# GIT ones and quivers

-

English
18 Pages

Description

Niveau: Supérieur, Doctorat, Bac+8
GIT- ones and quivers N. Ressayre ? June 8, 2009 Abstra t In this work, we improve results of [Res07, Res08a? about GIT- ones asso iated to the a tion of a redu tive group G on a proje tive variety X . These results are applied to give a short proof of the Derksen- Weyman theorem whi h parametrizes bije tively the fa es of a rational one asso iated to any quiver without oriented y le. An important example of su h a one is the Horn one. Contents 1 Introdu tion 2 2 Well overing pairs and GIT- ones 3 2.1 Well overing pairs . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Total ones and well overing pair . . . . . . . . . . . . . . . . 4 3 Appli ation to quiver representations 8 3.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Three ones . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Dominant pairs . . .

• linear group

• let

• lx ?t ?

• group

• g?-linearized line bundles

• tion morphism

• picg ?

• derksen-weyman theorem

Subjects

##### Matrix group

Informations

Report a problem

G
X

N.
Ressa
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June
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8,
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2009
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7].
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2ν Nνc = 1⇒c = 1 ∀N ≥ 1.λμ NλNμ
α◦β = 1 ⇒ α◦ (Nβ) = 1
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λ G P(λ) λ
λG
n o
−1P(λ) = g∈G : limλ(t).g.λ(t) G .
t→0
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+C C
+C ={x∈X | limλ(t)x∈C}.
t→0
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+G× C G× P(λ)
′ ′ −1(g,p).(g ,y) = (gg p ,py)
+ +G× C G×C {e}×P(λ)P(λ)
+ +(g,y)∈G×C G× C [g :y]P(λ)
+G×{e} G G× CP(λ)
+G×C −→G G π : G×P(λ)
+ +C −→ G/P(λ) C
G
+η : G× C −→X, [g :y] →gy.P(λ)
(C,λ) η (C,λ)
η
X C
and
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action
3
pair

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action
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y
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-equiv
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isomorphism
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∗K Lx
Lμ (x,λ)
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Lμ (C,λ)
Γ Γ = Γ⊗ Q ΓQ Z
Y(Γ) Γ F
E Span(F) E
F
G GTC (X) Pic (X)Q
GL Pic (X) G
ssX (L) x X
⊗nn G L σ(x) = 0
ss ss ⊗nL X (L) =X (L )
ss Gn X (L) L∈ Pic (X)Q
L(C,λ) L →μ (C,λ)
G LPic (X) Q μ (C,λ)Q
G LTC (X) μ (C,λ) ≤ 0
G LTC (X) μ (C,λ) = 0
GF(C) TC (X)
C
(C,λ) F(C) L ∈
G ssPic (X) X (L) CQ
GPic (X)
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GPic (X) /Span(F(C))Q
λ λG GPic (C) /Span(TC (C))Q
oin
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If
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b
e

a
The
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t
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pair.
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nite
an
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's
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the
xes
is
,

it
v

a
Consider
is
spanned
a
pr
group
es
morphism,
group,
it
w

onto
a
subgroups
linear
y
map
denote
from
e
linear
arian
action
-in
of
wing
the
impro
group
of
on
Theorem
the
assume
b
ank
to
non
Since
h
,

also
;
denoted
.
b
denoted
y
b
ample.
l
is
air.
er
ank
if
b
only
T
standard,
and
is
dimension
.
w
By
ell
[Res07,
pair
Lemma
quals
4],
dimension
denition

this
2.2.1
that
y
Note
of
.
Mor
.

is
morphism

isomorphism
tained
ab
in
e
the
denote

space
There
of6
is
exists
is
an
parameter
in
the
teger
group,
that
e
h
will

that
.
W
In
t
particular,
v
in

tersecting
.

follo
h
theorem
that:
an
of
v

t
t
[Res07,
with
4].
the
1
h
e
yp
that
erplane
r
arian
of
v
zero
-in
e
One
ha
a
is
exists
and
there
onsider
,
on
easily
it
,
b
one
L
obtains
e
a
in

b
teger
e
in
wel
e

ositiv
p
of
Then
p
r
some
of
for
y
that
y
h
2.2

otal
.
nite
Indeed,
the
the
o
follo
of
wing
and
lemma
in
sho
generated
ws

that
ering
the
Notation.

e
only
the
dep
o
ends
of
on
v

the
(see

[Res07,
in
Lemma
If
4]):
b
Lemma
is
1

L
.
et
e
in
e
's
the

estriction
the

of
an
b
fr
e
an
a
elian
dominant
w
p
set
air.
b
Then,
e
subset
.
en
ector
op
a
the
part
is
a
the
If
set
.
of
of
t
ks
that
one
do
set
es
an
not

dep
y
y
denote
b
w
Let
4πX -
λG Gp : Pic (X) −→ Pic (C)Q Q
Imp
p
λGPic (C)Q
λGAC (C)
Imp
Ω G X
+G× (C ∩ Ω) −→ Ω (C,λ)P(λ)
GΩ C Pic (Ω)
λGPic (C∩Ω)
E ,···,E D ,···,D1 s 1 t
λX−Ω C−Ω G G Ei
λD G Gi
λG G L LE Di i
G-⊕QL Pic (X)i E Qi
λG G -Pic (Ω) ≃ Pic (C∩Ω) 0.Q Q
?
λG-⊕QL Pic (C)i D Qi
X C π πX C
λGL TC (C) LD Ei i
G E Ei i

the

morphism
has

t.
and
(as
(resp.
ts
let
one
Let
is
.
map
denote
a
its
equiv
image.
By
A
Moreo
t
with
rst
if
)

b
b
e
en
the
Let

Since

the
onen
are
ts
no:
of
the

h
di-
v
mension
zero
one
In
of
surjectiv

the
it

is
the
(resp.
that
not
of
to
-stable
isomorphic
b
).
Pro
Since
es
is
and
and
smo
so
maps
ob
The
are

dierence
the
elongs
vious
and
to
one
see
er,
and
ws
the
follo
wh
t
y
in
's

are
4].
resp
particular,
ectiv
is
ely
e
5],
in
and
example
Lemma
in
[Res07,
the
-stable.
natural
W
e
e
Let

h
the
t
asso
subset

op
By
w
and
e
.
o
Theorem
of.
-linearized
statemen
line
not
bundles
do
tersecting
are
in
so
1
are
and
oth,
an

h
alen

and
.
main
Consider
sur-
the
e.
follo

wing
is
diagram:
b
nd
to

one
that
ering
rst
v
.

v
ell

w
allo
is
the
impro
a
v
-in
es
arian
[Res07,

Theorem
wing
Since
has
isomorphism.
lo
an
Since

's
Remark.
5
-
πCC L ∈ F(C)Ei
λGπ (F(C)) =π (TC (C))X C
λGssL∈F(C) X (L) C L TC (C)|C
λGπ (F(C))⊂π (TC (C))X C
λL G C
λGTC (C) L
λ ˜G σ L L G
˜Ω π (L) σ˜ G LC
G ˜σ M ∈ Pic (X) π (M) = L σ˜X
G M
P ⊗n′ ′ iG σ M =M+ Li Ei
′n E C σ Ci i
′ ′M ∈F(C) π (M) = π (M) = π (L)X X C
λGπ (F(C))⊃π (TC (C))X C

GAC (X) H
G X H
G G/HTC (X) Pic (X)
GL ∈ Pic (X) H
G G/HAC (X)⊂ Pic (X)Q
K G
G GX AC (X) Pic (X)Q
Y(Γ)
k·k : Y(G) −→ R
λ S G
k·k Y(T) QQ
Lμ (x,λ)Lμ (x,λ)
kλk
G
+G GPic (X) Pic (X)QQ
G G + GAC (X) = Pic (X) ∩TC (X).Q
G 0 G +F TC (X) F Pic (X)
Q
b
e
found
in
with
dened
a
the
er
pro
p
of
.
of
in
[Res07,
to
Theorem
of
4].
The

line
argue
a
e
arian
.
measures
e
of
are
norm
no
to
w
zero
in
an
terested
generate
in
in
the
b
span
.
of
it
v
asso
o
and
ab
b
the
,
out
an
ab
of
.
e
Let
a
details
This
b
um
e
v
a
there

dened
normal
parameter
subgroup
ositiv
of

that

acting
h
trivially
set:
on
v
Note
,
.
e
Then,
to
.
denotes
has
pro
to

act
-in
trivially
h
on
arian
an

y
length
line
Moreo
bundle
,
in
subtorus
that
the
ws
on
follo
-linearized
it
is
,

Since,
pro
.
Let
W
is
e
the
denote
er
b
t
y
.
.
a
particular,
w
in
,
;
[Kem78
on
destabilizing
zero
2.2.4

of
iden
line
not
op
the
ex
sub-
b
group
Up

elongs
of
on
the
W
is
a
,
,
tains
.

elongs
no
in
Since
.
.
v
tegers
of
in
sucien
on
So,
whic
in
h
es
e
,
acts
of
trivially
t
.
v
W
the
e
whic
ha
is
v
v
e:
t
negativ
y
non
and
some
the
for
of
of
.

v
t
precisely
arian
for
v
y
-in
to
regular

zero

non
asso
a
bundle
so
line
and
the
,
b
.
the
Consider
asso
no
to
w:
scalar
Denition.

Let
on
of
.
b
norm
e
used
the
normalize
neutral
n

b
onen
.
t

of
arian
the
-in
k
Using
ernel
non
of
exists
the
er,

o

e
on
Kempf
rational
(see
.

The
optimal

one
t
subgroup.
arian

v
set
-in
ample
zero
-linearized
non
bundles
is
an
said
en
to
v
b
p
e
y
non
hanging
degenerated
to
if
.
it
to
spans
b
a
whic

bundle

.
The
e
.
-linearized
that
e
h
let

ersely
.
Con
2.2.3
So,

to
Whereas
b
let
tersects
w,
Since
No
Let
.
that
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If
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group,
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one
t

is

,
(see
.
[MFK94
its

tersection
a
that
map
v
to
this

2.2.2
tains

W
6G GL ∈ Pic (X) AC (X)Q
(C,λ) k·k F(C)
GF(L) AC (X)
G◦F (L) L ∈AC(X)
GF AC (X) Δ(F)
G ◦L ∈AC(X) F =F (L)
L∈ Δ(F) (C,λ)
λGAC (C)
GΔ(F) Pic (X)Q
Δ(F)
ˆ ˆX = G/B×G/B
ˆG G
λG(C,λ) AC (C)
λGAC (C)
Σ(Q,β)
G P Q R
GG AC (G/P ×G/Q×G/R)
L ∈ Δ(F) (C,λ)
λ λK G C
λGPic (C)Q
λ λG /K λPic (C) S K T ⊃ SQ
λ ′G S T Y(S)Q
′ ′Y(S ) k·k S×S −→TQ
λ λ ′H G S
λ λλ λ λ G /KK × H −→ G Pic (C)Q
λHPic (C)Q
λ λ λG H G /Kp : Pic (X) −→ Pic (C) ≃ Pic (C) .Q Q Q
v
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pair
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d
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w
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one
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Ther
t
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and
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h
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group
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air

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Theorem
a
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of
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e
ell
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non
kind
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y
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rst
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t.
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v
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4
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e
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osition
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T
7λHp(L) AC (C)
λ λH Hp(L) AC (C) Pic (C)Q
• Gμ (C,λ) = 0 Pic (X)Q
+H L
GAC (X) Δ(F)
λ′ + ′ HL ∈H p(L)∈AC (C)
Im p p
λHp(L) AC (C)∩ Im p
λHIm p AC (C)
λHPic (C)Q
λHAC (C)∩Im p Im p
λHp(L) AC (C)∩Imp L Lǫ
λHp(L ) AC (C)∩Im p F(L ) = Fǫ ε
λHAC (C)∩Imp Imp
λHL L p(L ) AC (C)∩Impǫ ǫ
λHF(p(L )) =AC (C) F(L ) =Fǫ ε
(C ,λ ) L Cε ε ε ε
C Lε
C C ε
Q Q0
Q a ∈ Q ia ta1 1
R Q (V(s))s∈Q0
u(a) ∈ Hom(V(ia),V(ta))
Q0a∈Q R (dim(V(s))) ∈N1 s∈Q0
Q0α ∈N V(s) α(s)
s∈Q0 M
Rep(Q,α) = Hom(V(ia),V(ta)).
a∈Q1
has
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assume
w

terior
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to

graph)
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represen
Let
in
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tains
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w
h
b
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in
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tation

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rst
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w

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tains
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e
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rst
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By
8Y Y
GL(α) = GL(V(s)) and SL(α) = SL(V(s).
s∈Q s∈Q0 0
Rep(Q,α)
Q Q0 0GL(α) Γ =Z σ ∈ZQ
σ(s)χ χ (g(s)) )= det(g(s))σ σ s∈Q0 s∈Q0
C[Rep(Q,α)]
Q0Rep(Q,α) GL(α) σ∈Z C[Rep(Q,α)]σ
f ∈C[Rep(Q,α)] g∈ GL(α) g.f = χ (g)fσ
Q Q0 0Γ =Z Γ :=Q Σ(Q,α) ΓQ Q
σ ∈ Γ C[Rep(Q,α)]−σ
{0}
X = P(Rep(Q,α)⊕C)
g.(R,t) = (gR,t) ∀g∈ GL(α),R∈ Rep(Q,α) and t∈C,
GL(α)GL(α) X GL(α) L ∈ Pic (X)0
O(1) X
GL(α)AC (X) X
GL(α) GL(α)AC (X) =TC (X)
⊗n GL(α)n∈Z σ ∈ Γ L(n,σ) =L ⊗σ ∈ Pic (X)0
L(n,σ) =O(n)
GL(α)Z×Γ−→ Pic (X), (n,σ) →(Ln,σ)
L(n,σ) n
GL(α)AC (X) Q × Γ P(Q,α) =Q
GL(α) GL(α)AC (X) ∩{1} × Γ AC (X)Q
P(Q,α)
Z GL(α) Wt
Rep(Q,α)⊕C P(Q,α)
Wt P(Q,α) ΓQ
GL(α)AC (X) P(Q,α)
GL(α)AC (X) P(Q,α)
line
bundle.
W
e
.
p
the
ha
,
v
of
e
these
the
is
follo
.
wing
in
ob
e
vious
is
Lemma
2
e
The
h
map
a
space

e
b

pro
-linearization
the
is
Consider
of

set
3.2.2
for
.
line
p

oin

ts
the
to
a

in
non
e
is
are

groups:
h
v
that
the
a
w
as
The
the
and
y
olyhedral.
b
that
generated

of
ter

of
ex
and
v
non
is
eigh
an
action
isomor-
on
phism
an
of
.
gr
pro
oups.
dened
Mor
the
e
tained
over,
v

of
the
.
denote

that
in
Note
p
.
w
is
v
ample
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if
also
and
is
only
ted
if

Let
bundle
is
acts
p
h
ositive.
y
Lemma
on
2

allo
ties
ws
;
to
w
em
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b

ed
it
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in
Consider
ed

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The
action
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ula

b
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zero
e
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in
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ts
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h
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easily
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set
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denote
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and
w

set
in
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,
ex
or
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and

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asso
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is
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rational
ed
olytop
w
in
.
no
General
Moreo
prop
er,
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e
of
.
endo

on
the
functions
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regular
the
the
oin
of

algebra
ex
the
generated
Consider
y

line
imply
of
that
naturally
3.2.1

a
Three
a
3.2
the
.
of
denes
.
is

group

iden
v
with
ex
of
rational
to
and
,
p
e
on
ond
Consider
ely
.
9Rep(Q,α) X R →(R,1)
P(Rep(Q,α))
ssX (L ) = Rep(Q,α)0
0∈ Γ P(Q,α)Q
Γ P(Q,α) Σ(Q,α)Q
Q
ta ia
a ∈ Q λ GL(α)1 0
i
it
0 ∈ Rep(Q,α) ⊂ X λ0
+C = {0} C = Rep(Q,α) λ0 00
GL(α) (C ,λ ) F(C )0 0 0
GL(α)P(C ) TC (X) P(Q,α) (C ,λ )0 0 0
L∈F(C ) 0 L 0 GL(α)0
L 0
+F(C ) Q L0 0
ss(R,t) →t GL(α) L Rep(Q,α)⊂X (L )0 0
+F(C )=Q L0 0
ss0 GL(α) Rep(Q,α) X (L )//GL(α)0
ssX (L ) O0
ssGL(α).0 O ={0} X (L ) = Rep(Q,α)0

D =P(Rep(Q,α))
GL(α)GL(α) AC (D)
GL(α) GL(α)ρ : Pic (X) −→ Pic (D)D
ρ (L) LD
GL(α)AC (D) Q× Γ P(D,Q,α) =Q
GL(α)AC (D)∩{1}× Γ P(D,Q,α)Q
Γ P(D,Q,α)Q
P(Q,α)
,

t
orien
is
indexed
arian
hose
v
Consider
-in
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a
vious
is
is
Since
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.
is
ws:
tication

terested
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e
e
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umeration
tained
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Lemma
.

Then,
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is
Ob

e
as
w
an
are
op
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ering
implies
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This
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.

subset
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Since
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er
that
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to
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ector
in
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er
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ter
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,
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3
its
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;
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But
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easily
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tify

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v
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.
b
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Prop
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osition
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.
e
W
follo
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Lemma
that
The
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1
w
(i)
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W
one
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that
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:
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oups.
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e
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h
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and
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if
last
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ws
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em
prop
ed
osition
is
is
than
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index

for
application
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of
ks
[Res08a,

Theorem
.
4].
One
.
.
3.2.4
the

parameter
Consider
Set
no
of
w
of
the
oin
pro
xed

an
e
on
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space
The
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viously
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onding
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ate
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ed
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homothet
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as
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p
is
oin
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3.2.3
10

en expand_more