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Notation
Original
Image
Case 1:
P ’2
λ1
λ2
P ’1
λ ’3
P3
λ3
P ’3
P2
λ ’2
P1
λ ’1
Case 2:
P3
P2
P1
C 2’
P4
P6
C 3’
C 1’
P5
C 4’
C 6’
C 5’
C i is the conic not passing through P i .
P ’i is the conic not passing through C ’i .
Q
µ’
Case 3:
Q’
P1
λ ’1
P4
P2
λ1
λ2
P3 λ
4
λ ’4
λ3
P ’2
λ2 P1 λ3
Case 4:
Q3
λ1
P2
P ’3
Q 2’
Q 3’
µ1
λ 3’
Q2
Q 1’
P3
λ ’3 P ’
4
λ ’2
P ’1
µ ’2
λ 2’
µ ’3
λ 1’
Q1
µ i is the conic not passing through P i , Q i , but through the other P
j
,Q
P ’i is the conic not passing through λ ’i , µ i’, but through the other λ j ’ ,µ
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j
j
.
’ .
Case 1
Self Inverse ?
1.
No
2.
Yes
3.
Yes
4.
Cubic
Yes
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!
Case 2
Self Inverse ?
1.
Yes
2.
Yes
Cubic
3.
Yes
CXBK^NqM=g _bIKJM= Case 3
Self Inverse ?
1.
Yes
2.
No
Conic
3.
See
Comment
4.
Yes
5.
See
Comment
6.
Yes
Cubic
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Case 4
Self Inverse ?
1.
No
2.
Yes
i
No
3.
A line may pass through the Q
4.
if they are collinear.
Yes
Yes
5.
6.
Yes
Cubic
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k
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following plumbing
diagram.
0
0
k
l
-1
0
0
-1
Blowing down (R1)
yields
0
0
k
0
-1
Rearranging yields
l
0
0
0
0
k+1
0
l+1
0
0
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!
k
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Blowing up yields:
0
+
+
+ +
+
0
k
+
+1
0-Chain Absorption (R3)
0
-
+
0
- +
Oriented Handle Absorption (R5)

0
+
-
k-1
0
0
[k-1]
CFBD^Nqa=g` ! b CXEN=IkENm © =/ENz(CXW
k
Blowing up yields:
k >2
-1
0
0
0
k
0
Rearrange:
-1
0
k-1
0
Splitting (R6)
0
k-1
0
CXBD^Nqa=g` !K`b © =/ENz(CXW
Plumbing
Diagram
+1
+
+
+1
+
+1
Blow down (R1)
-
0
+
Oriented Handle
Absorption (R5)
0
[1]
CXBD^`qM=` ! b©qMCXIKENBDWF=
0
Plumbing Diagram
+1

Blow down (R1)
+1
0
CFBD^Nqa=` ! ! b CXEN= IKCXq
Plumbing Diagram
Blow down (R1)
(Leaves the empty
graph.)
+1

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§ i i ` M §Ã Z / Z · ·fÈA¹¹¢Ç`´K¹
sÍ u/ D eyZ Z M Ã § -u/ D eyZ! M Ã Í u/ D eyZ M Ã § -u/ D eyZ M Ã
HÍ D eyZ a Ã § n D 0yZ a Ã
AÍ !D eyZ a Ã § n !D 0yZ a Ã sÍ D ey a Ã § n D 0y a Ã Í !D ey a Ã § n !D 0y a Ã xQN=*hNqM=(JM=/E`z/=PUKS -CXE rIkLK=/JORQ`=*hNqaU z(=/JaJUKSOMQN=]hjIKJMJMIKBD=]?i=9OfYl=(=/E tu
IKE`m}t0yer@^`zQ&=/IKJMCF=/qORU^NE`mN=/qaJaOIkENm; =/=jBD^Nqa= F
<²QN=(EYl=*mNU&IKE IKENIKWFG_JMCFJ-IKJTCFE>Ja=/z9ORCXUKE ` }UkSxQNU!Y UKOMQN=/q-z/^Nq|D=(J@CXEsOR=(qMJa=/z(O
OMQN=-IKr@?NCFBD^NUD^NJxJa=(O¼Yl=-JM=(=-OMQjIkOxjqaJaOMWFGnUDE`WFGPOMQN=gORQNqa=/=-JMCFOM^jIkOMCXUDENJxCFEnjBD^Nqa= F!
z/IKEIKqMCFJM=@m`^N=@ORUOMQN=qM=(JaOMqMCXz9ORCp|D=@EjIOR^Nqa=UKS IKJM= !_v¯IKENm4OMQjIkOOMQN=/Ja=*IkqM=IKWFWJa=/WpS
CFEH|K=/qaJM=Kv¡BDCp|_CXENBqaCXJM=OMUJaG_r*r]=(OMqMCFzghNWX^Nr?NCXENBm`CIKBDqMIKr]J/
/¤ ey9 Ã § ey9 Ã d OOIkLK=/J&WXCpOMORWF=4=(oiUDqO}OMU Ja=/=ORQjIkOOMQN=>BKqRIKhNQ r@^`JaO}?i=ORQ`=>UDE`=>JaQNU!Y[E CXE
jBK^NqM= p
<²QN=(EUDEN=]mNU_=/JORQN=*IKEjIkWFG_JMCFJ-UkSQNU!Y UKOMQN=/qz(^Nqa|K=/J@r*I2G7CXEsOR=(qMJa=/z(O-ORQ`=IKr
?NCFBD^NUD^`JJM=9O-OMQN=ENUDE`JM=/WpSfCFEH|K=/qaJM=z2IKJa=mNU_=/J-IKqMCFJM=kTxQN=Ja^Nr*r*IKqGUKSY[QjIkOr*I2G
U_z/z(^Nq[CXJxJMQ`UZY[E&CXEPjBK^NqM= p !
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B2
B1
-2
E*
R1
R2
R2
B1
, E*
B2
R1
B2
R2
R1
E*
B1
R
1
B2
E*
B2
R1
B1
E*
B1
B2
CXBD^`qM= p b

B1
R2
R2
R1 ,E*
R2
1.
2.
3.
CXBD^`qM= p !`b
Î
-1
B1
B2
-1
-1
B3
-1
B4
E*
-2
-1
R1
-1
R2
-1
R3
B4
B2
-1
R4
R1
B3
R3
E*, B1
R4
R2
CXBD^`qM= p b
B4
B2
B3
E*, B1
R1
R3
R4
R4
R2
R3
E*, R1
R2
B1
B3
B4
Self Inverse under correspondence
which exchanges the 4-tuples:
(R1,R2,R3,R4) and (B1,B2,B4,B3) ?
Yes
B2
No
Yes
Yes
CXBD^`qM= p ! b
D jqMUKr Q`=/qM=xUKE*IkENmCXE@Ja^Nz/z(=/=(mNCXENB-JM=(z(OMCXUDENJ(vsCFOY[CXWXW_?A=^NJM=9SV^NWNORUeJM=/=¼Ä&ÁTÂtua IKE`m
Ä&ÁTÂt0yw TIkJ?A=/CXE`BORQN=*JRIKr]=BKqRIKhNQ?N^Ò@WFU UKLK=/mIkOOMQNqMUK^NBDQ mNCpoc=(qM=(EHOBDWIkJMJM=(J/
jUDqg= NIkr*hNWF=CXE7ORQ`=]h`qM=(|_CFUD^NJz2IKJa=*UkS ¼Ã Ã !nUD^NqIKEjIkWFG_JMCFJg?iUDCFWX=(m7mNU!Y[E>ORU
Y[QN=(qM=TCXEORQN=-mNCIkBDqRIKrORQN=OfYlU]jBD^Nqa=/J0CXEjBD^Nqa= p Yl=(qM=K CXBD^Nqa= F JaQNU!Y[J
IKWFW[hiUDJaJMCF?NWX=&qM=/WXIkORCp|D=&hiUDJaCFOMCXUDENJ(v[IKENm UDE`WFG ORQN=&ORQNCFqMm{qa=/WIORCF|K=hAUDJMCpORCFUDEz/IK^NJM=(m
JaUDr*=9ORQNCFENBEN=9YY[QNCXzQ}QjIKmORU?A=CXEs|D=(JaORCFBsIkOM=/m;
0
0
0
0
CXBD^`qM= p b
1. (Disjoint)
0
2. (Coincident)
0
3. (Overlapping)
0
0
0
0
0
0
CXBD^`qM= p b
0
fd E4UD^`q0hNqa=/Ja=/EsOgz2IkJM=TY=QNI!|K=ORQ`=TOfYUjBD^Nqa=/JeIKJ0JaQNU!Y[E7CXE}jBD^Nqa= p Y[CFORQ
IKWFWiOMQN=/CFqxhiUDJaJMCF?NWX=gqa=/WIORCF|K=ghiUKJMCFOMCXUDE`J/
xQN=jqMJO¼hiUDJaCFOMCXUDEPY[QN=(qM=ORQN=9GIKqa=-m`CXJ aUKCXEsO[CXJx=/IKJMCF=/JaO/K^NJaO¼?NWFUZY ^Nh&IKJxYIKJ
mNUKEN=&CFE z/IKJM=>UKSeJM=(z(ORCFUDE `pORUqa=/z(UZ|K=/qIKWXW[IKr?NCXBD^ÙD^NJz/^Nq|D=(JY[QNCXzQ BDUKO
?NWFU!Y[EmNU!Y[E7CXE}ORQN=ThjIKJaJRIKBD=TORUEÙDqMr*IKW¯SVUDqMr} EN=z/IKE4zQN=(zL&ORQNIkOgIKr?NCXBD^ÙD^NJ
Ja^N?NBDqMIKhNQ UKSlOMQN=IkWFOR=(qM=(mhNWF^Nr@?NCFENBmNCIKBKqRIKr
Y[CXWXWJRIkOMCXJaS¢GI `CFUDr]J ( nUkS
OMQN=/UDqa=/r FD IKENm}JMU]ENUKOMQNCXENBE`=(Y QjIkhNhi=(ENJ/
xQN=]JM=(z/UDENm>hiUKJMCFOMCXUDE7CFJgCXr]hiUKJMJMCF?NWX=T?i=(z2IK^`JM=]CFOCXJgCFWXWF=/BsIKW SVqMUKr ORQ`=]hAUDCFEHOUkS
K?
DD
0
1. (Disjoint)
0
0
0
+1
2. (Overlapping)
0
3. (Overlapping)
0
0
+1
0
0
+1
CXBD^`qM= p `b
|_CX=9Y UkS§Ä&ÁTÂt0y9 9vNENIKr*=(WFG]ORQN=gOfYU*= z/=/h ORCXUKEjIKW¯mNCp| CFJMUDqaJxUKS Yl=(CXBDQsO z/UD^`WXmENUKO
QjI2|K=T?A=/=(E aUDCFEN=/m?sGIKE}=/m`BD=K
xQN=ORQ`CXqMmghAUDJaCFORCFUDEz/IKETQNIKhNhA=/E; <=^`JM=ORQN=JMIKr*=hNqMU_z/=(JMJIKJ¯CXEOMQN=hNqM=9|_CXUD^NJ
Ja^N?NJM=(z(OMCXUDE?sG²?NWFU!Y[CXENB^NhIkWXUDENBqM=(WX=9|KIkEHO&=/m`BD=/J}CXEÅUDqMmN=(q}OMU qa=/JORUDqa=?NWFUZY[E
mNU!Y[EIKr@?NCFBD^NUD^NJz(^Nqa|K=/J/ EN=}z2IkEzQN=(zLOMQjIkO]ORQN=&IKr@?NCFBD^NUD^`JJM^N?NBKqRIKhNQ
UKSORQN=IkWFOR=(qM=(m4hNWX^`r@?NCXE`B*mNCXIKBDqMIKrJRIkOMCXJj=/JeOMQN=@I `CFUDr]J ( UKS `CFUDr ` CFE JM^N?`JM=/z9ORCFUDE FY[Q`=/qM=nOMQN=nI CXUKr*JTEN=/=(mN=/m IKWFOM=/qaCXENBN&xQN=nUDENWpGm`CFoi=/qM=(ENz/=
CFJOMQjIkO]ORQN=(qM=&CXJ]UDEN=}r*UDqa=&m`CXJaOMCXENBK^NCXJaQN=/m |D=(qaOR= IkENm OMQjIkO]ORQN=&IKr@?NCF=/EsO*ENUDE
IKr?NCXBD^ÙD^NJz(^Nqa|K=/JeJMIkORCFJaS¢G´K¹ÉX·M´»9¹cOMQN=JRIKr]=gz/UDE`mNCFOMCXUDENJ[IkJ¼CFE CXUDr ` _
xQH^NJTIKJgSIKqIKJ ¼Ã ! ÃPBDU =(JgYl=mNUDEcO-JM=(=IkEHGHOMQNCXENBORQNIkOgYl=]QjI2|D=(E; O
Ja=/=/E&?i=9SVUDqM=k
?
A
*
xQN=(qM=>CXJUDENWpG UDEN=qM=(WIkOMCF|D=hAUDJMCpORCFUDE CXE²Y[QNCXzQ ORQN=qM=(WX=9|KIkEHO&JMr*IKWFWmNCIkBDqRIKr]J
¢jBD^Nqa= F!Î] z2IkE&U_z/z(^Nq¼IKENmORQNCFJxCXJx?sG?i=(CXENBmNCFJ aUDCFEsO2§<>=-?`WXU!Y ^`h}OMU*qa=/JaOMUDqM=
IKEsGWFUDJaOIKr@?`CXBD^NUK^NJz/^Nq|D=/JIKENm]JM=/=¼ORQjIOOMQN=0qa=/Ja^NWFOMCXENB-IKr@?`CXBD^NUK^NJJM^N?NBKqRIKhNQ
UKSORQN=TIKWpOR=/qa=/m}hNWX^`r@?NCXE`B]m`CIKBDqMIKr JMIkORCFJaj=(JeI `CFUDr]J ( UkSORQ`=/UDqa=/r `pD ]JaU
OMQjIkO[ENUKOMQNCXE`B]E`=(YQNIKhNhA=/ENJ(
K?
D +1
+1
1. (Disjoint)
0
0
+1
CXBD^`qM= p !Î`b
0WXWlOMQN=qM=/WXIkORCp|D=hAUDJMCpORCXUKENJ@UKS0ORQN=qM=(WX=9|KIkEHO]JarIKWFWmNCXIKBDqRIkr*JIkqM=JMQNU!Y[E CXE jBk
^Nqa=
0
1. (Disjoint)
0
+2
0
0
0
0
0
2. (Overlapping)
+2
0
0
0
3. (Overlapping)
0
+2
0
CXBD^`qM= b
x QN=]jqMJOThAUDJaCFORCFUDE mNCFJ aUDCFEsOw 0Y=PIKBsIkCXE=2IKJaCXWpG7mN=2IkW§Y[CFORQ?sG?NWXU!Y[CXE`B}^`h>ORU
qa=/JaOMUDqM=&IKEsGWXUDJO*IKr@?NCFBD^NUD^NJz(^Nqa|K=/JnIkENmJa=/=&OMQjIkOORQ`=&qa=/JM^`WFORCFENBJM^N?NBKqRIKhNQ
UKSORQN=TIKWpOR=/qa=/m}hNWX^`r@?NCXE`B]m`CIKBDqMIKr JMIkORCFJaj=(JeI `CFUDr]J ( UkSORQ`=/UDqa=/r `pD ]JaU
OMQjIkO[ENUKOMQNCXE`B]E`=(YQNIKhNhA=/ENJ(
xQN=TJM=/z(UDENm7hAUDJMCpORCFUDECXJ¼CXr]hiUDJaJMCF?NWX=?A=/CFENBCFWXWF=/BsIKWASVqMUDrOMQN=hAUDCFEHO¼UKS| CF=(Y UkS
Ä&ÁTÂt0yw 9
xQN=gORQ`CXqMmnhAUDJaCFORCFUDEPz/IKE&U_z/z(^Nq[IKENmnY=mN=2IKWiY[CFORQCpO[IKBsIKCFEP?sGn?NWFUZY[CFENB^NhnORU
qa=/JaOMUDqM=@IKEsGWFUDJaOeIKr@?NCFBD^NUD^`J0z(^Nqa|K=/J/xQN=qM=/Ja^NWFOMCXENB JRIkOMCXJaN=/JgI CFUDr*J ( UKS CFUDr ` @JMUORQjIkO[EÙKORQNCFENBEN=(YQjIKhNhA=/ENJ(
K?
D!
0WXWlOMQN=qM=/WXIkORCp|D=hAUDJMCpORCXUKENJ@UKS0ORQN=qM=(WX=9|KIkEHO]JarIKWFWmNCXIKBDqRIkr*JIkqM=JMQNU!Y[E CXE jBk
^Nqa= _D
0
1. (Disjoint)
0
0
0
+1
+1
0
0
2. (One Vertex
in Common)
+1
0
0
0
0
3. (One Vertex
in Common)
+1
+1
0
4. (Two Vertices
in Common)
0
0
+1
0
+1
5. (Two Vertices
in Common)
0
0
+1
6. (Three Vertices
in Common)
0
0
+1
CXBD^`qM= Db
x QN=@jqaJaO-IKE`m4ORQN=WXIKJaOehiUDJaCFOMCXUDENJeIKqM==2IKJaCX=/JO2TlWXU!Y ^Nh4OMUnqM=/JORUDqa=WXUDJOgIKr
?NCFBD^NUD^`Jxz/^Nq|D=/J(xQN=-qM=(JM^NWpORCXE`B 5JRIkOMCXJj=/J0I CXUDr]J ( UKS©ORQN=(UDqM=(r `pD JaU
OMQjIkO[ENUKOMQNCXE`B]E`=(YQNIKhNhA=/ENJ(
xQN=eOMQNCXqamnIKENmPOMQN=eSVUD^`qaORQhiUDJaCFOMCXUDENJIKqa=IKWFJMU@=/IKJaGIKENmnqa=/JM^`WFOCFEORQ`=gI CXUDr]J
UKS CFUDr `
xQN=PJa=/z(UDENm hAUDJMCpORCXUKECXJ-mNCpoc=(qM=/EsO/7lWXU!Y[CFENB&^`h ORUqa=/JORUDqa=PWXUDJOIKr?NCXBD^ÙD^NJ
z(^Nqa|K=/J[G CF=/WFmNJxjBD^Nqa= _
D -1
-1
R3
B1
-1
-1
R4
B2
-3
E*
-1
-1
R1
R2
-1
-1
B3
CXBD^`qM= b
B4
x QN=-qM=(JM^NWpORCXE`BI CXUDr]JSVUDqORQ`=-qa=/Ja^NWFOMCXENB IKqM=IKJxSVUDWFWXU!Y[J/b
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OMU f jqaUDr OMQNCXJUDEN=xz2IkE]=/IKJMCFWFGTmN=(OM=/qar*CFEN=lORQNIkO i w Ã c -uM IKE`m i Ã
i 9 gE`=@IKWFJMUP=/IKJMCFWFGJM=(=/JOMQjIkOCpS§UDE`=@UKS QjIKJ¼|KIkWX=/E`z/= ORQN=(E7JMUPmNU_=/J
OMQN=UKORQN=(q/xQNCFJ[CXENmN^`z/=/J[ORQ`=OfYU]z2IKJa=/J[ORQjIkOxSVUKWXWXU!Yb
/¤ ° i 9 Ã
jqMUKrORQ`CXJxUDEN=\H^NCFzL_WpGnmN=9OR=/qar*CFEN=/J $ORU]?i=IkJ¼CFEnjBD^Nqa= K
<²QN=(EYl=mNUeOMQN=lIkEjIKWFG_JaCXJ©UKSNQNU!Y UKOMQN=/qz/^Nq|D=(J§rI2GCFEHOM=/qaJM=/z9OOMQN=IKr?NCXBD^ÙD^NJ
Ja=(OYl=¼JM=(=xORQjIkOOMQN=(GIkqM=¼IKWXW JM=(WFS;CFEs|D=/qaJM=¼JMCpOR^jIkOMCXUDE`J§JMUORQjIOENUKORQ`CXENBEN=9Y U z(z/^NqaJ/
=(=gjBD^Nqa=
/¤ i 9 Ã d O[CFJ¼EÙKOxORU_UQjIKqMmnOMU*mN=9OR=(qMr]CXEN=OMQjIkO r^NJaO[?A=-IKJ[CXEnjBD^Nqa= !
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+*
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N*
G
J
G
J
D B1
B2
B3
B4
R1
R2
R3
R4
E*
CXBD^`qM= K`b
1.
B4
E*, B1 , B2
R4
B4
R3
B3
B2
R2
R4
R1
E*,R1 ,R2
B3
Self Inverse?
Yes
B1
R3
2.
Yes
3.
Yes
CXBD^`qM= b
D B1
B2
B3
B4
R1
R2
R3
R4
E*
CXBD^`qM= ! b
x QN=¼z/WXIKJMJaCFjz/IkORCFUDEIKJORU-QNU!Y{UKOMQN=/qz(^Nqa|K=/Jlr*I2G@CFEHOM=/qaJM=/z9OOMQN=¼IKr@?NCFBD^NUD^`JJM=(O
CFJJMQÙZY[E CFEjBD^Nqa= _U7JMUDr]=(OMQNCXENBEN=(Y U_z/z/^`qMJQN=(qM=P?A=/z/IK^NJM=PY=nQjI2|D=I
ENUKENJM=(WFSfCFEs|D=/qaJM=JMz(=/EjIkqMCXU`
Self-Inverse?
1.
Yes
2.
No
3.
Yes
CXBD^`qM= b
D x QN=NS¢ORQhAUDJMCpORCFUDEgCXJ mNCpoc=(qM=/EsO/lWFU!Y[CXENBx^NhORU¼qM=(JaOMUDqM=WXUKJaO©IKr@?`CXBD^NUK^NJ¯z/^Nq|D=(J
G_CX=(WXmNJOMQN=mNCIkBDqRIKr CXEnjBK^NqM=
B3
-1
-1
R1
R2
-1
E*2
-2
E*1
-2
-1
B1
B2
-1
-1
R3
CXBD^`qM= b
x QN=TI CXUDr]JlSVUDq 5CXEnORQ`CXJxCXE`JaOIkENz/=-IKqa=IKJxSVUDWXWFUZY[J(
¨V« ®X¬;®
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CXBD^`qM= `b

Self-Inverse?
1.
Yes
2.
Yes
3.
Yes
CXBD^`qM= KÎ`b

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+1
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3. (Two vertices
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4. (Three vertices
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Links and Complements of Arrangements of Complex Projective Plane Algebraic Curves

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