Quick description
There are many situations in algebraic geometry in which one wishes to compute dimensions of various spaces, such as the space of all lines contained in a given surface in projective space. One approach to doing this is via so-called incidence varieties. An incidence variety is essentially the graph of a relation (for example, the collection of all ordered pairs of lines and surfaces of a given degree , where the line lies on the surface), but thought of itself as a variety. Since the incidence variety is a set of ordered pairs, it admits two natural projections, onto the first or second member of each ordered pair. Playing off the information that one obtains from considering these two projections,
it is often possible to make non-trivial deductions.
Prerequisites
Basic algebraic geometry
Example 1
Let denote the space of all lines contained in
(
-dimensional projective space). (The letter
is chosen to stand for Grassmannian.)
What is the dimension of
? There are many ways to compute this dimension; we present here two approaches via incidence varieties.
In the first approach, we use the fact that a line is a -dimensional collection of points, together with
the fact that if
is any point of
, then the space of lines through
is isomorphic
to
, and hence is
-dimensional. (To see this, note that by choosing our coordinates
on projective space appropriately, we can imagine that
is the origin of the affine space
sitting inside projective space
. Now just recall that one definition of
is that it is the space of lines through the origin of
-dimensional affine space.)
Consider the incidence variety

that is, is the incidence variety consisting of pairs of points
and lines
, with
lying on
.
We have maps
and
given by projecting onto the
first or second coordinate respectively (i.e.
while
).
We use the map to elucidate the structure of
, as follows: we have already noted that for a fixed
, the space of lines
passing through
is a
. We can rephrase
this by saying that the fibre of
over each point
of
is a
. (Or again,
we could say that the map
realizes
as a
-bundle over
.) In particular,
the dimension of
must be
(i.e.
).
Now consider the map . The fibre of any point (i.e. line!)
is precisely the
set of points
that lie on
, i.e. the line
itself, but now thought of, not as a point in
, but
as a subset of
. As already remarked,
is
-dimensional. (We could say that
realizes
as a
-bundle over
.) Thus the map
has
-dimensional fibres, and a
-dimensional domain. Its target thus has dimension
(i.e.
).
So the answer to the question is that has dimension
.
Here is another way to make the same computation, using a different incidence variety. In this
argument, we again use the fact a line is a -dimensional collection of points. But rather than
using any information about the space of lines passing through a fixed point, we instead use the
fact that a line is determined by any two distinct points on it, together with the converse, namely
that any two distinct points determine a line.
This time we consider a different incidence variety (which we will again label as ), namely

so consists of pairs of points
, together with a line
containing both of them.
We consider two projections, as above:
defined by
and
defined by
.
If is a point of
with
then there is a unique line
containing both
and
. Thus
is generically 1-1,
i.e. it is a birational map, and so we see that its source and target have the same dimension.
Considering the target, we see that this dimension equals
(i.e.
). Thus
has dimension
.
Now the fibre of over any line
consists of pairs
with both
and
lying on
,
i.e. this fibre is the product
thought of as lying in
.
In particular, this fibre is
-dimensional. Since
has a
-dimensional source, and
-dimensional
fibres, its target has dimension
(i.e.
). Thus
has dimension
.
General discussion
Suppose that and
are two algebraic varieties, and that
is a relation between
and
.
Thinking of
as being a set of ordered pairs, we can regard
as being a subset of
.
(Sometimes, to be explicit, we refer to this subset as the graph of the relation; but in most theoretical
treatments of the notion of relation, a relation is simply identified with the collection of ordered pairs
that it determines, i.e. with its graph, and that is what we are doing here.) Now suppose that the relation
is described by an algebraic condition. Then we can think of
(which is a priori just a subset
of
) as being an algebraic subvariety of
. (And indeed, we could take this
as being the definition of what it means for the relation to be described by an algebraic condition.)
The subvariety comes equipped with two important structures, namely the projection maps
onto the first and second factor, i.e.
and
One can
then hope to study these maps (in particular, their fibres), to pass geometric information back
and forth between
and
, via the intermediary of
.
An incidence variety is just a particular instance of the above general set-up, when and
are varieties that themselves parameterize other geometric objects (e.g. spaces of points, lines, curves,
surfaces, etc.), and the relation
is given by some geometric condition of ``incidence'',
e.g. ``lies on'',
``has non-empty intersection with'',
``is tanget to'', etc. In our first example,
was the space either of points in
(i.e.
-itself) or else the
space of pairs of points in
(i.e.
), and
was the space
of lines in
(an example of a Grassmannian).
Example 2
Suppose that we wish to compute the dimension of the space of lines contained in a surface of some
given degree in projective space
. Again, let
denote the space of all lines
contained in
; this is a
-dimensional space,
as we saw above.
The space of surfaces of degree in
is itself a projective space, of dimension
. (There are
monomials of degree
in
-variables.)
We take this space to be
, and will in fact denote it by
. We take
to be the space
of lines. We define the incidence variety
as follows:

thus consists of the ordered pairs of a surface
of degree
and a line
,
for which the line
lies on the surface
.
We have the projections and
.
Our goal is to compute the dimensions of the fibres of
; i.e. the dimension of the space
of lines lying on a given degree
surface
. Since we know the dimension
of
, what we need is the dimension of
.
To compute the dimension of , we turn to a consideration of the map
. It has a
-dimensional
target. What are the dimensions of its fibres? To answer this, we have to compute how many surfaces
of degree
pass through a given line
. We may as well choose our (homogeneous) coordinates
so that
is cut out by the equations
For
to contain
, we
need every monomial appearing in the equation of
to be divisible by at least one of
or
.
There are
monomials of degree
divisible by neither
nor
(i.e. involving
just
or
), and so there are
monomials divisible by at least one of
or
.
Thus each fibre of
has dimension equal to
and so
has dimension equal to this
fibre dimension plus
(the dimension of
), i.e. the dimension of
equals
.
We now return to the map . We now know that is maps from a variety of dimension
to a variety of dimension
. What can we conclude?
Well, if , the dimension of the source is less than that of the target, and hence the map
cannot be surjective. Thus a generic surface of degree
contains no lines on it at all.
(Of course, some
will, since
contains
which are reducible,
e.g. which are the union of
planes, and such a
contains plenty of lines.)
When , we see that the source and target have the same dimension, and so we might
expect that a generic fibre of
is
-dimensional, i.e. finite, and hence that a generic cubic surface contains a finite
number of lines. This is indeed true: a smooth cubic surface contains exactly
lines (if we are working over an algebraically closed field).
When , we see that the source has dimension
greater than the target, and so we might
expect that the typical degree
surface (or quadric, as one says) contains a one-dimensional
family of lines. This is indeed true of any smooth quadric: such a quadric is a
double-ruled surface. (Again, over an algebraically closed field; but one can see this for a one-sheeted hyperboloid plotted over the real numbers.)
When , we see that the source has dimension
greater than the target. And in this
case we know that the fibres are all of dimension
: a degree
surface is just a plane,
and a plane contains a
-dimensional family of lines on it (parameterized by the
dual plane); one can prove this, for example, following the method of Example 1 above.
This example brings out a general feature that will tend to arise in any careful analysis of a situation
involving incidence varieties: namely, the maps and
need not always have fibres of constant
dimension (this is a general feature of maps between varieties), and so to draw precise conclusions,
one sometimes has to investigate the structure of these maps in more detail. (This is discussed
at greater length on the page How to compute the dimensions of the fibres of a map of varieties.)
This is a general feature of the method of counting constants: often to make precise conclusions, one has to consider more of the geometry of the situation than just the dimensions of the varieties involved. But even without this extra work, rough computations of the type given in this example can be very suggestive, and can be well-worth making before one proceeds with more careful geometric analyses.
Remark
The method of incidence varieties can be thought of as an algebro-geometric version of the method of double counting in combinatorics (but one is now counting dimensions, rather than actual numbers of elements).