We classify smooth complex projective varieties
X
⊂
P
N
X\subset \mathbb {P}^N
of dimension
n
≥
2
n\geq 2
admitting a divisor of the form
A
+
B
A+B
among their hyperplane sections, both
A
A
and
B
B
of codimension
≤
1
\leq 1
in their respective linear spans. In this setting, one of the following holds: 1)
X
X
is either the Veronese surface in
P
5
\mathbb {P}^5
or its general projection to
P
4
\mathbb {P}^4
, 2)
n
≤
3
n\leq 3
and
X
⊂
P
n
+
2
X\subset \mathbb {P}^{n+2}
is contained in a quadric cone of rank
3
3
or
4
4
, 3)
n
=
2
n=2
and
X
⊂
P
3
X\subset \mathbb {P}^3
.