For the complex topology, any connected component of an affine variety
is either a single point or non-compact.
Indeed, if is a connected component of an affine variety,
the functions can't have a maximum on unless
they are constant so if is compact then each is equal to some
constant on , i.e. is the point .
This non-compactness of affine varieties is a motivation
to introduce projective spaces and projective varieties.
Let be the space of complex lines in .
We denote the points of .
As a line
is determined by some direction vector, well-defined up to scaling by non-zero
scalars, can be identified with the quotient
Making use of the natural Hermitian structue on
, i.eof the norm
, we see that a line is well-defined
by a unitary vector up to scaling by some scalar of unit norm. This gives an identification
with the quotient
is compact for the complex topology. The space
is called the "projective space of dimension
In order to describe the projective space in terms of algebra, it is enough to work on
in a -equivariant way.
More precisely, we consider the
action of on with
acting on a degree homogeneous polynomial by multiplication by
. Let us denote and the finite dimensional
vector space of
degree homogeneous polynomials. As any polynomial can be uniquely decomposed in
homogeneous pieces, we have the direct sum decomposition
This decomposition is the eigenspace decomposition for the
is the eigenspace of the action of
for the eigenvalue
. The space
is called the "weight space of weight
" of the
"'Exercise"': We say that a finite dimensional representation of
is "algebraic" if the corresponding group homomorphism
Show that the data of a -graded vector space
with finite-dimensional graded components is equivalent to the data of a vector space
with a -action such that every is contained in a finite-dimensional -invariant subspace
on which the -action is algebraic.
Let be a degree homogeneous polynomial.
Then, for every ,
we have if and only if .
So it makes sense to say that vanishes at some point . An ideal of is called
or equivalently if is generated by homogeneous polynomials.
It makes sense to say that all the functions in a given homogeneous ideal of
vanish at some point and so every homogeneous ideal of defines a subvariety of .
In other words, the subvarieties of defined by the homogeneous ideals of
are precisely the -invariant subvarieties of and so define subvarieties of
. The -invariant subvariety of associated to a subvariety
of is called the "affine cone" of this subvariety.
A subvariety of is called a "projective variety".
Let be a homogeneous ideal of . Then the quotient ring is also graded:
We define the "homogeneous spectrum" of the graded ring
, as being the set of homogeneous
which are maximal amongst homogeneous ideals, except the
so called "irrelevant ideal"
which is the ideal of the origin in the
-invariant subvariety of
, and thus corresponds to the empty set after passing to projective space.
It is possible to show that
can be naturally identified with the
set of points of the projective variety defined by
In other words, if
, we have
"'Example"': Let be the projective curve in defined by the degree 3 homogeneous polynomial . Its affine cone (with real coordinates) is a union of lines through the origin. Consider its intersection with the plane . We can identify with the plane affine curve via the map . We think of the remaining points on as "at infinity" with respect to this affine set.
"'Exercise"': Rewrite the table of dualities by
replacing by , ideals by homogeneous ideals,
rings by graded rings, by ...
"'Exercise"': Work out the details of .
Pass back and forth between homogeneous polynomials in
and polynomials in .
A degree homogeneous polynomial is a function on
but is not a function on . A natural question is: what is the interpretation
of the degree homogeneous polynomials in terms of the projective space ?
For , a degree homogeneous polynomial defines a function on
the line which is of degree . For example, if ,
the coordinates are linear functionals on the line .
We call the line
and the union of these lines, i.e.
As the fibres of the natural map from
, are copies of
, i.e. lines, we see that
is a "line bundle"
. As the fibre over a point of
, i.e. over a line in
, is precisely this line,
is called the
"tautological" line bundle on
Let us consider the natural map from to ,
. The fibre of over
is the set of lines passing through . If is non-zero, there is a unique such
line and so is one-to-one above . If , any
line passes through zero and so the fibre is .
This map is called the "blow-up
of the origin in ": one goes from to
by separating the various lines intersecting each other at the origin and this effectively
replaces the origin of by the projective space .
As the coordinates are linear functionals on the lines ,
the natural interpretation of the 's is as sections of the dual line bundle
of , called . More generally, degree
homogeneous polynomials naturally define sections of the line bundle obtained by the
-th tensor product of the dual of , i.e. of . This line bundle
is called .
"'Exercise"': What is the interpretation of the quotient of an affine variety by the
action of a finite group in terms of coordinate rings?
"'Exercise"': Prove the degree-genus formula, a smooth degree curve in
is of genus , by degenerating the degree curve to a union
"'Exercise"': Let be a "reasonable" topological space.
Show that the closed subsets of
are exactly the subsets of the form where the
are continuous functions on .
Now invent the Zariski topology by replacing continuous functions by polynomials.