Contraction morphism

In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, a contraction morphism is a surjective <a href="/facts/Projective_morphism/eh7NLhmS">projective morphism</a> 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 between normal projective varieties (or projective schemes) such that 
 
 
 
 
 f
 
 ∗
 
 
 
 
 
 O
 
 
 
 X
 
 
 =
 
 
 
 O
 
 
 
 Y
 
 
 
 
 {\displaystyle f_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{Y}}
 
 or, equivalently, the geometric fibers are all connected (<a href="/facts/Zariski%27s_connectedness_theorem/zgL5X8Ml">Zariski's connectedness theorem</a>). It is also commonly called an algebraic fiber space, as it is an analog of a <a href="/facts/Fiber_space/4g6exYHA">fiber space</a> in <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>.
By the <a href="/facts/Stein_factorization/097PaJwk">Stein factorization</a>, any surjective projective morphism is a contraction morphism followed by a finite morphism.
Examples include <a href="/facts/Ruled_surface/uAGM9b1X">ruled surfaces</a> and <a href="/facts/Mori_fiber_space/JMKJnzqw">Mori fiber spaces</a>.

Contraction morphism open-in-new

Contraction morphism