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A closed subvariety  $  Y $
of an [[Algebraic variety|algebraic variety]]  $  X $, 
defined over an algebraically closed field, that can be mapped onto a subvariety  $  Y _ {1} $
of lesser dimension by a proper [[Birational morphism|birational morphism]]  $  f :  X \rightarrow X _ {1} $
such that  $  f :  X \setminus  Y \rightarrow X _ {1} \setminus  f ( X) $
is an isomorphism (cf. also [[Proper morphism|Proper morphism]]). The morphism  $  f $
is called a contraction of  $  Y $
onto  $  Y _ {1} = f ( Y) $; 
this concept is a particular case of that of a [[Modification|modification]] of algebraic spaces [[#References|[3]]]. If  $  X $, 
$  X _ {1} $, 
$  Y $, 
$  Y _ {1} $
are smooth irreducible varieties,  $  Y $
is called an exceptional subvariety of the first kind. If  $  Y $
has codimension 1 in  $  X $, 
it is an exceptional divisor. Exceptional divisors on an algebraic surface are called exceptional curves.

The notion of an exceptional subvariety can be naturally generalized to schemes, algebraic and complex-analytic spaces. The corresponding morphisms are called contractions; the notion of an exceptional subvariety of the first kind can also be naturally generalized. An exceptional subvariety in a complex-analytic space is also called an [[Exceptional analytic set|exceptional analytic set]].

Characterizing the exceptional subvarieties within the ambient variety is one of the basic problems in birational geometry. Historically, the first example of such a characterization is the Enriques–Castelnuovo criterion: An irreducible complete curve  $  Y $
on a smooth surface  $  X $
is an exceptional subvariety of the first kind if and only if it is isomorphic to the projective line  $  P  ^ {1} $
and if its self-intersection index  $  ( Y \cdot Y ) $
on  $  X $
is equal to  $  - 1 $(
cf. [[#References|[1]]], [[#References|[9]]]). This criterion can be generalized to one-dimensional subschemes of a two-dimensional regular scheme (cf. [[#References|[6]]], [[#References|[10]]]). If  $  Y = \sum_{i=1}^ {m} Y _ {i} $
is an arbitrary connected complete curve with irreducible components  $  Y _ {i} $
on a smooth projective surface  $  X $, 
then a necessary (but not sufficient) condition for  $  Y $
to be exceptional is that the matrix  $  ( Y _ {i} \cdot Y _ {j} ) $
is negative definite (cf. [[#References|[2]]]). In the case of a connected complex curve on a smooth complex surface, or a connected complete curve on a smooth two-dimensional algebraic space, the analogous condition is necessary and sufficient for exceptionality.

The multi-dimensional analogue of the Enriques–Castelnuovo criterion for contractions to a point has the following form [[#References|[5]]]: An irreducible complete subvariety  $  Y $
in a smooth algebraic variety  $  X $
is exceptional of the first kind relative to a contraction to a point if the following two conditions hold: a)  $  Y \cong P  ^ {r} $, 
where  $  r =  \mathop{\rm dim}  X - 1 $; 
and b) the normal bundle  $  N _ {Y / X }  $
to  $  Y $
in  $  X $
is defined by a divisor  $  - H $, 
where  $  H $
is a hyperplane in  $  P  ^ {r} $. 
In this case  $  X _ {1} $
is projective. The corresponding contraction  $  f $
is a [[monoidal transformation]] with centre at the point  $  f ( Y) $(
cf. [[#References|[7]]], [[#References|[8]]]).

In the analytic case, necessary and sufficient conditions for a connected compact complex submanifold  $  Y $
in a complex manifold  $  X $
to be exceptional of the first kind have been found; the corresponding contraction  $  f $
must be a monoidal transformation with centre at  $  Y _ {1} = f ( Y) $(
cf. [[Exceptional analytic set|Exceptional analytic set]]). For algebraic varieties the corresponding conditions are necessary, but not always sufficient.

For a contraction  $  f :  X \rightarrow X _ {1} $
of an exceptional subvariety of the first kind  $  Y $
in a projective algebraic variety  $  X $
onto a zero-dimensional subvariety  $  Y _ {1} $
in an algebraic variety  $  X _ {1} $, 
$  X _ {1} $
need not be projective. Moreover, if the algebraic varieties  $  X $
and  $  Y $
are defined over the field of complex numbers, under an analytic contraction  $  f $
of the exceptional subvariety of the first kind  $  Y $, 
the variety  $  X _ {1} $
need not be algebraic at any point, in general.

Regarding the question of contractibility of an exceptional subvariety (not necessarily of the first kind) to a point, a necessary condition of exceptionality of a complete connected algebraic subspace  $  Y $
in a smooth algebraic space  $  X $
is that the normal bundle  $  N _ {Y / X }  $
is negative (this condition is not sufficient for  $   \mathop{\rm dim}  X > 2 $). 
The analogous fact holds for complex spaces.

In the case of algebraic spaces, the most general criterion of exceptionality states that in the category of Noetherian algebraic spaces a subspace  $  Y $
in  $  X $
is an exceptional subvariety if and only if the formal completion  $  \widehat{Y}  $
in  $  \widehat{X}  $
is an exceptional subvariety in the category of formal algebraic spaces [[#References|[3]]]. In other words, contraction of algebraic subspaces is possible if and only if it is possible for the corresponding formal completions.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> "Algebraic surfaces" ''Proc. Steklov Inst. Math.'' , '''75''' (1967) ''Trudy Mat. Inst. Steklov.'' , '''75''' (1975) {{MR|}} {{ZBL|0172.37901}} {{ZBL|0153.22401}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> M. Artin, "Some numerical criteria for contractability of curves on algebraic surfaces" ''Amer. J. Math.'' , '''84''' (1962) pp. 485–496 {{MR|0146182}} {{ZBL|0105.14404}} </TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top"> M. Artin, "Algebraization of formal moduli. II Existence of modifications" ''Ann. of Math.'' , '''91''' : 1 (1970) pp. 88–135 {{MR|0260747}} {{ZBL|0185.24701}} {{ZBL|0177.49003}} </TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top"> H. Grauert, "Ueber Modificationen und exceptionelle analytische Mengen" ''Math. Ann.'' , '''146''' (1962) pp. 331–368</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top"> K. Kodaira, "On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties)" ''Ann. of Math.'' , '''60''' (1954) pp. 28–48</TD></TR>
<TR><TD valign="top">[6]</TD> <TD valign="top"> S. Lichtenbaum, "Curves over discrete valuation rings" ''Amer. J. Math.'' , '''90''' : 2 (1968) pp. 380–405 {{MR|0230724}} {{ZBL|0194.22101}} </TD></TR>
<TR><TD valign="top">[7]</TD> <TD valign="top"> S. Nakano, "On the inverse of monoidal transformations" ''Publ. Res. Inst. Math. Sci.'' , '''6''' : 3 (1971) pp. 483–502</TD></TR>
<TR><TD valign="top">[8]</TD> <TD valign="top"> A. Fujiki, S. Nakano, "Supplement to "On the inverse of monoidal transformations" " ''Publ. Res. Inst. Math. Sci.'' , '''7''' : 3 (1972) pp. 637–644 {{MR|0294712}} {{ZBL|0234.32019}} </TD></TR>
<TR><TD valign="top">[9]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> I.R. Shafarevich, "Lectures on minimal models and birational transformations of two-dimensional schemes" , Tata Inst. (1966) {{MR|0217068}} {{ZBL|0164.51704}} </TD></TR></table>