In this paper, we prove that if
X
⊂
P
n
X\subset \mathbb {P}^n
,
n
≥
4
n\ge 4
, is a locally complete intersection of pure codimension
2
2
and defined scheme-theoretically by three hypersurfaces of degrees
d
1
≥
d
2
≥
d
3
d_1\ge d_2\ge d_3
, then
H
1
(
P
n
,
I
X
(
j
)
)
=
0
H^1(\mathbb {P}^n,\mathcal {I}_X(j))=0
for
j
>
d
3
j>d_3
using liaison theory and the Arapura vanishing theorem for singular varieties. As a corollary, a smooth threefold
X
⊂
P
5
X\subset \mathbb {P}^5
is projectively normal if
X
X
is defined by three quintic hypersurfaces.