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A closed subvariety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e0367801.png" /> of an [[Algebraic variety|algebraic variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e0367802.png" />, defined over an algebraically closed field, that can be mapped onto a subvariety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e0367803.png" /> of lesser dimension by a proper [[Birational morphism|birational morphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e0367804.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e0367805.png" /> is an isomorphism (cf. also [[Proper morphism|Proper morphism]]). The morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e0367806.png" /> is called a contraction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e0367807.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e0367808.png" />; this concept is a particular case of that of a [[Modification|modification]] of algebraic spaces [[#References|[3]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e0367809.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678012.png" /> are smooth irreducible varieties, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678013.png" /> is called an exceptional subvariety of the first kind. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678014.png" /> has codimension 1 in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678015.png" />, it is an exceptional divisor. Exceptional divisors on an algebraic surface are called exceptional curves.
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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]].
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678016.png" /> on a smooth surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678017.png" /> is an exceptional subvariety of the first kind if and only if it is isomorphic to the projective line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678018.png" /> and if its self-intersection index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678019.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678020.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678021.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678022.png" /> is an arbitrary connected complete curve with irreducible components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678023.png" /> on a smooth projective surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678024.png" />, then a necessary (but not sufficient) condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678025.png" /> to be exceptional is that the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678026.png" /> 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.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678027.png" /> in a smooth algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678028.png" /> is exceptional of the first kind relative to a contraction to a point if the following two conditions hold: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678030.png" />; and b) the normal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678031.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678032.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678033.png" /> is defined by a divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678034.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678035.png" /> is a hyperplane in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678036.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678037.png" /> is projective. The corresponding contraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678038.png" /> is a [[Monoidal transformation|monoidal transformation]] with centre at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678039.png" /> (cf. [[#References|[7]]], [[#References|[8]]]).
+
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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678040.png" /> in a complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678041.png" /> to be exceptional of the first kind have been found; the corresponding contraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678042.png" /> must be a monoidal transformation with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678043.png" /> (cf. [[Exceptional analytic set|Exceptional analytic set]]). For algebraic varieties the corresponding conditions are necessary, but not always sufficient.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678044.png" /> of an exceptional subvariety of the first kind <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678045.png" /> in a projective algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678046.png" /> onto a zero-dimensional subvariety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678047.png" /> in an algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678049.png" /> need not be projective. Moreover, if the algebraic varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678051.png" /> are defined over the field of complex numbers, under an analytic contraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678052.png" /> of the exceptional subvariety of the first kind <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678053.png" />, the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678054.png" /> need not be algebraic at any point, in general.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678055.png" /> in a smooth algebraic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678056.png" /> is that the normal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678057.png" /> is negative (this condition is not sufficient for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678058.png" />). The analogous fact holds for complex spaces.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678059.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678060.png" /> is an exceptional subvariety if and only if the formal completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678061.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036780/e03678062.png" /> 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.
+
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====
 
====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. Kodairae, "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 monodial 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 monodial 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>
 
<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. Kodairae, "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 monodial 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 monodial 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>

Revision as of 19:38, 5 June 2020


A closed subvariety $ Y $ of an 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 $ f : X \rightarrow X _ {1} $ such that $ f : X \setminus Y \rightarrow X _ {1} \setminus f ( X) $ is an isomorphism (cf. also 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 of algebraic spaces [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.

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. [1], [9]). This criterion can be generalized to one-dimensional subschemes of a two-dimensional regular scheme (cf. [6], [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. [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 [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. [7], [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). 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 [3]. In other words, contraction of algebraic subspaces is possible if and only if it is possible for the corresponding formal completions.

References

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[4] H. Grauert, "Ueber Modificationen und exceptionelle analytische Mengen" Math. Ann. , 146 (1962) pp. 331–368
[5] K. Kodairae, "On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties)" Ann. of Math. , 60 (1954) pp. 28–48
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[7] S. Nakano, "On the inverse of monodial transformations" Publ. Res. Inst. Math. Sci. , 6 : 3 (1971) pp. 483–502
[8] A. Fujiki, S. Nakano, "Supplement to "On the inverse of monodial transformations" " Publ. Res. Inst. Math. Sci. , 7 : 3 (1972) pp. 637–644 MR0294712 Zbl 0234.32019
[9] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[10] I.R. Shafarevich, "Lectures on minimal models and birational transformations of two-dimensional schemes" , Tata Inst. (1966) MR0217068 Zbl 0164.51704
How to Cite This Entry:
Exceptional subvariety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Exceptional_subvariety&oldid=46869
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article