∗
G
X
∗
N.
Ressa
yre
ers
.
.
.
June
.
8,
.
2009
quiv
.
In
pairs
this
theorem
w
ork,
.
w
.
e
.
impro
.
v
.
e
.
results
.
of
.
[Res07
ellier
,
Cedex
Res08a
.
℄
4
ab
8
out
.
.
asso
8
.
to
.
the
.
action
9
of
.
a
.
.
e
3.4
group
.
quiv
.
on
Univ
a
t
pro
34095
e
.
v
.
ariet
.
y
Application
and
represen
.
Denitions
.
.
.
.
These
.
results
.
are
.
applied
.
to
.
giv
Three
e
.
a
.
short
.
pro
.
of
.
of
.
the
.
Derksen
Dominan
W
.
eyman
.
theorem
.
whic
.
h
.
parametrizes
.
.
ely
DerksenW
the
.
.
of
.
a
.
rational
15
Mon
asso
I,
Math?matiques,
to
Eug?ne
an
tp
y
F
quiv
.
er
.
without
.
orien
.
ted
.
.
An
.
imp
3
ortan
to
t
er
example
tations
of
3.1
.
h
.
a
.
.
is
.
the
.
Horn
.
.
Con
.
ten
.
ts
.
1
.
In
.
tro
.
.
2
3.2
2
W
.
ell
.
.
v
.
ering
.
pairs
.
and
.
.
3
.
2.1
.
W
.
ell
.
.
v
.
ering
3.3
pairs
t
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
11
.
The
.
eyman
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3
.
2.2
T
ersit?
otal
tp
I
and
D?partemen
w
de
ell
Case
051Place
v
Bataillon,
ering
Mon
pair
ellier
.
5,
.
rance.
.
1C Q = (Q ,Q )0 1
Q Q0 1
β = (β(s)) Q Rep(Q,β)s∈Q0
β
Q
GL(β) = GL(β(s)) Rep(Q,β)s∈Q0
Q0Γ GL(β) Z
Σ(Q,β) Γ⊗Q σ∈ Γ
f ∈C[Rep(Q,β)] g.f =σ(g)f
g ∈ GL(β)
Σ(Q,β)
Σ(Q,β)
Σ(Q,β)
Rep(Q,β)
group
of
of
ma
v
v
the
ey
as
of
the
space
use
ector
the
v
as
the
Res07
e
the
;
pro
it
The
is
ts
isomorphic
on
to
in
b
enligh
and
The
of
group
.
w
W
GIT
e
w
of
the
see
eyman
dimension
hnical
ector
of
v
that
a
role.
e
the
b
ts
in
w
Let
yself
ws.
of
arro
h
generated
studied
b
also
y
action
the
elemen
oin
ts
[BS00
of
in
set
the
the
quiv
to
h
v
that
o
there
not
exists
et
a
here
non
ob
zero
whic
regular
w
function
the
and
more
ertexes
fundamen
v
℄
of
pla
set
w
the
that
is
v
Here,
acts
the
ted
t.
orien
v
without
for
er
see
represen
h
v
that
phenomenons
quiv
not
a
the
e
[Res07,
b
Let
e
ers.
for
b
the
um
a
n
for
This
an
of
y
used
BK06
of
℄
eld
literature
the
h
er
problem
v
from
o
one,
.
in
It
w
is
are
a
useful
t
v
hes.
ex
b
p
easy
olyhedral
dierences
een
In
presen
[D
the
W00,
The
D
one
W06],
text
Derksen
the
W
of
eyman
is
sho
guess
w
ed
mak
that
the
the
A
Horn
dierence
and
p
b
the
e
a
obtained
v
in
do
the
h
base
a
of
w
t
a
on
y
group
from
w
impro
ell
emen
It
hosen
as
quiv
ery
er
teresting
and
m
v
to
ector
that
dimension.
example
This
tations
is
dimension
an
ector
imp
tens
ortan
whic
t
did
motiv
o
ation
in
to
example
the
in
study
℄
ork
Horn
w
e
b
W
obtained
tro
the
.
of
Here,
linear
w
on
e
pro
use
of
general
ag
metho
arieties.
ds
p
of
t
view
In
as
v
in
arian
,
t
,
Theory
,
to
Whereas,
giv
the
e
this
a
pro
of
of
Horn
of
w
the
DerksenW
the
eyman
er
theorem
this
(see
ork
[D
tends
W06
sho
℄
that
whic
h
a
parametrizes
ery
generalization
ely
these
the
w
of
It
In
y
as
e
so
e
to
W
the
.
b
.
w
In
the
of
2,
ted
w
and
e
DerksenW
impro
one.
v
most
e
vious
results
is
of
[Res07
in
℄
h
ab
out
part
the
in
ork
general.
made.
In
author
particular,
that
Theorem
general
1
text
is
GIT
an
es
impro
v
k
emen
argues.
t
most
of
tal
[Res07,
is
Theorem
here
4],
in
and
the
Theorem
oin
2
outside
of
[Res07,
y
Theorem
7].
Moreo
After
er,
e
h
not
statemen
here
t
the
a
the
of
kno
space
The
semiin
ulton
arian
ab
functions
the
.
o
naturally
The
w
e
is
ha
wn.
v
F
e
out
a
Littlew
remark
o
to
hardson
mak
e
is
1
2ν Nνc = 1⇒c = 1 ∀N ≥ 1.λμ NλNμ
α◦β = 1 ⇒ α◦ (Nβ) = 1
G
λX λ G G
λ G P(λ) λ
λG
n o
−1P(λ) = g∈G : limλ(t).g.λ(t) G .
t→0
λC X λ X
+C C
+C ={x∈X  limλ(t)x∈C}.
t→0
λ +C G C P(λ)
+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
of
action
3
pair
the
that
tersecting
on
.
2
action
also
of
pair
y
ws
b
W
stable
is
allo
.
bration
Consider
used.
o
W06
v
to
er
to
Then,
op
:
ering
to
er,
pairs
the
arian
action
v
of
d
asso
a
Theorem
kiBirula
arian
Bia?ynic
this
the
v
giv
.
en
if
b
generalization
y
l
the
o
form
of
ula
Let
(with
ob
Moreo
vious
rst
notation):
also
v
e
ell
W
map
.
this
in
to
of
in
set
the
t
h
oin
p
b
xed
.
the
of
map
t
is
onen
er.
and
an
in
e
Denition.
b
generalization
Let
is
in
e
exists
is.
:
quiv
.
is
Consider
e
the
overing
quotien
an
t
er
subgroup
subset
Levi
e
with
b
to
pairs
v
asso
ell
subgroup
.
olic
v
of
the
parab
pro
usual
2.1
the
and
ering
b
y
a
the
equiv
action
t
of
W
e
result.
W
e
.
repro
in
w
of
℄
tralizer
used
metho
the
and
.
whic
The
is
lo
of
trivial
a
with
pair
er
denote
trast,
Let
Consider
.
the
of
equiv
subgroup
t
parameter
In
one
not
a
result
e
Here,
b
pap
in
for
Let
is
.
I
y
℄
ariet
[D
v
ed
e
pro
The
pro
is
is
This
denoted
is
b
said
y
b
oth
dominant
smo
setting
a
The
on
er
acting
to
.
The
The
said
action
b
of
wel
group
e
if
isomorphism
a
v
an
an
en
action
of
b
in
follo
y
the
3GL∈ Pic (X) x C λ x
∗K Lx
Lμ (x,λ)
L∗ −μ (x,λ)∀x˜∈L ∀t∈C λ(t)x˜ =t x˜.x
Lμ (x,λ) x∈C
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)
GTC (X) (C,λ)
λGPic (C) F(C)
λ λG G GPic (X) TC (C) Pic (C)Q Q
GPic (X) /Span(F(C))Q
λ λG GPic (C) /Span(TC (C))Q
oin
p
y
a
the
an
the
e
in
if
v
b
is
Let
whic
.
.
dene
ex
om
.
of
one
interse
),
Theorem
in
v
teger
et
.
the
Let
in
e
.
ositiv
r
p
y
y
If
b
b
e
a
The
dominan
emen
t
W
pair.
ends
Since
nite
an
will
(for
's
.
overing
Since
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
is
If
not
is
a
group,
pro
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
ering
pair
e
d
.
w
W
theorem,
e
Kempf
the
this
mainly
statemen
using
dep
,
on
to
e
elong
w
b
obtain
not
.
has
of
non
Theorem
empt
a
y
of
in
Theorem
terior;
b
and
y
so,
asserts
this
one
impro
has
v
w
emen
Ther
t
b
is
useful
not
y
useful.
assertion
W
impro
e
that
will
Let
see
is
that
a
this
Let
prop
7]
ert
.
y
is
of
that
es
interior
do
Let
h
.
whic
that
ample
an
y
gener
an
tify
explains
ample
the
the
role
In
of
an
the
v
Sc
h
e
ur
of
ro
is
ots
start
relativ
a
ely
to
The
the
es
to
o
(ii)
is
ell
empt
w
b
ery
torus
ev
and
.
b
Let
mal
for
an
b
asserts
e
subtorus
a
h
simple
[Res07
group
orthogonal
and
The
℄
for
,
.
in
pro
and
isogen
as
non
three
[Res07,
parab
(ii)
olic
subgroup
subgroups
of
)
pro
.
isogen
Theorem
2
d
asks
No
for
an
non
algorithm
for
to
air
if
asso
need
taining
w
group
exists
e
Then,
t
Theorem
a
L
is
a
of
in
the
(here
to
℄
on
a
.
w
e
ell
b
has
non
kind
empt
pro
y
from
in
rst
terior.
t.
In
this
t
v
yp
on
e
2
A,
of
Prop
Assertion
osition
4
It
giv
y
es
.
a
not
in
e
terpretation
maximal
of
of
v
(and
in
ending
terms
e
of
maxi
the
torus
y
.
osition
that
of
b
a
the
v
of
ector
dimension
that
of
,
a
Remark.
quiv
is
er.
with
In
).
[D
W02
also
℄
denoted
DerksenW
in
eyman
Note
giv
the
es
an
an
algorithm
y
whic
empty
h
has
answ
.
ers
.
the
Theorem
question.
b
In
the
general,
of
it
7]
seems
taining
to
that
b
h
e
the
unkno
wn.
an
Pro
y
of.
y
Let
of
ering
pair
ate
dened
.
up
w,
to
e
if
iden
and
de
,
a
equals
A
some
inequalities.
.
useful
p
b
with
e
asso
an
that
asso
Here,
e
w
.
ell
particular,
e
v
a
ering
map:
pair.
impro
Let
e
un
(i)
elimenate
.
denote
emen
the
of
neutral
result.
that
onen
2
t
et
of
e
the
k
ample
ernel
of
of
Consider
action
the
set
question
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
empt
y
in
assume
w
terior
is
in
to
graph)
assertion.
represen
Let
in
.
one
One
and
w
nd
ws
a
to
neigh
terior
b
and
or
erplane
in
nd
terior
the
of
Let
in
y
tains
h
;
that
are
y
An
empt
initial
non
the
has
in
denote
The
do
dimensional
es
linear
not
b
b
elong
b
to
dimension
the
,
image
and
of
other
.
.
Note
space
that
non
that
that
ws
ample
sho
assertion
1
This
Theorem
v
;
and
and,
and
w
h
b
that
ertex
terminal
this
.
in
.
But,
a
.
in
the
as
ab
of
o
ector
in
family
v
of
e
to
pro
h
of
that
of
of
Asser
b
tion
neigh
.
.
By
ector
[Res07,
the
Lemma
half
℄
(ii)
also
b
w
orks
x
of
.
terior
v
in
terior
.
dimension
Let
Lemma
the
℄
to
.
b
set
elongs
that
b
Lemma
elongs
the
if
that
b
ted
e
with
the
ertexes
pair
e
asso
arro
w
to
.
to
arro
.
rst
.
that
Up
has
to
v
elongs
,
and
Let
one
us
Then,
is
A
tation
of
tained
is
in
family
that
of
.
.
No
rst
assume
sertion
w
has
nite
to
v
restart
spaces
the
a
pro
of
of
maps
with
terior
no
the
e
elongs
.
The
The
yp
pro
h
will
in
nish
t
since
or
W
indexed
y
is
a
The
tained
v
in
of
preceding
is
.
family
the
o
3
spaces,
Application
one
to
tains
quiv
the
er
of
represen
elongs
tations
the
3.1
one
Denitions
.
In
us
this
Since
tains
w
[Res07,
e
a
x
ector
some
in
in
notation
of
ab
empt
out
has
quiv
for
er
h
represen
asserts
tations.
Let
Set
in
the
b
of
the
[Res07,
assume
e
no
ws.
e
℄
a
quiv
W
er
follo
(that
rst
is,
and,
a
h
nite
.
orien
w,
implies
one
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
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
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
W
if
also
and
is
only
ted
if
Let
bundle
is
acts
p
h
ositive.
y
Lemma
on
2
allo
ties
ws
;
to
w
em
W
b
ed
it
.
in
Consider
ed
.
an
The
action
form
ample,
ula
b
the
em
zero
e
of
in
w
W
ts
.
its
,
on
all
y
.
bundle
Set
y
for
Since
that
b
h
One
easily
of
v
set
that
the
denote
on
e
and
w
set
in
,
,
ex
or
ull
F
the
and
Finally
the
asso
with
the
or
terested
F
is
.
rational
ed
olytop
w
in
.
no
General
Moreo
prop
er,
erties
e
of
.
endo
on
the
functions
They
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
of
a
vious
is
is
Since
of
.
is
ws:
tication
terested
W
e
e
ell
in
umeration
tained
Mor
Lemma
.
Then,
acting
is
Ob
e
as
w
an
are
op
(resp.
en
ering
implies
the
This
r
.
subset
ws
.
ertexes
Since
isomorphism
er
that
is
only
the
to
only
the
Set
v
subgroup
o
p
of
ector
in
to
er
a
orbit
.
in
the
b
y
the
an
on
w
trivially
the
act
has
to
Let
has
.
ter
v
,
wing
3
its
map
particular
a
in
;
n
b
it
y
the
;
tral
and
is
w
gr
y
over,
b
that
xed
if
is
index
e
allo
But
b
of
greater
all
of
as
in
t.
easily
So,
.
.
one
iden
.
for
t
semistable
isolated
is
the
tify
Then,
,
v
tains
oin
only
p
one
p
the
orbit
Lemma
.
b
whic
.
h
is
and
e
tained
no
in
in
the
in
the
Since
no
t
ted
with
pair.
.
v
Prop
W
osition
ha
.
e
W
follo
e
ob
Lemma
that
The
Let
estriction
.
1
w
(i)
is
W
one
and
that
a
e
that
have:
follo
.
:
(ii)
of
to
v
in
asso
)
The
an
p
of
oint
oups.
is
e
a
h
vertex
and
(resp.
the
of
ample
.
and
The
if
last
is.
assertion
3
of
ws
the
em
prop
ed
osition
is
is
than
a
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
space
v
(iii)
space
The
.
viously
one
onding
of
the
gener
ertex
ate
t
d
is
of
rational
olytop
the
in
denote
The
)
Via
endo
iden
w
of
ed
3,
with
relation
the
et
by
een
is
b
.
t
Pro
of.
y
homothet
W
is
as
a
p
is
oin
follo
3.2.3
10