6 JEFFREY D. ACHTER
The proof of Lemma 3.3 is modeled closely on [17, Prop. 4.5], and we focus on
the differences.
LEMMA
3.3. Let A be a hyperplane arrangement in an n-dimensional vector
space V over an algebraically closed field k, and let¢: Y
-t
M(A) be an irreducible
etale Galois cover of degree m which is invertible ink. Then
o"c(Y) :::; mo"c(A).
PROOF. By Poincare duality, it suffices to prove the analogous result for the
sum of Betti numbers a(Y,
Q£).
Since m is invertible in
k,
the cover ¢ is tamely
ramified on the boundary of M(A) in V. A result of Deligne and Lusztig [14, 2.6,
Cor. 2.8] shows that x(Y, Qt)
=
mx(M(A),
Q£).
Our
proof is by induction on dim V.
If n
=
1, let
b1
=
h
1
(M(A),
Q£).
Then
x(M(A),
Q£)
=
1 -
b1,
so that x(Y, Qt)
=
m · (1 -
b1),
and a(Y,
Q£)
=
1
+
(1
+
m(b1-
1)) :::; m · (1
+
b1)
=
ma(M(A),
Q£).
Now assume that the lemma holds for any arrangement in a vector space of
dimension n - 1. Using [17, Prop. 4.6], one may choose a hyperplane H
C
V,
generic with respect to A, such that the pullback YH
-t
M(AH) is an irreducible
Galois cover of M(AH ).
As in the proof of [17, Prop. 4.5], we find that
a(Y, Qt) :::; m (( -1tx(M(A),
Q£)
+ (
-1)n- 1x(M(AH),
Qt))
+
a(YH,
Q£).
By Lemma 3.2, the first term on the right-hand side is simply m · hn(M(A),
Q£),
while by the inductive hypothesis the other term is at most m · a(M(AH),Q£).
A second application of Lemma 3.2 shows that a(M(AH),Qt)
=
a(M(A),Qt)-
hn(M(A),
Q£).
Taken together, this shows that a(Y,
Q£) :::;
m · a(M(A),
Q£).
0
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