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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