6 JEFFREY D. ACHTER
The proof of Lemma 3.3 is modeled closely on [17, Prop. 4.5], and we focus on
3.3. Let A be a hyperplane arrangement in an n-dimensional vector
space V over an algebraically closed field k, and let¢: Y
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,
Since m is invertible in
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)
proof is by induction on dim V.
so that x(Y, Qt)
m · (1 -
1)) :::; m · (1
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
generic with respect to A, such that the pullback YH
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),
By Lemma 3.2, the first term on the right-hand side is simply m · hn(M(A),
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)
Taken together, this shows that a(Y,
m · a(M(A),
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