Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
on
the
axio
matiz
atio
nof
double
pse
udolin
earrangements
Mich
elP
occhiola,
Lau
sanne,
29Sep
tember
2010
Universite
Pierre
etM
arieC
urie,
Paris
1
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
Fig
ure
1:
Scu
lptu
red’A
ngel
DU
ART
E,Lausa
nne,
Suisse.
2
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
summary
3
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
pse
udolin
es
and
double
pse
udolin
es
P≈
D2/{x
∼−
x|x∈
∂D
2}
4
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
Hil
bert,D
.[1
899]G
rundlagen
derG
eometrie.
«It
was
Hilb
ert’saim
togive
asim
ple
axiomatic
characterization
ofreal(E
uclid
ean)geom
etries.H
e
expressed
the
necessary
continuity
assum
ption
sin
terms
ofprop
ertiesof
anord
er.In
deed
,th
ereal
pro
jectiveplan
eis
the
only
desargu
esianord
eredpro
jectiveplan
ew
here
everym
onoton
esequ
ence
ofpoints
has
alim
it,see
the
elegantexp
ositionof
Coxeter
[61].»
Hilb
ert1899,
Kolm
ogoroff1932,
Koth
e1939,
Skorn
jakov1954,
Salzm
ann
1955,Freu
denth
al1957,
and
others
[1]H
elmut
Salzm
ann,
Dieter
Betten
,T
heo
Gru
ndhofer,
Herm
ann
Hah
l,R
ainer
Low
en,
and
Marku
sStrop
pel.
Com
pact
pro
jectiveplan
es.N
um
ber
21in
De
Gru
yterexp
ositions
inm
ath-
ematics.
Walter
de
Gru
yter,1995.
5
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
real
tw
o-d
imensio
nal
proje
ctiv
egeometrie
s
u∗
ℓ
ℓ
u
point
space
line
space
DF
1(H
ilbert
etal.).
Areal
two-dim
ension
alprojective
geometry
isa
topologicalpoin
t-line
projectivein
cidence
geometry
whose
pointspace
isa
projectiveplan
ean
dwhose
line
spaceis
asu
bspaceof
thespace
ofpseu
dolines
ofthe
pointspace.
TH
1(H
ilbert
et
al.).
The
line
spaceof
areal
two-dim
ension
alprojective
geometry
isa
projectiveplan
ean
dthe
pencil
oflin
esthrou
gha
pointis
apseu
doline
ofthe
line
space
(P,L
)→
(L,P
∗)→
(P∗,L
∗)≈
(P,L
)
6
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
arrangements
of
pse
udolin
es
Levi
1926,R
ingel
1956,G
runb
aum
1972,etc.
DF
2.Let
Pbe
aprojective
plane.
An
arrangem
entof
pseudolin
esin
Pis
afinite
family
of
pseudolin
esin
Pwith
theproperty
thatan
ytw
oin
tersectexactly
once.
TH
2(G
oodm
ann,Polla
ck,W
enger
&R
am
fire
scu94).
Every
arrangem
entof
pseu-
dolines
canbe
extended
toa
(realtw
o-dimen
sional)
projectivegeom
etry.
7
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
arrangements
of
double
pse
udolin
es
γ
M(γ
)
DF
3.Let
Pbe
aprojective
plane.
An
arrangem
entof
double
pseudolin
esin
Pis
afinite
family
ofdou
blepseu
dolines
inP
,with
theproperty
thatan
ytw
oin
tersect(tran
sversally)
inexactly
four
points
and
indu
cea
cellstru
cture
onP
,that
is,the
connected
compon
ents
of
thecom
plemen
tof
theunion
ofthe
double
pseudolin
esare
two-cells.
8
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
iso.
classe
ssim
ple
arrang.
three
double
pse
udolin
es
04
24
07
618
437
615
243
2
22
433
232
225
225∗
136
12
64
24
9
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
connectedness
under
mutatio
ns
movin
gcu
rve
TH
3(H
abert
&P.06).
The
spaceof
arrangem
ents
ofn
double
pseudolin
esis
connected
under
mutation
s.
TH
4(F
erte
,P
ilaud
&P.
08).
The
one-exten
sionspace
ofan
arrangem
ent
ofdou
ble
pseudolin
esis
connected
under
mutation
s.
10
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
04
07
18
181
37
15
151
43
431
22
221
33
331
32
321
322
25
251
25∗
25∗1
25∗2
36
64
11
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
04
07
18
181
37
15
151
43
431
251
33
331
25
25∗
25∗1
25∗2
32
321
322
22
221
36
64
12
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
numbers
of
arrangements
n2
34
5
aSn
113
6570181
403533
an
146
153528
nc
bSn
116
11502238
834187
bn
159
245351
nc
J.Ferte,
V.P
ilaud
and
P.2010
–tw
odou
ble
corew
orkstations
at2G
Hz,
three
weeks
-
13
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
sub-a
rrangements
of
size
three
replacem
en
1
11
1
2
22
2
3
33
3
4
44
4
TH
5(H
abert
&P.06).
The
isomorphism
classof
anin
dexedarran
gemen
tof
oriented
double
pseudolin
esdepen
dson
lyon
itschirotope,
i.e.,on
them
apthat
assigns
toecah
triple
ofdistin
ctin
dicesthe
isomorphism
classesof
thesu
barrangem
entin
dexedby
thistriple.
14
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
examples
of
double
pse
udolin
es
U
U∗
u∗
ℓ
ℓ
u
DF
4(H
abert
&P.09).
Acon
vexbody
ofa
projectivegeom
etryis
aclosed
subset
ofpoin
ts
with
non
empty
interior
whose
intersection
with
any
line
isan
interval
ofthat
line.
TH
6(H
abert
&P.09).
Let
Ube
acon
vexbody
ofa
projectivegeom
etry(P
,L).
Then
the
boundary
ofU
isa
double
pseudolin
ein
Pan
dthe
dual
ofU
isa
double
pseudolin
ein
L.
15
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
examples
of
arrangements
of
double
pse
udolin
es
U
V
U∗
V∗
u∗
v∗
u
v
TH
7(H
abert
&P.06-0
9).
The
dual
family
ofa
finite
family
ofpairw
isedisjoin
tcon
vex
bodiesof
a(real
two
dimen
sional)
projectivegeom
etryis
anarran
gemen
tof
double
pseudo-
lines.
Con
versely,an
yarran
gemen
tof
double
pseudolin
esis
isomorphic
tothe
dual
family
ofa
finite
family
ofdisjoin
tcon
vexbodies
ofa
projectivegeom
etry.
16
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
cocycles
of
afa
mily
of
bodie
s
thrcd
ℓ
12
3
4
{2,4,13•
3}∼
{2,4,13•
3}
DF
5.Let
∆be
afinite
indexed
family
ofpairw
isedisjoin
torien
tedcon
vexbodies
ofa
projectivegeom
etry.The
cocyclesof
∆are
theisom
orphismclasses
ofthe
arrangem
ents
∆∪{ℓ}
asℓ
ranges
overthe
spaceof
lines
ofthe
projecctivegeom
etry.
17
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
example
AB
CD
A
B
C
D
12
31
1
11
2
2
2
2
3
3
3
3
{{1•
2•3•},{
1•3•
2•},{1•
2•3•},{
1•3•
2•}}
18
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
dualit
y
AB
CD
A
A
B
B
CC
D
D1
2
31
2
3
11
11
2
2
2
2
3
3
3
3
TH
8(H
abert
&P.09).
Two
finite
indexed
families
ofpairw
isedisjoin
torien
tedcon
vex
bodieshave
isomorphic
dual
arrangem
ents
ifan
don
lyif
theyhave
thesam
eset
ofcocycles
iffthey
havethe
same
setof
extremal
cocyclesiff
theyhave
thesam
echirotope.
19
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
chir
otopes
and
vis.
graph
algorit
hms
TH
9(H
abert
&P.07,A
ngelie
r&
P.03,P.&
Vegte
r96).
The
kedges
ofthe
visibility
graphof
aplan
arfam
ilyof
npairw
isedisjoin
tcon
vexbodies
presented
byits
chirotopeis
compu
tablein
time
O(k
+n
logn)
and
linear
workin
gspace.
20
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
axio
matiz
atio
ntheorem
Let
Ibe
afin
itein
dexin
gset.
DF
6.A
k-chirotope
(ofdou
blepseu
doline
arrangem
ents)
isa
map
χdefi
ned
onthe
setof
triplesof
Isu
chthat
foran
ysu
bsetJ
ofI
ofsize
atm
ostk
therestriction
ofχ
tothe
setof
triplesof
Jis
thechirotope
ofa
double
pseudolin
earran
gemen
tin
dexedby
J.W
eden
oteby
Ck
theset
ofk-chirotopes.
TH
10
(Habert
&P.06).
C3
)C
4)
C5
=C
6=C
7=···
.
21
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
axio
matiz
atio
ntheorem
Let
Ibe
afin
itein
dexin
gset.
DF
6.A
k-chirotope
(ofdou
blepseu
doline
arrangem
ents)
isa
map
χdefi
ned
onthe
setof
triplesof
Isu
chthat
foran
ysu
bsetJ
ofI
ofsize
atm
ostk
therestriction
ofχ
tothe
setof
triplesof
Jis
thechirotope
ofa
double
pseudolin
earran
gemen
tin
dexedby
J.W
eden
oteby
Ck
theset
ofk-chirotopes.
TH
10
(Habert
&P.06).
C3
)C
4)
C5
=C
6=C
7=···
.
n0
12
34
5
aSn
11
113
6570
180403
533
ρSn
11
1214
2415
112nc
2nn
!aSn
6242
822580
692749
566720
bSn
11
116
11502238
834187
τSn
11
1118
541820
nc
2n!b
Sn192
552096
57320
204880
22
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
a4-c
hir
otope
2
22
2
3
33
3
44
4
4
5
55
5
1
11
1
12341245
1253
13 542354
23
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
k-c
hir
otopes
–k-a
rrangements
DF
7.A
k-arran
gemen
tof
double
pseudolin
esis
afinite
indexed
byI
family
τof
simple
closed
oriented
curves
embedded
ina
compact
surface
Sτ
with
theproperties
that(1)
τin
duces
a
regular
celldecom
positionX
τof
Sτ ;
and
(2)an
ysu
bfamily
νof
τof
sizeat
least2
and
at
most
k,con
sideredas
embedded
not
inS
τbu
tin
thecom
pactsu
rfaceS
νdefi
ned
bythe
local
embeddin
gof
νin
Sτ ,
isan
arrangem
entof
double
pseudolin
es.W
eden
oteby
Ak
theset
of
k-arran
gemen
tsan
dby
Ak→
Ck
them
apthat
assigns
toa
k-arran
gemen
tits
chirotope.
TH
11.The
graphof
mutation
son
A5
iscon
nected.
TH
12.A
5→
C5
ison
e-to-one
and
onto.
24
Conferen
ceon
Geo
metric
Gra
ph
Theo
ryM
ichel
Pocch
iola
Cen
treIn
terfacu
ltaire
Bern
oulli
short
bib
lio
graphy
[1]Ju
lienFerte,
Vin
centP
ilaud,
and
Mich
elP
occhiola.
On
the
num
ber
ofsim
ple
arrangem
ents
offive
dou
ble
pseu
dolin
es.D
iscreteC
omput.
Geom
.(to
appear),
2010.Special
issue
devoted
toth
eW
orkshop
onTran
sversalan
dH
elly-type
Theorem
sin
Geom
etry,C
ombin
atoricsan
d
Top
ology,B
anff
Station
,Sep
tember
2009.A
prelim
inary
versionap
peared
inth
eab
stracts
18thfall
Worksh
opC
omput.
Geom
.,Troy,
2008(F
WC
G2008).
[2]L.H
abert
and
M.P
occhiola.
Arran
gements
ofdou
ble
pseu
dolin
es.Subm
ittedto
Disc.C
omput.
Geom
.A
bbreviated
versionin
Proc.
25thA
nnu
.A
CM
Sym
pos.
Com
put.
Geom
.(S
CG
09),
pages
314–323,Ju
ne
2009,A
ahru
s,D
enm
ark.A
partial
abbreviated
versionap
pears
inth
e
Abstracts
12thE
urop
eanW
orkshop
Com
ut.G
eom.p
ages211–214,2006,D
elphes,an
da
poster
versionw
aspresented
atth
eW
orkshop
onG
eometric
and
Top
ologicalC
ombin
atorics(satellite
conferen
ceof
ICM
2006),Sep
tember
2006,A
lcalade
Hen
ares,Spain
.,octob
er2006.
[3]J.
E.
Good
man
,R
.P
ollack,R
.W
enger,
and
T.
Zam
firescu.
Arran
gements
and
topological
plan
es.A
mer.
Math
.M
onthly,
101(9):866–878,N
ovember
1994.
[4]H
elmut
Salzm
ann,
Dieter
Betten
,T
heo
Gru
ndhofer,
Herm
ann
Hah
l,R
ainer
Low
en,
and
Marku
sStrop
pel.
Com
pact
pro
jectiveplan
es.N
um
ber
21in
De
Gru
yterexp
ositions
inm
ath-
ematics.
Walter
de
Gru
yter,1995.
25