Proof.
We know by Corollary 2.4 that Ψ
Γ
is a sum of monomials with
coefficients +1. The lemma is clear for the zero set of any polynomial
with coefficients
>
0.
/
GRAPH POLYNOMIALS
19
Remark 7.2.
(i) The assertion of the lemma is true for any graph
polynomial.
We do not need hypotheses about numbers of edges or
loops.
(ii) By Proposition 3.1, coordinate linear spaces
L
⊂
X
Γ
correspond to
subgraphs
G
⊂
Γ such that
h
1
(
G
)
>
0.
Proposition 7.3.
Let
Γ
be as above. Define
(7.2)
η
=
η
Γ
=
Ω
2
n

1
(
A
)
Ψ
2
Γ
as in
(5.2)
. There exists a tower
P
=
P
r
π
r,r

1
→
P
r

1
π
r

1
,r

2
→
. . .
π
2
,
1
→
P
1
π
1
,
0
→
P
2
n

1
;
(7.3)
π
=
π
1
,
0
◦ · · · ◦
π
r,r

1
where
P
i
is obtained from
P
i

1
by blowing up the strict transform of a
coordinate linear space
L
i
⊂
X
Γ
and such that
(i)
π
*
η
Γ
has no poles along the exceptional divisors associated to the
blowups.
(ii) Let
B
⊂
P
be the total transform in
P
of the union of coordinate
hyperplanes
Δ
2
n

2
:
A
1
A
2
·
A
2
n
= 0
in
P
2
n

1
.
Then
B
is a normal
crossings divisor in
P
. No face (= nonempty intersection of compo
nents) of
B
is contained in the strict transform
Y
of
X
Γ
in
P
.
(iii) the strict transform of
σ
2
n

1
(
R
)
in
P
does not meet
Y
.
Proof.
Our algorithm to construct the blowups will be the following.
Let
S
denote the set of coordinate linear spaces
L
⊂
P
2
n

1
which are
maximal, i.e.
L
∈
S, L
⊂
L
⊂
X
Γ
⇒
L
=
L
. Define
(7.4)
F
=
{
L
⊂
X
Γ
coordinate linear space

L
=
\
L
(
i
)
, L
(
i
)
∈
S
}
.
Let
F
min
⊂ F
be the set of minimal elements in
F
. Note that elements
of
F
min
are disjoint. Define
P
1
π
1
,
0
→
P
2
n

1
to be the blowup of elements
of
F
min
. Now define
F
1
to be the collection of strict transforms in
P
1
of elements in
F \ F
min
. Again elements in
F
1
,
min
are disjoint, and we
define
P
2
by blowing up elements in
F
1
,
min
. Then
F
2
is the set of strict
transforms in
P
2
of
F
1
\ F
1
,
min
, etc. This process clearly terminates.
Note that to pass from
P
i
to
P
i
+1
we blow up strict transforms of
coordinate linear spaces
L
contained in
X
Γ
. There will exist an open
set
U
⊂
P
2
n

1
such that
P
i
×
P
2
n

1
U
∼
=
U
and such that
L
∩
U
=
∅
.
It follows that to calculate the pole orders of
π
*
η
Γ
along exceptional
divisors arising in the course of our algorithm it suffices to consider
the simple blowup of a coordinate linear space
L
⊂
X
Γ
on
P
2
n

1
.
Suppose
L
:
A
1
=
. . . A
p
= 0.
By assumption, the subgraph
G
=
{
e
1
, . . . , e
p
} ⊂
Γ is convergent, i.e.
p >
2
h
1
(
G
).
As in Proposition
20
SPENCER BLOCH, H
´
EL
`
ENE ESNAULT, AND DIRK KREIMER
3.5, if
I
= (
A
1
, . . . , A
p
)
⊂
K
[
A
1
, . . . , A
2
n
], then Ψ
Γ
∈
I
h
1
(
G
)

I
h
1
(
G
)+1
so the denominator of
η
Γ
contributes a pole of order 2
h
1
(
G
) along the
exceptional divisor.
On the other hand, writing
a
i
=
A
i
A
2
n
, a typical
open in the blowup will have coordinates
a
i
=
a
i
a
p
, i < p
together with
a
p
, . . . , a
2
n

1
and the exceptional divisor will be defined by
a
p
= 0.
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 Spring '13
 MatildeMarcolli
 Linear Algebra, Polynomials, Vector Space, subgraph, SPENCER BLOCH, DIRK KREIMER, Graph Hypersurfaces