GIT ones and quivers

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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


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,

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v


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morphism,
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w

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a
subgroups
linear
y
map
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from
e
linear
arian
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-in
of
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the
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ank
to
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Since
h
,

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;
denoted
.
b
denoted
y
b
ample.
l
is
air.
er
ank
if
b
only
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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
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W
In
t
particular,
v
in

tersecting
.

follo
h
theorem
that:
an
of
v

t
t
[Res07,
with
4].
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1
h
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yp
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r
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v
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a
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exists
and
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easily
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one
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e
a
in

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Then
p
r
some
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for
y
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2.2

otal
.
nite
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generated
ws

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ering
the
Notation.

e
only
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o
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(see

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Lemma
If
4]):
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Lemma
is
1

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.
et
e
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estriction
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of
an
b
fr
e
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elian
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p
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air.
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e
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en
ector
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If
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of
of
t
ks
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one
do
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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

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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

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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
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a
,
,
tains
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elongs
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in
Since
.
.
v
tegers
of
in
sucien
on
So,
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in
h
es
e
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acts
of
trivially
t
.
v
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the
e
whic
ha
is
v
v
e:
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negativ
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non
and
some
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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.

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on
of
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b
norm
e
used
the
normalize
neutral
n

b
onen
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t

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arian
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-in
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ernel
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er,

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rational
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.

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optimal

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arian

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set
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zero
-linearized
non
bundles
is
an
said
en
to
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b
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e
y
non
hanging
degenerated
to
if
.
it
to
spans
b
a
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.
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e
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e
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let

ersely
.
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.
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its

tersection
a
that
map
v
to
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2.2.2
tains

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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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t
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group
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d
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a
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rst
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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)
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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
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(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))
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Q0α ∈N V(s) α(s)
s∈Q0 M
Rep(Q,α) = Hom(V(ia),V(ta)).
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assume
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terior
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to

graph)
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v
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w
h
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in
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to
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By
8Y Y
GL(α) = GL(V(s)) and SL(α) = SL(V(s).
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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.
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e
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p
the
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follo
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vious
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Lemma
admitting
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
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the
and
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olyhedral.
b
that
generated

of
ter

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ex
and
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eigh
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phism
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of
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oups.
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e
tained
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of
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.
denote

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p
.
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if
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and
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p
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ositive.
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on
2

allo
ties
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b

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ts
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ed
olytop
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on
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algebra
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y

line
imply
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that
naturally
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a
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a
3.2
the
.
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group

iden
v
with
ex
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rational
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and
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e
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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
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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
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ws:
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umeration
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subset
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p
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;
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But
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.
b
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osition
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e
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follo
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1
w
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one
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that
a
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follo
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:
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in
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)

The
an
p
of
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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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of
ws
the
em
prop
ed
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is
is
than
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index

for
application
that
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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v
(iii)
space
The
.

viously
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onding
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ertex
ate
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d
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olytop
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denote
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ed
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t
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y

homothet
W
is
as
a
p
is
oin
follo
3.2.3
10