We prove the following:
Theorem. Let
X
⊂
P
5
X\subset \mathbf {P}^5
be a smooth, subcanonical threefold. If
h
0
(
I
X
(
4
)
)
≠
0
h^0(\mathcal {I}_X(4))\ne 0
, then
X
X
is a complete intersection.
Let
X
⊂
P
6
X\subset \mathbf {P}^6
be a smooth, codimension two subvariety, if
h
0
(
I
X
(
5
)
)
≠
0
h^0(\mathcal {I}\!_X(5))\!\ne 0
or
deg
(
X
)
≤
73
\operatorname {deg}(X)\le 73
, then
X
X
is a complete intersection.
This improves, for
5
≤
n
≤
6
5\le n\le 6
, earlier results on Hartshorne’s conjecture for codimension two subvarieties of
P
n
\mathbf {P}^n
.