Smooth morphism

In algebraic geometry, a morphism ${\displaystyle f:X\to S}$ between schemes is said to be smooth if

• (i) it is locally of finite presentation
• (ii) it is flat, and
• (iii) for every geometric point ${\displaystyle {\overline {s}}\to S}$ the fiber ${\displaystyle X_{\overline {s}}=X\times _{S}{\overline {s}}}$ is regular.

(iii) means that each geometric fiber of f is a nonsingular variety (if it is separated). Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties.

If S is the spectrum of an algebraically closed field and f is of finite type, 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.