathcal{L}}$ to a slightly smaller open
subset. Hence, although the intermediate extension
$K=j_{!*}({\mathcal{L}})[2]$ satisfies the property $P,$ it follows from Prop.
1.4(2) that $\mathcal{R}(K)=\widetilde{j}_{!*}\mathcal{R}({\mathcal{L}})[2]$
does not have the property $P.$

Another example comes from the theory of Zariski pairs (see [16]): Recall the
classical Zariski pair, consisting of a six-cuspidal plane sextic $C$ having
its cusps on a conic and a six-cuspidal plane sextic $C^{\prime}$ whose cusps
do not lie on a conic.

Zariski showed that the fundamental group of the complement of $C$ is
isomorphic to the free group on two generators of order $2,$ resp. $3,$
whereas the fundamental group of the complement of $C^{\prime}$ is isomorphic
to a cyclic group of order $6.$

The braid monodromy of the Zariski pair is determined in [15], Example 3, (cf.
[3], Prop. 7.3) to be as follows: for $C$ one has twice the tuple:

$\left(\beta_{1}^{3},\beta_{1}^{\beta_{2}^{-1}\beta_{1}},\beta_{3}^{3},\beta_{3}^{\beta_{4}^{-1}\beta_{3}},\beta_{5}^{3},\beta_{2}^{\beta_{3}\beta_{3}\beta_{1}},\beta_{4}^{\beta_{5}\beta_{5}\beta_{3}},\beta_{3}^{\beta_{2}^{-1}},\beta_{5}^{\beta_{4}^{-1}}\right).$

For $C^{\prime}$ one has:

$\left(\beta_{3}^{\beta_{2}^{-1}\beta_{1}\beta_{2}^{-1}},\beta_{4},\beta_{5},\beta_{2}^{3},(\beta_{2}^{\beta_{1}^{-1}})^{3},\beta_{2}^{\beta_{3}^{-1}},\beta_{2}^{\beta_{1}^{-1}\beta_{3}^{-1}\beta_{4}^{-1}},\beta_{1}^{3},\beta_{1}^{\beta_{2}^{-1}\beta_{3}\beta_{4}^{-1}},\beta_{5}^{\beta_{4}^{-1}\beta_{4}^{-1}}\right.,$
$\left.\beta_{3}^{\beta_{2}^{-1}\beta_{4}^{-1}\beta_{4}^{-1}},\beta_{4}^{3},\beta_{1}^{3},\beta_{3}^{\beta_{2}^{-1}\beta_{1}\beta_{2}^{-1}},\beta_{3}^{\beta_{2}^{-1}\beta_{1}\beta_{2}^{-1}},\beta_{1}^{3},\beta_{1}^{\beta_{2}^{-1}\beta_{2}^{-1}},\beta_{2}\right).$

Let now ${\mathcal{L}},$ resp. ${\mathcal{L}}^{\prime},$ be the rank-one
$\mathbb{C}$-local system on ${\mathbb{P}}^{2}\smallsetminus C,$ resp.
${\mathbb{P}}^{2}\smallsetminus C^{\prime},$ defined by the obvious
homomorphism of fundamental groups onto the group of 6-th roots of unity
inside $\mathbb{C}^{\times}=\operatorname{GL}_{1}(\mathbb{C}),$ so that in
both cases, the monodromy tuple is a $6$-tuple $(\zeta_{6},\ldots,\zeta_{6})$
whose entries consist of the same primitive sixth root of unity
$\zeta_{6}\in\mathbb{C}^{\times}.$

It is astonishing that the Radon transforms $\mathcal{R}({\mathcal{L}}),$
resp. $\mathcal{R}({\mathcal{L}}^{\prime}),$ descend to local systems on the
complement of the dual curves
$\widetilde{{\mathbb{P}}}^{2}\smallsetminus{C}^{*},$ resp.
$\widetilde{{\mathbb{P}}}^{2}\smallsetminus{C^{\prime}}^{*}$ (with ${C}^{*}$
being the dual curve of $C$ and ${C^{\prime}}^{*}$ being the dual curve of
$C^{\prime}$) since, a priori, they are defined on the complement of the
larger curves $\widetilde{C},$ resp. $\widetilde{C}^{\prime},$ defined in
Section 4.

An application of the explicit braid monodromy for the curves $C,$ resp.
$C^{\prime},$ in combination with Thm. 4.6 leads to the following result:

###### Theorem 5.3

The monodromy group of $\mathcal{R}({\mathcal{L}})$ is a finite solvable group
$G\leqslant\operatorname{GL}_{4}(\mathbb{C})$ of order $648$ acting
irreducibly on a direct summand of dimension $3.$ The monodromy group
$G^{\prime}\leqslant\operatorname{GL}_{4}(\mathbb{C})$ of
$\mathcal{R}({\mathcal{L}}^{\prime})$ is a direct product of the cyclic scalar
group of order $3$ with the finite symplectic group
$\operatorname{Sp}_{4}(3)\leqslant\operatorname{GL}_{4}(\mathbb{C}).$

###### Remark 5.4

  1. (1)

It is interesting, that the local system ${\mathcal{L}}$ does not have the
property $P$ since its Radon transform splits into a simple subfactor of rank
$3$ and a constant subfactor of rank $1.$

  2. (2)

In the Appendix to this article, we have listed the commented source code
which computes the monodromy tuple of the Radon transform, written in Magma-
code [5]. It can easily be pasted (from the arxiv-latex-source) to a separate
file and then read into Magma.

## References

  * [1] W.A. Arvola, _The fundamental group of the complement of an arrangement of complex hyperplanes_ , Topology 31 (1992), 757–765.
  * [2] E.A. Bartolo, _Topology of arrangements and position of singularities_ , Annales de la Faculté des Sciences de Toulouse 23 (2014), 223–265.
  * [3] E.A. Bartolo, J.I. Cogolludo-Agustín and J. Ortigas-Galindo, _Kummer covers and braid monodromy_ , J. Inst. Math. Jussieu 13 (2014), 633–670.
  * [4] A. A. Beilinson, J. Bernstein, and P. Deligne, _Faisceaux pervers_ , Analysis and topology on singular spaces, I (Luminy, 1981), Soc. Math. France, Astérisque 100 (1982), 5–171.
  * [5] W. Bosma, J. Cannon, and C. Playoust, _The Magma algebra system. I. The user language_ , Journal of Symbolic Computation 24 (1997), 35–265.
  * [6] J.-L. Brylinsky, _Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques_ , Géométrie et analyse microlocales (Editors: J.-L. Brylinski and T. Monteiro Fernandes), Soc. Math. France, Astérisque 474 (1986), 3–134.
  * [7] M. Dettweiler and S. Wewers, _Variation of local systems and parabolic cohomology_ , Israel J. Math. 156 (2006), 157–185.
  * [8] A. Dimca, _Singularities and Topology of Hypersurfaces_ , Springer, Universitext (1992).
  * [9] A. Dimca, _Sheaves in topology_ , Springer, Universitext (2004).
  * [10] M. Goresky and R. MacPherson, _Stratified Morse Theory_ , Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete 14 (1988).
  * [11] R. Kiehl and R. Weissauer, _Weil conjectures, Perverse Sheaves and l’adic Fourier Transform_ , Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete 42 (2001).
  * [12] G. Laumon, _Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil_ , Publ. Math. IHES 65 (1987), 131–210.
  * [13] D. Patel and K.V. Shuddhodan, _Brylinski-Radon transformation in characteristic $p>0$_, arXiv:2307.04156 (2023).
  * [14] J. Radon, _Über die Bestimmung von Funktionen längs gewisser Mannigfaltigkeiten_ , In: Berichte über die Verhandlungen der Königlich-Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Mathematisch-Physische Klasse 69 (1917), 262–277.
  * [15] L. Rudolph, Some knot theory of complex plane curves, L’Enseignement Mathémathique 29 (1983), 185–208.
  * [16] O. Zariski, The topological discriminant group of a Riemann surface of genus $p$, Amer. J. Math. 59 (1937), 335–358.

## 6\. Appendix - the Radon program

The following lists the MAGMA code which is used in the proof of Thm. 5.3.

⬇

H_g1:=function(g,K)

//Description: Computes the vector space

//$H_{g,1}:=\\{(v_1,\ldots, v_r)\mid v_i\in\Im(g_i-I)\\}$.

local r,n,W,I,Kn,i;

r:=#g;

n:=Nrows(g[1]);

W:=VectorSpace(K,0);

I:=IdentityMatrix(K,n);

Kn:=VectorSpace(K,n);

for i in [1..r] do

W:=DirectSum(W,Image(g[i]-I));

end for;

return W;

end function;

H_g2:=function(g,K)

//Description: Computes the vector space

// $H_{g,2}:=\\{(v_1,\ldots, v_r)\mid

// v_1\cdot g_2\cdot\ldots\cdot g_r+v_2\cdot g_3\cdots g_r+\ldots+v_r=0\\}$.

local r,n,gg,i,A;

r:=#g;

n:=Nrows(g[1]);

gg:=[];

gg[1]:=g[r];

A:=ZeroMatrix(K,n*r,n*r);

for i in [2..r] do

gg[i]:=g[r-i+1]*gg[i-1];

end for;

for i in [1..(r-1)] do

A:=InsertBlock(A,gg[r-i],(i-1)*n+1,1);

end for;

A:=InsertBlock(A,IdentityMatrix(K,n),(r-1)*n+1,1);

return Kernel(A);

end function;

H_g:=function(g,K)

//Description: Computes the vector space $H_g = H_{g,1}\cap H_{g,2}$

return (H_g1(g,K) meet H_g2(g,K));

end function;

E_g:=function(g,K)

//Description: Computes the vector space

// $E_g:=\\{(v\cdot(g_1-I),\ldots,v\cdot (g_r-I))\mid v\in V\\}$.

local h,r,n,i,I,Im,Mat;

r:=#g;

n:=Nrows(g[1]);

I:=IdentityMatrix(K,n);

Mat:=ZeroMatrix(K,n,n*r);

h:=g;

for i in [1..r] do

h[i]:=g[i]-I;

end for;

Mat:=Hom(VectorSpace(K,n),VectorSpace(K,n*r))!HorizontalJoin(h);

Im:=Image(Mat);

return Im;

end function;

Beta_i:=function(g,i)

//Description: Computes the formula for

//the effect of the standard generators $\beta_i$

// of the Artin braid group $\mathcal{A}_r$ on $\mathcal{E}_r$

//Input: List $g=[g_1,…,g_r]$, where $g_k \in \GL_n(K)$

//s.t. $\prod_k g_k =1$,

// Integer i, s.t. $|i|\le\\#g-1,\,i \neq 0$, representing $\beta_i$

local r,k,gg;

r:=#g;

gg:=g;

if i ge 1 then

gg[i]:=g[i+1];

gg[i+1]:=(g[i+1])^-1*g[i]*g[i+1];

end if;

if i le -1 then

gg[-i]:=g[-i]*g[-i+1]*(g[-i])^-1;

gg[-i+1]:=g[-i];

end if;

return gg;

end function;

phi:=function(g,i,K)

//Description: Computes the map

//$\Phi(g,\beta_i): H_g \rightarrow H_{g^{\beta_i}}$,

// where $\beta_i$ is a standard generator of $\A_r$

// (see Section 3).

local Mat,n,r,gg;

r:=#g;

n:=Nrows(g[1]);

if i ge 1 then

Mat:=IdentityMatrix(K,n*r);

Mat:=InsertBlock(Mat,ZeroMatrix(K,n,n),n*(i-1)+1,n*(i-1)+1);

Mat:=InsertBlock(Mat,IdentityMatrix(K,n),n*i+1,n*(i-1)+1);

if i le r-1 then

Mat:=InsertBlock(Mat,g[i+1],n*(i-1)+1,n*i+1);

Mat:=InsertBlock(Mat,IdentityMatrix(K,n)-(g[i+1])^-1*g[i]*g[i+1],

n*i+1,n*i+1);

end if;

end if;

if i le (-1) then

gg:=Beta_i(g,i);

Mat:=IdentityMatrix(K,n*r);

Mat:=InsertBlock(Mat,ZeroMatrix(K,n,n),n*(-i-1)+1,n*(-i-1)+1);

Mat:=InsertBlock(Mat,IdentityMatrix(K,n),n*-i+1,n*(-i-1)+1);

if -i le r-1 then

Mat:=InsertBlock(Mat,gg[-i+1],n*(-i-1)+1,n*(-i)+1);

Mat:=InsertBlock(Mat,IdentityMatrix(K,n)-(gg[-i+1])^-1*gg[-i]*

gg[-i+1],n*(-i)+1,n*(-i)+1);

end if;

Mat:=Mat^-1;

end if;

return Mat;

end function;

ListenProdukt:=function(L)

//Description: Help function to compute the product over a list

//Input: List L with entries over a ring

//Output: Product of the entries of L

local r,i,LL;

r:=#L;

LL:=L[1];

for i in [2..r] do

LL:=LL*L[i];

end for;

return LL;

end function;

Phi:=function(g,I,K)

//Description: Computes the function $\Phi(g,\beta)

//Input: List $g=[g_1,…,g_r]$, where $g_k \in \GL_n(K)$

// s.t. $\prod_k g_k =1$,

// List $I$ of Integers representing the braid $\beta$

// (e.g. [-1,2,1] represents

// the braid ${\beta_1}^{-1}\cdot\beta_2\cdot\beta_1$

// Field $K$

//Output: matrix defining the map $\Phi(g,\beta)$

local k,L,gg,j;

k:=#I;

L:=[];

gg:=g;

L[1]:=phi(g,I[1],K);

for j in [2..k] do

gg:=Beta_i(gg,I[j-1]);

L[j]:=phi(gg,I[j],K);

end for;

return ListenProdukt(L);

end function;

Trafodat:=function(g,K)

local B1,B2,B3,M,n,r;

r:=#g;

n:=Nrows(g[1]);

B1:=Basis(E_g(g,K));

B2:=ExtendBasis(B1,H_g(g,K));

B3:=ExtendBasis(B2,VectorSpace(K,n*r));

M:=Matrix(K,n*r,n*r,B3);

return [*Dimension(E_g(g,K)),Dimension(H_g(g,K)),M*];

end function;

Phibar:=function(g,I,K)

//Description: Computes the isomorphism

// $\bar{\Phi}(g,\beta):

//H_g/E_g \rightarrow H_{g^\beta}/E_{g^\beta}$,

// induced by the map $\Phi(g,\beta)$

//Input: List $g=[g_1,…,g_r]$, where

// $g_k \in \GL_n(K)$ s.t. $\prod_k g_k =1$,

// List $I$ of Integers representing the braid $\beta$

// Field $K$

//Output: matrix, representing the

// isomorphism $\bar{\Phi}(g,\beta)$

local n,r,P,T,M;

r:=#g;

n:=Nrows(g[1]);

P:=Phi(g,I,K);

T:=Trafodat(g,K);

M:=T[3]*P*(T[3])^-1;

return ExtractBlock(M,T[1]+1,T[1]+1,T[2]-T[1],T[2]-T[1]);

end function;

Radon:=function(g,W,K)

//Computes the monodromy tuple of the Radon transform

// Input: $g=[g_1,…,g_r]$, where

//$g_k \in \GL_n(K)$ s.t. $\prod_k g_k =1$

//$W=[W[1],…,W[s]] \in \cB_r$

//braid monodromy of the singular locus $C$

// of the local system belonging to $g$.

local n,r,s,TT,i;

r:=#g;

s:=#W;

n:=Nrows(g[1]);

TT:=[];

for i in [1..s] do

TT[i]:=Phibar(g,W[i],K);

end for;

return TT;

end function;