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Definition. Suppose that
and
are
two irreducible varieties and that their intersection has irreducible
components
. We say that
and
intersect generically
transversally if, for each
,
and
intersect transversally at
a general point
, i.e., are smooth at
with tangent spaces spanning
(the tangent space to
at
.)
Theorem 5 (Bezout)
Let
![$ X$](img2.gif)
and
![$ Y\subset \mathbb{P}^n$](img187.gif)
be
subvarieties of pure dimensions
![$ k$](img192.gif)
and
![$ l$](img193.gif)
with
![$ k+l\geq n$](img194.gif)
, and
suppose they intersect generically transversely. Then
In particular, if
![$ k+l=n$](img198.gif)
, this says that
![$ X\cap Y$](img199.gif)
will consist of
deg
![$ (X)\cdot$](img200.gif)
deg
![$ (Y)$](img201.gif)
points.
A pair of pure-dimensional varieties
and
intersect properly if their intersection has the expected
dimension, i.e.,
Theorem 6
If
![$ X$](img2.gif)
and
![$ Y$](img95.gif)
intersect properly,
where the sum is over
all irreducible subvarieties
![$ Z$](img207.gif)
of the appropriate dimension (in
effect, over all irreducible components
![$ Z$](img207.gif)
of
![$ X\cap Y$](img199.gif)
). Here
![$ m_Z(X,Y)$](img208.gif)
is the
intersection multiplicity of
![$ X$](img2.gif)
and
![$ Y$](img95.gif)
along
![$ Z$](img207.gif)
:
- 1.
-
for all
(
otherwise.)
- 2.
-
if
and
intersect transversely at a general
point of
.
- 3.
is additive, i.e.
for any
and
as long as
all three numbers are defined and
and
have no common
components.
In particular, for any subvarieties
and
of
pure dimension in
intersecting properly:
Next: About this document ...
Up: Algebraic Geometry
Previous: Zariski Topology
Zvezdelina Stankova-Frenkel
2001-01-24