In mathematics, Castelnuovo's contraction theorem is used in the classification theory of algebraic surfaces to construct the minimal model of a given smooth algebraic surface.
More precisely, let
X
be a smooth projective surface over
C
and
C
a (−1)-curve on
X
(which means a smooth rational curve of self-intersection number −1), then there exists a morphism from
X
to another smooth projective surface
Y
such that the curve
C
has been contracted to one point
P
, and moreover this morphism is an isomorphism outside
C
(i.e.,
X
∖
C
is isomorphic with
Y
∖
P
).
This contraction morphism is sometimes called a blowdown, which is the inverse operation of blowup. We also call curve
C
exceptional curve of the first kind.
