Smooth morphism

In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, a <a href="/facts/Morphism_of_schemes/55CKqHHo">morphism</a> 
 
 
 
 f
 :
 X
 →
 S
 
 
 {\displaystyle f:X\to S}
 
 between <a href="/facts/Scheme_(mathematics)/c7uSqGqL">schemes</a> is said to be smooth if

<ul><li>(i) it is <a href="/facts/Finitely_presented_scheme/eh7NLhmS">locally of finite presentation</a></li>
<li>(ii) it is <a href="/facts/Flat_morphism/CLM7e8d6">flat</a>, and</li>
<li>(iii) for every <a href="/facts/Geometric_point/eh7NLhmS">geometric point</a> 
 
 
 
 
 
 s
 ¯
 
 
 →
 S
 
 
 {\displaystyle {\overline {s}}\to S}
 
 the <a href="/facts/Fiber_(mathematics)/rL3pwFwa">fiber</a> 
 
 
 
 
 X
 
 
 s
 ¯
 
 
 
 =
 X
 
 ×
 
 S
 
 
 
 
 s
 ¯
 
 
 
 
 {\displaystyle X_{\overline {s}}=X\times _{S}{\overline {s}}}
 
 is regular.</li></ul>
(iii) means that each geometric fiber of f is a <a href="/facts/Smooth_scheme/Y76o7b9T">nonsingular variety</a> (if it is <a href="/facts/Separated_morphism/0BjhsmC7">separated</a>). Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties.
If S is the <a href="/facts/Spectrum_of_a_ring/2k2nDE5w">spectrum</a> of an <a href="/facts/Algebraically_closed/HuVQClok">algebraically closed</a> <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> and f is <a href="/facts/Morphism_of_finite_type/djrynAmJ">of finite type</a>, then one recovers the definition of a nonsingular variety.
A singular variety is called smoothable if it can be put in a flat family so that the nearby fibers are all smooth. Such a family is called a smoothning of the variety.

Smooth morphism open-in-new

Smooth morphism