In algebraic geometry, the Zariski closure of the union of the secant lines to a projective variety
X
⊂
P
n
is the first **secant variety** to
X
. It is usually denoted
Σ
1
.

The **
k
t
h
secant variety** is the Zariski closure of the union of the linear spaces spanned by collections of k+1 points on
X
. It is usually denoted
Σ
k
. Unless
Σ
k
=
P
n
, it is always singular along
Σ
k
−
1
, but may have other singular points.

If
X
has dimension d, the dimension of
Σ
k
is at most kd+d+k.