Projective Varieties, Algebraic Set
Projective Algebraic Set
A set of polynomials S, in x1 through xn, vanishes on a certain region S′ in Kn.
Conversely, given a region R in Kn, a set of polynomials R′ vanishes on R.
A region R becomes an algebraic set if R = S′ for a set of polynomials S.
It is sometimes convenient to ratchet S up to S′′, which is all the polynomials that vanish on R.
You should review the properties of the
algebraic set before continuing.
If S contains only homogeneous polynomials, S′ becomes a region in projective space.
This is a projective algebraic set.
Note that our projective space could be based on K, or an algebraic extension of K, or its algebraic closure,
while the polynomials of S could have coefficients in K, or any subring of K.
Conversely, a region R in projective space defines a set of homogeneous polynomials R′
that vanish on R.
If R = S′ for some S, then R is a projective algebraic set,
and R′ contains all the polynomials in S,
and perhaps some additional homogeneous polynomials that vanish on R.
If S vanishes on R, the functions of S can be added together, and multiplied by any other polynomial,
homogeneous or not, to build more functions that vanish on R.
In other words, S can be expanded to a homogeneous ideal in the polynomial ring.
Conversely, a homogeneous ideal establishes a set of homogeneous polynomials
which serve to generate that ideal.
It is therefore equivalent to view S as a homogeneous ideal in the polynomial ring.
This is more general than a set of homogeneous elements,
and it allows us to use some of the machinery we have already developed.
A homogeneous ideal S is closed if S′′ = S,
and a region R in projective space is closed if R′′ = R.
These are the same definitions we saw before.
A region R is closed iff R = S′, iff R is an algebraic set.
This was true in normal space; and it's true in projective space.
Topology
A homogeneous ideal is also an ideal,
hence, using the same set of polynomials S, an algebraic set in projective space is an algebraic set in normal space.
We already demonstrated the zariski topology in normal space.
This applies in projective space as well.
We only need show the arbitrary union and finite product of homogeneous ideals produces another homogeneous ideal,
so that the arbitrary intersection and finite union of projective algebraic sets is indeed a projective algebraic set,
not just a normal algebraic set.
Fortunately these relationships among homogeneous ideals
have already been demonstrated.
Hence the zariski topology remains valid.
Let R′ be the homogeneous ideal that vanishes on a projective region R.
Mod out by this ideal, and the quotient ring is, by definition, the coordinate ring.
This is still a graded ring,
generated by homogeneous polynomials.
Select one, and evaluate it on a point in R.
The result is scaled by ld, where l is the distance from the origin and d is the degree of the polynomial.
Thus the coordinate ring is not a well defined set of functions on R,
as it was in normal space.