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Algebraic Geometry Lecture 14 – Intersections in Projective Space Part II
We need to recap/know the following ring theory.
Def n. A multiplicative system in a ring S is a subset A containing 1 and closed under multiplication. The localisation A−^1 S is defined to be the ring formed by equivalence classes of fractions s/a with s ∈ S, a ∈ A, where
s/a ∼ s′/a′^ ⇔ there exists a′′^ ∈ A such that a′′(a′s − as′) = 0.
If p is a prime ideal in S then A = S\p is a multiplicative system. The localisation A−^1 S is then denoted Sp.
Example Consider the ring Z. For any prime number p, A = Z \ pZ is a multi- plicative system, for if ab ∈ pZ then either a or b must be in pZ, so if a, b ∈ A then ab ∈ A. The localisation is then
ZpZ = {a/b | a ∈ Z, b ∈ Z \ pZ}
where a/b ∼ a′/b′^ if and only if there is a c ∈ Z \ pZ such that
c(b′a − ba′) = 0.
But c 6 = 0 so this just means b′a − ba′^ = 0, or simply a/b = a′/b′, noting that neither b nor b′^ can be zero. So we have
ZpZ =
{ (^) a b
: p - b
( = Q ∩ Zp).
The Hilbert Polynomial ctd.
Recall we’re trying to generalise B´ezout’s theorem on how many times two curves intersect counting multiplicities. By the end of this lecture we will be able to state how many times a variety intersects a hypersurface, but we still need to define the degree of a polynomial and how to count the multiplicity of an intersection. First we need to introduce graded rings and modules.
p Recall: A module over a ring S, or S-module, is an abelian group M such that we can multiply by elements of S, so for all s, s 1 , s 2 ∈ S, m, m 1 , m 2 ∈ M :
- s 1 (s 2 m) = (s 1 s 2 )m
- (s 1 + s 2 )m = s 1 m + s 2 m
- s(m 1 + m 2 ) = sm 1 + sm 2.
We say the module is finitely generated if there is a finite set of elements in M , say m 1 ,... , mr such that any element of M can be written as a linear combination of these r elements over the ring S. 1
y
Def n. A graded ring is a ring S together with a decomposition as a direct sum
S =
d> 0
Sd
with Sd abelian groups, such that if a ∈ Sd and b ∈ Se then ab ∈ Sd+e.
Def n. (i) An element of Sd is called a homogeneous element of degree d.
(ii) An ideal A ⊆ S is a homogeneous ideal if A =
d> 0
(A ∩ Sd).
Def n. If S is a graded ring then a graded S-module is an S-module M with a decomposition
M =
d∈Z
Md
such that if s ∈ Sd and m ∈ Me then sm ∈ Md+e.
Given an algebraic set Y we will be taking S to be k[x 0 ,... , xn], and M to be k[Y ] = S/I(Y ), the homogeneous coordinate ring of Y. We have
S =
d> 0
Sd
where Sd are the homogeneous polynomials in S of degree d, and
M =
d∈Z
Sd/Id(Y ),
where Id(Y ) = I(Y ) ∩ Sd.
Now for a few more definitions.
Def n. For any graded S-module M and any integer the twisted module M () is formed by shifting the decomposition of M ` places to the left. So if
M =
d∈Z
Md
then M (`) =
d∈Z
Md+`.
So the Hilbert polynomial is just PM (z) = z + 1, and thus deg(Y ) = 1.
Def n. Let M be a module over the ring S. A filtration of M is an ascending chain of submodules of 0 = M 0 ⊂ M 1 ⊂... ⊂ M n^ = M.
We define the length of the filtration to be n. The length of M is the maximum length of any of its filtrations.
We are now in a position to start defining the multiplicity of an intersection. The definition when we get to it would seem to be not well defined, but the following proposition takes care of that.
Proposition 1. Let M be a finitely generated graded module over a noetherian graded ring S (noetherian means all prime ideals are finitely generated). Then there exists a filtration
0 = M 0 ⊂ M 1 ⊂... ⊂ M r^ = M
by graded submodules such that for each i,
M i/M i−^1 ∼= (S/pi)(`i)
for a homogeneous prime ideal pi of S and some integer `i. The filtration is not necessarily unique, but to each such filtration we get a collection of (not necessarily distinct) prime ideals {p 1 ,... , pr }. And while the filtration is not unique, for any such filtration the collection of prime ideals is the same.
With this in the bag we can now define:
Def n. If p is a minimal prime ideal of a graded S-module M then the multiplicity of M at p, denoted μp(M ), is defined to be the number of times that p appears in any filtration as above.
And now at last we can define the following.
Def n. Let Y ⊆ Pn^ be a projective variety of dimension r. Let H be a hypersurface not containing Y , i.e. Y * H. Then by the projective dimension theorem Y ∩ Z = Z 1 ∪... ∩ Zs, where each Zj is a variety of dimension r − 1. Let pj = I(Zj ) be the homogeneous prime ideal of Zj. We define the intersection multiplicity of Y and H along Zj to be i(Y, H; Zj ) = μpj (S/(I(Y ) + I(H))).
Theorem. Let Y ⊆ Pn^ be a variety of dimension > 1 , and let H be a hypersurface not containing Y. Let Z 1 ,... , Zs be the irreducible components of Y ∩ H. Then
∑^ s
j=
i(Y, H; Zj ) · deg Zj = (deg Y )(deg H).
As you may have noticed working out the Hilbert polynomials is a big challenge in most cases, though Groebner bases can help with it. But we can work out the Hilbert polynomial of a point and so give a proof of B´ezout’s theorem.
Corollary (B´ezout’s theorem). Let Y, Z be distinct curves in P^2 having degrees d, e respectively. Let Y ∩ Z = {P 1 ,... , Ps}. Then
∑^ s
j=
i(Y, Z; Pj ) = de.
Proof. By the projective dimension theorem the Pj ’s must be points and it’s an easy exercise to show that the Hilbert polynomial of a point is just 1, and so deg Pj = 1 for each j, and so the result follows from the theorem.
