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''on an algebraic variety''
 
''on an algebraic variety''
  
The theory of intersections of algebraic subvarieties and cycles. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i0520401.png" /> be a smooth [[Algebraic variety|algebraic variety]] of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i0520402.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i0520403.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i0520404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i0520405.png" /> are subvarieties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i0520406.png" /> of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i0520407.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i0520408.png" />, respectively. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i0520409.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204010.png" /> intersect transversally, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204011.png" /> is a smooth subvariety of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204012.png" />, which is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204013.png" />. In the general case, the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204014.png" /> constitutes an [[Algebraic cycle|algebraic cycle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204015.png" /> of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204016.png" />. The idea behind this definition is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204018.png" /> are replaced by cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204020.png" /> that are equivalent in some sense but that are in general position, and one then takes the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204022.png" />; of course, the cycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204023.png" /> is also defined up to equivalence.
+
The theory of intersections of algebraic subvarieties and cycles. Let $  X $
 +
be a smooth [[Algebraic variety|algebraic variety]] of dimension $  n $
 +
over a field $  k $,  
 +
while $  Y $
 +
and $  Z $
 +
are subvarieties of $  X $
 +
of codimension i $
 +
and $  j $,  
 +
respectively. If $  Y $
 +
and $  Z $
 +
intersect transversally, then $  Y \cap Z $
 +
is a smooth subvariety of codimension i+ j $,  
 +
which is denoted by $  Y \cdot Z $.  
 +
In the general case, the pair $  ( Y, Z) $
 +
constitutes an [[Algebraic cycle|algebraic cycle]] $  Y \cdot Z $
 +
of codimension i + j $.  
 +
The idea behind this definition is that $  Y $
 +
and $  Z $
 +
are replaced by cycles $  Y  ^  \prime  $
 +
and $  Z  ^  \prime  $
 +
that are equivalent in some sense but that are in general position, and one then takes the intersection of $  Y  ^  \prime  $
 +
and $  Z  ^  \prime  $;  
 +
of course, the cycle $  Y  ^  \prime  \cdot Z  ^  \prime  $
 +
is also defined up to equivalence.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204024.png" /> be the group of classes of algebraic cycles of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204025.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204026.png" /> modulo rational equivalence; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204027.png" />. Chow's intersection theory consists of the construction:
+
Let $  A  ^ {i} ( X) $
 +
be the group of classes of algebraic cycles of codimension i $
 +
on $  X $
 +
modulo rational equivalence; let $  A( X) = \oplus _ {i \geq  0 }  A  ^ {i} ( X) $.  
 +
Chow's intersection theory consists of the construction:
  
a) of a graded commutative ring structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204028.png" /> for each smooth quasi-projective variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204029.png" />;
+
a) of a graded commutative ring structure on $  A( X) $
 +
for each smooth quasi-projective variety $  X $;
  
b) of a homomorphism of graded rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204030.png" /> for each morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204031.png" /> (inverse image); and
+
b) of a homomorphism of graded rings $  f ^ { \star } : A( Y) \rightarrow A( X) $
 +
for each morphism $  f: X \rightarrow Y $(
 +
inverse image); and
  
c) of a homomorphism of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204032.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204033.png" /> for each [[Proper morphism|proper morphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204034.png" /> (direct image).
+
c) of a homomorphism of groups $  f _  \star  : A( X) \rightarrow A( Y) $
 +
of degree $  \mathop{\rm dim}  Y -  \mathop{\rm dim}  X $
 +
for each [[Proper morphism|proper morphism]] $  f: X \rightarrow Y $(
 +
direct image).
  
 
Between the constructions a), b), c) there are a number of relations, of which the main ones are as follows:
 
Between the constructions a), b), c) there are a number of relations, of which the main ones are as follows:
  
the projection formula: For a proper morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204035.png" /> and cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204037.png" />,
+
the projection formula: For a proper morphism $  f: X \rightarrow Y $
 +
and cycles $  \alpha \in A( X) $
 +
and $  \beta \in A( Y) $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204038.png" /></td> </tr></table>
+
$$
 +
f _  \star  ( \alpha \cdot f ^ { \star } ( \beta ))  = f _  \star  ( \alpha )
 +
\cdot \beta ;
 +
$$
  
reduction to the diagonal: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204039.png" /> is the diagonal morphism, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204040.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204041.png" />.
+
reduction to the diagonal: If $  \Delta : X \rightarrow X \times X $
 +
is the diagonal morphism, and $  \alpha , \beta \in A( X) $,  
 +
then $  x \cdot y = \Delta  ^  \star  ( \alpha \times \beta ) $.
  
 
Also, there exists a natural homomorphism
 
Also, there exists a natural homomorphism
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204042.png" /></td> </tr></table>
+
$$
 +
c _ {1} :   \mathop{\rm Pic} ( X)  \rightarrow  A  ^ {1} ( X),
 +
$$
  
 
which enables one to construct the theory of Chern classes (cf. [[Chern class|Chern class]]) with values in Chow rings, and in particular the [[Chern character|Chern character]]
 
which enables one to construct the theory of Chern classes (cf. [[Chern class|Chern class]]) with values in Chow rings, and in particular the [[Chern character|Chern character]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204043.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm ch} : K( X)  \rightarrow  A( X) \otimes \mathbf Q ,
 +
$$
  
 
which is a ring homomorphism.
 
which is a ring homomorphism.
  
It is simplest to determine the direct image homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204044.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204045.png" /> be an irreducible subvariety; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204046.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204047.png" />, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204048.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204049.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204050.png" /> is the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204051.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204052.png" />. By linearity, the definition is extended to cycles and classes of cycles. The inverse image homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204053.png" /> amounts to multiplication of cycles in accordance with
+
It is simplest to determine the direct image homomorphism $  f _  \star  $.  
 +
Let $  Z \subset  X $
 +
be an irreducible subvariety; if $  \mathop{\rm dim}  f( Z) < \mathop{\rm dim}  Z $,  
 +
then $  f _  \star  ( Z) = 0 $,  
 +
while if $  \mathop{\rm dim}  f( Z) = \mathop{\rm dim}  Z $,  
 +
then $  f _  \star  ( Z) = d \cdot f ( Z) $,  
 +
where $  d $
 +
is the degree of $  Z $
 +
over $  f( Z) $.  
 +
By linearity, the definition is extended to cycles and classes of cycles. The inverse image homomorphism $  f ^ { \star } $
 +
amounts to multiplication of cycles in accordance with
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204054.png" /></td> </tr></table>
+
$$
 +
f ^ { \star } ( \alpha )  = p _  \star  ( \Gamma _ {f} \cdot ( X \times
 +
\alpha )),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204055.png" /> is the projection and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204056.png" /> is the graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204057.png" />. The definition of multiplication of cycles is given in two stages. Let, initially, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204059.png" /> be irreducible subvarieties in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204060.png" /> that intersect properly (i.e. the codimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204061.png" /> is equal to the sum of the codimensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204063.png" />). Each component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204064.png" /> of the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204065.png" /> is ascribed some positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204066.png" />, which is the local multiplicity of the intersection. There are several definitions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204067.png" />, for example, Serre's Tor-formula:
+
where $  p: X \times Y \rightarrow X $
 +
is the projection and $  \Gamma _ {f} \subset  X \times Y $
 +
is the graph of $  f $.  
 +
The definition of multiplication of cycles is given in two stages. Let, initially, $  Y $
 +
and $  Z $
 +
be irreducible subvarieties in $  X $
 +
that intersect properly (i.e. the codimension of $  Y \cap Z $
 +
is equal to the sum of the codimensions of $  Y $
 +
and $  Z $).  
 +
Each component $  W $
 +
of the intersection $  Y \cap Z $
 +
is ascribed some positive integer i( Y, Z;  W) $,  
 +
which is the local multiplicity of the intersection. There are several definitions of i( Y, Z;  W) $,  
 +
for example, Serre's Tor-formula:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204068.png" /></td> </tr></table>
+
$$
 +
i( Y, Z; W)  = \sum _ { k\geq  } 0 (- 1)  ^ {k} l(  \mathop{\rm Tor} _ {k}  ^ {A} ( A /
 +
\mathfrak a , A / \mathfrak b )),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204069.png" /> is the local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204070.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204071.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204072.png" /> are ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204074.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204075.png" /> is the length of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204076.png" />-module. After this, one puts
+
where $  A $
 +
is the local ring $  {\mathcal O} _ {X,W} $,
 +
$  \mathfrak a $
 +
and $  \mathfrak b $
 +
are ideals of $  Y $
 +
and $  Z $,  
 +
and $  l $
 +
is the length of the $  A $-
 +
module. After this, one puts
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204077.png" /></td> </tr></table>
+
$$
 +
Y \cdot Z  = \sum _ { W } i( Y, Z; W) \cdot W,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204078.png" /> runs through the irreducible components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204079.png" />.
+
where $  W $
 +
runs through the irreducible components of $  Y \cap Z $.
  
The second stage is Chow's moving lemma: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204081.png" /> on a quasi-projective variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204082.png" /> there exists a cycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204083.png" /> that is rationally equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204084.png" /> and that intersects properly with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204085.png" />; moreover, the rational equivalence class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204086.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204087.png" />.
+
The second stage is Chow's moving lemma: For any $  Y $
 +
and $  Z $
 +
on a quasi-projective variety $  X $
 +
there exists a cycle $  Z  ^  \prime  $
 +
that is rationally equivalent to $  Z $
 +
and that intersects properly with $  Y $;  
 +
moreover, the rational equivalence class of $  Y \cdot Z  ^  \prime  $
 +
is independent of $  Z  ^  \prime  $.
  
The most interesting case is that of a projective variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204088.png" />; applying the direct image functor to the structure morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204089.png" /> one obtains the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204090.png" />. In essence, the degree of a cycle is the number of points in its zero-dimensional component. The composition of multiplication with the degree enables one to measure an intersection numerically. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204091.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204092.png" /> have complementary dimensions, then one obtains the [[Intersection index (in algebraic geometry)|intersection index (in algebraic geometry)]] (the intersection number) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204094.png" />. Similarly, one obtains an intersection index for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204095.png" /> divisors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204096.png" />:
+
The most interesting case is that of a projective variety $  X $;  
 +
applying the direct image functor to the structure morphism $  X \rightarrow  \mathop{\rm Spec}  k $
 +
one obtains the mapping $  \mathop{\rm deg} : A( X) \rightarrow \mathbf Z $.  
 +
In essence, the degree of a cycle is the number of points in its zero-dimensional component. The composition of multiplication with the degree enables one to measure an intersection numerically. For example, if $  Y $
 +
and $  Z $
 +
have complementary dimensions, then one obtains the [[Intersection index (in algebraic geometry)|intersection index (in algebraic geometry)]] (the intersection number) of $  Y $
 +
and $  Z $.  
 +
Similarly, one obtains an intersection index for $  n $
 +
divisors $  D _ {1} \dots D _ {n} $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204097.png" /></td> </tr></table>
+
$$
 +
( D _ {1} \dots D _ {n} )  =   \mathop{\rm deg}  ( D _ {1} \dots D _ {n} ).
 +
$$
  
For example, the [[Chow ring|Chow ring]] for the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204098.png" /> is generated by the class of a hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i05204099.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040100.png" />. Therefore, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040101.png" /> are hypersurfaces of degrees <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040102.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040103.png" /> (Bezout's theorem). The degree of a projective variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040104.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040105.png" /> is defined as the intersection index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040106.png" /> with a linear subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040107.png" /> of complementary dimension; if the varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040109.png" /> intersect transversally, then the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040110.png" /> is the product of the degrees of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040111.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040112.png" />.
+
For example, the [[Chow ring|Chow ring]] for the projective space $  P  ^ {n} $
 +
is generated by the class of a hyperplane $  H $,  
 +
where $  ( H  ^ {n} ) = ( H \dots H) = 1 $.  
 +
Therefore, if $  D _ {1} \dots D _ {n} $
 +
are hypersurfaces of degrees $  d _ {1} \dots d _ {n} $,  
 +
then $  ( D _ {1} \dots D _ {n} ) = d _ {1} \dots d _ {n} $(
 +
Bezout's theorem). The degree of a projective variety $  Y \subset  P  ^ {n} $
 +
of dimension $  k $
 +
is defined as the intersection index of $  Y $
 +
with a linear subspace $  P  ^ {n-} k $
 +
of complementary dimension; if the varieties $  Y $
 +
and $  Z $
 +
intersect transversally, then the degree of $  Y \cap Z $
 +
is the product of the degrees of $  Y $
 +
and $  Z $.
  
For properly-intersecting effective divisors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040113.png" />, but in the general case this is not true. For example, for an exceptional curve (cf. [[Exceptional subvariety|Exceptional subvariety]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040114.png" /> on a surface, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040115.png" />.
+
For properly-intersecting effective divisors $  ( D _ {1} \dots D _ {n} ) \geq  0 $,  
 +
but in the general case this is not true. For example, for an exceptional curve (cf. [[Exceptional subvariety|Exceptional subvariety]]) $  E $
 +
on a surface, $  ( E, E) = - 1 $.
  
Other theories have many of the formal properties of the theory of Chow rings: cycles modulo algebraic or numerical equivalence, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040116.png" />-theory, singular cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040117.png" /> (in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040118.png" />), and [[L-adic-cohomology|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040119.png" />-adic cohomology]] theory (see also [[Weil cohomology|Weil cohomology]]). This leads to the axiomatic construction of intersection theory by putting each variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040120.png" /> (from some category) into correspondence with a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040121.png" /> and homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040122.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040123.png" /> related by axioms of the type of projection or reduction-to-the-diagonal formulas (see [[#References|[1]]]). The comparison of different intersection theories leads to useful results. For example, in the complex case, the concept of a [[Fundamental cycle|fundamental cycle]] enables one to define an intersection-theory homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040124.png" />, which enables one to use transcendental methods. Comparison of [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040125.png" />-theory]] with Chow's theory leads to the Riemann–Roch–Grothendieck theorem (cf. [[Riemann–Roch theorem|Riemann–Roch theorem]]). An important part is played here by the intersection theory for monoidal transformations [[#References|[2]]], [[#References|[6]]]. Another application of intersection theory is related to the foundations of Schubert's geometric calculus [[#References|[3]]]. This branch of geometry can be regarded as the theory of Chow rings for various varieties that classify geometric objects: Grassmann manifolds, flag manifolds, etc.
+
Other theories have many of the formal properties of the theory of Chow rings: cycles modulo algebraic or numerical equivalence, $  K $-
 +
theory, singular cohomology theory $  H  ^  \star  (  , \mathbf Z ) $(
 +
in the case $  k = \mathbf C $),  
 +
and [[L-adic-cohomology| $  l $-
 +
adic cohomology]] theory (see also [[Weil cohomology|Weil cohomology]]). This leads to the axiomatic construction of intersection theory by putting each variety $  X $(
 +
from some category) into correspondence with a ring $  C( X) $
 +
and homomorphisms $  f ^ { \star } $
 +
and $  f _  \star  $
 +
related by axioms of the type of projection or reduction-to-the-diagonal formulas (see [[#References|[1]]]). The comparison of different intersection theories leads to useful results. For example, in the complex case, the concept of a [[Fundamental cycle|fundamental cycle]] enables one to define an intersection-theory homomorphism $  A( X) \rightarrow H  ^  \star  ( X, \mathbf Z ) $,  
 +
which enables one to use transcendental methods. Comparison of [[K-theory| $  K $-
 +
theory]] with Chow's theory leads to the Riemann–Roch–Grothendieck theorem (cf. [[Riemann–Roch theorem|Riemann–Roch theorem]]). An important part is played here by the intersection theory for monoidal transformations [[#References|[2]]], [[#References|[6]]]. Another application of intersection theory is related to the foundations of Schubert's geometric calculus [[#References|[3]]]. This branch of geometry can be regarded as the theory of Chow rings for various varieties that classify geometric objects: Grassmann manifolds, flag manifolds, etc.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> , ''Anneaux de Chow et applications'' , ''Sem. Chevalley'' (1958) {{MR|}} {{ZBL|0098.13101}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> Yu.I. Manin, "Lectures on algebraic geometry" , '''1''' , Moscow (1970) (In Russian) {{MR|0284434}} {{ZBL|0204.21302}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> "Hilbert's problems" ''Bull. Amer. Math. Soc.'' , '''8''' (1902) pp. 437–479 (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> M. Baldassarri, "Algebraic varieties" , Springer (1956) {{MR|0082172}} {{ZBL|0995.14003}} {{ZBL|0075.15902}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J.-P. Serre, "Algèbre locale. Multiplicités" , ''Lect. notes in math.'' , '''11''' , Springer (1965) {{MR|0201468}} {{ZBL|0142.28603}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> P. Berthelot (ed.) A. Grothendieck (ed.) L. Illusie (ed.) et al. (ed.) , ''Théorie des intersections et théorème de Riemann–Roch (SGA 6)'' , ''Lect. notes in math.'' , '''225''' , Springer (1971)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> R. Hartshorne (ed.) , ''Algebraic geometry (Arcata, 1974)'' , ''Proc. Symp. Pure Math.'' , '''29''' , Amer. Math. Soc. (1982)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> W. Fulton, "Intersection theory" , Springer (1984) {{MR|0735435}} {{MR|0732620}} {{ZBL|0541.14005}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> , ''Anneaux de Chow et applications'' , ''Sem. Chevalley'' (1958) {{MR|}} {{ZBL|0098.13101}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> Yu.I. Manin, "Lectures on algebraic geometry" , '''1''' , Moscow (1970) (In Russian) {{MR|0284434}} {{ZBL|0204.21302}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> "Hilbert's problems" ''Bull. Amer. Math. Soc.'' , '''8''' (1902) pp. 437–479 (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> M. Baldassarri, "Algebraic varieties" , Springer (1956) {{MR|0082172}} {{ZBL|0995.14003}} {{ZBL|0075.15902}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J.-P. Serre, "Algèbre locale. Multiplicités" , ''Lect. notes in math.'' , '''11''' , Springer (1965) {{MR|0201468}} {{ZBL|0142.28603}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> P. Berthelot (ed.) A. Grothendieck (ed.) L. Illusie (ed.) et al. (ed.) , ''Théorie des intersections et théorème de Riemann–Roch (SGA 6)'' , ''Lect. notes in math.'' , '''225''' , Springer (1971)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> R. Hartshorne (ed.) , ''Algebraic geometry (Arcata, 1974)'' , ''Proc. Symp. Pure Math.'' , '''29''' , Amer. Math. Soc. (1982)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> W. Fulton, "Intersection theory" , Springer (1984) {{MR|0735435}} {{MR|0732620}} {{ZBL|0541.14005}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
W. Fulton has defined Chow groups for singular varieties [[#References|[a1]]]. A more refined intersection theory has been developed by Fulton and R. MacPherson: given cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040126.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040127.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040128.png" />, it gives a well-defined element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040129.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052040/i052040130.png" /> [[#References|[a2]]].
+
W. Fulton has defined Chow groups for singular varieties [[#References|[a1]]]. A more refined intersection theory has been developed by Fulton and R. MacPherson: given cycles $  Y $
 +
and $  Z $
 +
on $  X $,  
 +
it gives a well-defined element $  X \cdot Y $
 +
of $  A ( Y \cap Z ) $[[#References|[a2]]].
  
 
A new development is the intersection theory for arithmetic varieties, i.e. flat schemes over the ring of integers of number fields with suitable extra data for the infinite places, [[#References|[a3]]]–.
 
A new development is the intersection theory for arithmetic varieties, i.e. flat schemes over the ring of integers of number fields with suitable extra data for the infinite places, [[#References|[a3]]]–.

Latest revision as of 22:13, 5 June 2020


on an algebraic variety

The theory of intersections of algebraic subvarieties and cycles. Let $ X $ be a smooth algebraic variety of dimension $ n $ over a field $ k $, while $ Y $ and $ Z $ are subvarieties of $ X $ of codimension $ i $ and $ j $, respectively. If $ Y $ and $ Z $ intersect transversally, then $ Y \cap Z $ is a smooth subvariety of codimension $ i+ j $, which is denoted by $ Y \cdot Z $. In the general case, the pair $ ( Y, Z) $ constitutes an algebraic cycle $ Y \cdot Z $ of codimension $ i + j $. The idea behind this definition is that $ Y $ and $ Z $ are replaced by cycles $ Y ^ \prime $ and $ Z ^ \prime $ that are equivalent in some sense but that are in general position, and one then takes the intersection of $ Y ^ \prime $ and $ Z ^ \prime $; of course, the cycle $ Y ^ \prime \cdot Z ^ \prime $ is also defined up to equivalence.

Let $ A ^ {i} ( X) $ be the group of classes of algebraic cycles of codimension $ i $ on $ X $ modulo rational equivalence; let $ A( X) = \oplus _ {i \geq 0 } A ^ {i} ( X) $. Chow's intersection theory consists of the construction:

a) of a graded commutative ring structure on $ A( X) $ for each smooth quasi-projective variety $ X $;

b) of a homomorphism of graded rings $ f ^ { \star } : A( Y) \rightarrow A( X) $ for each morphism $ f: X \rightarrow Y $( inverse image); and

c) of a homomorphism of groups $ f _ \star : A( X) \rightarrow A( Y) $ of degree $ \mathop{\rm dim} Y - \mathop{\rm dim} X $ for each proper morphism $ f: X \rightarrow Y $( direct image).

Between the constructions a), b), c) there are a number of relations, of which the main ones are as follows:

the projection formula: For a proper morphism $ f: X \rightarrow Y $ and cycles $ \alpha \in A( X) $ and $ \beta \in A( Y) $,

$$ f _ \star ( \alpha \cdot f ^ { \star } ( \beta )) = f _ \star ( \alpha ) \cdot \beta ; $$

reduction to the diagonal: If $ \Delta : X \rightarrow X \times X $ is the diagonal morphism, and $ \alpha , \beta \in A( X) $, then $ x \cdot y = \Delta ^ \star ( \alpha \times \beta ) $.

Also, there exists a natural homomorphism

$$ c _ {1} : \mathop{\rm Pic} ( X) \rightarrow A ^ {1} ( X), $$

which enables one to construct the theory of Chern classes (cf. Chern class) with values in Chow rings, and in particular the Chern character

$$ \mathop{\rm ch} : K( X) \rightarrow A( X) \otimes \mathbf Q , $$

which is a ring homomorphism.

It is simplest to determine the direct image homomorphism $ f _ \star $. Let $ Z \subset X $ be an irreducible subvariety; if $ \mathop{\rm dim} f( Z) < \mathop{\rm dim} Z $, then $ f _ \star ( Z) = 0 $, while if $ \mathop{\rm dim} f( Z) = \mathop{\rm dim} Z $, then $ f _ \star ( Z) = d \cdot f ( Z) $, where $ d $ is the degree of $ Z $ over $ f( Z) $. By linearity, the definition is extended to cycles and classes of cycles. The inverse image homomorphism $ f ^ { \star } $ amounts to multiplication of cycles in accordance with

$$ f ^ { \star } ( \alpha ) = p _ \star ( \Gamma _ {f} \cdot ( X \times \alpha )), $$

where $ p: X \times Y \rightarrow X $ is the projection and $ \Gamma _ {f} \subset X \times Y $ is the graph of $ f $. The definition of multiplication of cycles is given in two stages. Let, initially, $ Y $ and $ Z $ be irreducible subvarieties in $ X $ that intersect properly (i.e. the codimension of $ Y \cap Z $ is equal to the sum of the codimensions of $ Y $ and $ Z $). Each component $ W $ of the intersection $ Y \cap Z $ is ascribed some positive integer $ i( Y, Z; W) $, which is the local multiplicity of the intersection. There are several definitions of $ i( Y, Z; W) $, for example, Serre's Tor-formula:

$$ i( Y, Z; W) = \sum _ { k\geq } 0 (- 1) ^ {k} l( \mathop{\rm Tor} _ {k} ^ {A} ( A / \mathfrak a , A / \mathfrak b )), $$

where $ A $ is the local ring $ {\mathcal O} _ {X,W} $, $ \mathfrak a $ and $ \mathfrak b $ are ideals of $ Y $ and $ Z $, and $ l $ is the length of the $ A $- module. After this, one puts

$$ Y \cdot Z = \sum _ { W } i( Y, Z; W) \cdot W, $$

where $ W $ runs through the irreducible components of $ Y \cap Z $.

The second stage is Chow's moving lemma: For any $ Y $ and $ Z $ on a quasi-projective variety $ X $ there exists a cycle $ Z ^ \prime $ that is rationally equivalent to $ Z $ and that intersects properly with $ Y $; moreover, the rational equivalence class of $ Y \cdot Z ^ \prime $ is independent of $ Z ^ \prime $.

The most interesting case is that of a projective variety $ X $; applying the direct image functor to the structure morphism $ X \rightarrow \mathop{\rm Spec} k $ one obtains the mapping $ \mathop{\rm deg} : A( X) \rightarrow \mathbf Z $. In essence, the degree of a cycle is the number of points in its zero-dimensional component. The composition of multiplication with the degree enables one to measure an intersection numerically. For example, if $ Y $ and $ Z $ have complementary dimensions, then one obtains the intersection index (in algebraic geometry) (the intersection number) of $ Y $ and $ Z $. Similarly, one obtains an intersection index for $ n $ divisors $ D _ {1} \dots D _ {n} $:

$$ ( D _ {1} \dots D _ {n} ) = \mathop{\rm deg} ( D _ {1} \dots D _ {n} ). $$

For example, the Chow ring for the projective space $ P ^ {n} $ is generated by the class of a hyperplane $ H $, where $ ( H ^ {n} ) = ( H \dots H) = 1 $. Therefore, if $ D _ {1} \dots D _ {n} $ are hypersurfaces of degrees $ d _ {1} \dots d _ {n} $, then $ ( D _ {1} \dots D _ {n} ) = d _ {1} \dots d _ {n} $( Bezout's theorem). The degree of a projective variety $ Y \subset P ^ {n} $ of dimension $ k $ is defined as the intersection index of $ Y $ with a linear subspace $ P ^ {n-} k $ of complementary dimension; if the varieties $ Y $ and $ Z $ intersect transversally, then the degree of $ Y \cap Z $ is the product of the degrees of $ Y $ and $ Z $.

For properly-intersecting effective divisors $ ( D _ {1} \dots D _ {n} ) \geq 0 $, but in the general case this is not true. For example, for an exceptional curve (cf. Exceptional subvariety) $ E $ on a surface, $ ( E, E) = - 1 $.

Other theories have many of the formal properties of the theory of Chow rings: cycles modulo algebraic or numerical equivalence, $ K $- theory, singular cohomology theory $ H ^ \star ( , \mathbf Z ) $( in the case $ k = \mathbf C $), and $ l $- adic cohomology theory (see also Weil cohomology). This leads to the axiomatic construction of intersection theory by putting each variety $ X $( from some category) into correspondence with a ring $ C( X) $ and homomorphisms $ f ^ { \star } $ and $ f _ \star $ related by axioms of the type of projection or reduction-to-the-diagonal formulas (see [1]). The comparison of different intersection theories leads to useful results. For example, in the complex case, the concept of a fundamental cycle enables one to define an intersection-theory homomorphism $ A( X) \rightarrow H ^ \star ( X, \mathbf Z ) $, which enables one to use transcendental methods. Comparison of $ K $- theory with Chow's theory leads to the Riemann–Roch–Grothendieck theorem (cf. Riemann–Roch theorem). An important part is played here by the intersection theory for monoidal transformations [2], [6]. Another application of intersection theory is related to the foundations of Schubert's geometric calculus [3]. This branch of geometry can be regarded as the theory of Chow rings for various varieties that classify geometric objects: Grassmann manifolds, flag manifolds, etc.

References

[1] , Anneaux de Chow et applications , Sem. Chevalley (1958) Zbl 0098.13101
[2] Yu.I. Manin, "Lectures on algebraic geometry" , 1 , Moscow (1970) (In Russian) MR0284434 Zbl 0204.21302
[3] "Hilbert's problems" Bull. Amer. Math. Soc. , 8 (1902) pp. 437–479 (Translated from German)
[4] M. Baldassarri, "Algebraic varieties" , Springer (1956) MR0082172 Zbl 0995.14003 Zbl 0075.15902
[5] J.-P. Serre, "Algèbre locale. Multiplicités" , Lect. notes in math. , 11 , Springer (1965) MR0201468 Zbl 0142.28603
[6] P. Berthelot (ed.) A. Grothendieck (ed.) L. Illusie (ed.) et al. (ed.) , Théorie des intersections et théorème de Riemann–Roch (SGA 6) , Lect. notes in math. , 225 , Springer (1971)
[7] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
[8] R. Hartshorne (ed.) , Algebraic geometry (Arcata, 1974) , Proc. Symp. Pure Math. , 29 , Amer. Math. Soc. (1982)
[9] W. Fulton, "Intersection theory" , Springer (1984) MR0735435 MR0732620 Zbl 0541.14005

Comments

W. Fulton has defined Chow groups for singular varieties [a1]. A more refined intersection theory has been developed by Fulton and R. MacPherson: given cycles $ Y $ and $ Z $ on $ X $, it gives a well-defined element $ X \cdot Y $ of $ A ( Y \cap Z ) $[a2].

A new development is the intersection theory for arithmetic varieties, i.e. flat schemes over the ring of integers of number fields with suitable extra data for the infinite places, [a3]–.

References

[a1] W. Fulton, "Rational equivalence on singular varieties" Publ. Math. IHES , 45 (1975) pp. 147–167 MR0404257 Zbl 0332.14002
[a2] W. Fulton, R.D. MacPherson, "Defining algebraic intersections" L.D. Olson (ed.) , Algebraic geometry , Lect. notes in math. , 687 , Springer (1978) pp. 1–30 MR0527228 Zbl 0405.14003
[a3] G. Faltings, "Calculus on arithmetic surfaces" Ann. of Math. , 119 (1984) pp. 387–424 MR0740897 Zbl 0559.14005
[a4] S. Arakelov, "Intersection theory of divisors on an arithmetical surface" Math. USSR Izv. , 8 (1974) pp. 1167–1180 Izv. Akad. Nauk. SSSR Ser. Mat. , 38 : 6 (1974) pp. 1179–1192 MR472815
[a5] H. Gillet, "An introduction to higher dimensional Arakelov theory" K.A. Ribet (ed.) , Current trends in arithmetical algebraic geometry , Contemp. Math. , 67 , Amer. Math. Soc. (1987) pp. 209–228 MR0902594 Zbl 0621.14004
How to Cite This Entry:
Intersection theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Intersection_theory&oldid=23867
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article