BEZOUT’S THEOREM
CHRISTIAN KLEVDAL
A weaker version of Bézout’s theorem states that if C, D are projective plane curves of
degrees c and d that intersect transversally, then |C ∩ D| = cd. The goal of this talk is
to give a deformation theoretic proof of Bézout’s theorem. Many of the tools used in this
proof are overkill for proving something as simple as Bézout’s theorem (especially since
we consider only transverse intersection) but the important part is the techniques used,
not so much the result. For example, a modified version of this argument can be used to
show there are 27 lines on a cubic.
Throughout the notes, k will be a fixed algebraically closed field.
1. Parameter Space of Curves
Projective plane curves are defined by homogenous polynomials in k[x, y, z], which are in
turn determined by their coefficients. To consider the parameter space of polynomials of a
given degree, we should think of the coefficients as being coordinates. Let k[x, y, z]ddenote
the homogenous polynomials of degree d. As a k vector space it has dimension d+2
2 . Note
that if λ ∈ C is non zero and f ∈ k[x, y, z]d then λf and f determine the same planar curve,
d+2
d+2
so the parameter space of curves of degree d should be (C( 2 ) − {0})/C× = P( 2 )−1 .
− 2, points of PN are pairs (f, g) where f is a polynomial of
+ e+2
Letting N = d+2
2
2
degree d and g a polynomial of degree e. Consider the projection map:
π : PN × P2 → PN
Points of PN × P2 are triples (f, g, p) with f, g as above and p a point in the projective
plane. The condition that f (p) − g(p) = 0 (i.e. that p is a point on V (f ) ∩ V (g)) is a
polynomial condition, hence X = {(f, g, p) | p ∈ V (f ) ∩ V (g)} ⊆ PN × P2 is a closed
subvariety, and π restricts to the projection map X → PN . The fiber X(f,g) over a point
(f, g) ∈ PN is then just the intersection V (f ) ∩ V (g) so in this context Bézout’s theorem
states that if V (f ) and V (g) intersect transversally then |X(f,g) | = de. The rough idea
of the proof is to show that the subset U ⊆ PN of points with transverse intersection is
open and that π −1 U → U is a “covering space”. Then we show that the cardinality of
the fiber is constant over points in U and show for one point (f, g) ∈ U that |X(f,g) | = de.
Of course π −1 U → U is not a covering space in the topological sense because the Zariski
topology is too coarse so we first need to come up with the algebraic analog of a covering
space.
2. Algebraic Covers
If p : Y → X is a covering space (is locally trivial), then for any x, x0 ∈ X there is a
bijection p−1 (x) → p−1 (x0 ). This can be constructed by taking a path γ with γ(0) = x
1
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CHRISTIAN KLEVDAL
and γ(1) = x0 , and a point y ∈ p−1 (x) is sent to γ
e(1) where γ
e is a lift of γ to Y starting
at x. In a nutshell,
• Existence of path lifts defines the map.
• Local homeomorphism gives that it is injective (different lifts end at different
points).
• Uniqueness of path lifts allows for the inverse to be defined.
So we see that the two important properties that covering spaces satisfy for this are:
(1) They are local homeomorphisms.
(2) They have unique lifting of paths.
The locally trivial definition does not work well for varieties, but the above definitions
can be transferred to varieties. We will see that local homeomorphisms correspond to
étale maps and unique path lifting corresponds to valuatively proper maps. Using these
definitions, we can modify the argument that shows covering spaces have constant fiber
size to show that valuatively proper étale maps have constant fiber size.
2.1. Étale maps. First, we start out with an example to show the Zariski topology is too
coarse to detect maps which should be considered local homeomorphisms.
Example 2.1. The map C× → C× induced by the algebra map C[x, x−1 ] → C[x, x−1 ]
sending x 7→ xn is not a local homeomorphism in the Zariski topology, but it is a topological
covering space in the analytic topology.
A more useful idea comes from differential geometry. It is a consequence of the inverse
function theorem that a smooth map f : M → N is a local diffeomorphism at p ∈ M if
and only if the map on tangent spaces Dp f : Tp M → Tf (p) N is an isomorphism.
Definition 2.1. A map of schemes Z → Z 0 is an infinitesimal thickening if it is a closed
embedding and the sheaf of ideals IZ/Z 0 is nilpotent.
Example 2.2. The important examples of an infinitesimal thickening are Spec k[t]/(tn ) →
Spec k[t]/(tn+1 ) induced by the quotient k[t]/(tn+1 ) → k[t]/(tn ). We write k[ε] = k[t]/(t2 )
which is called the ring of dual numbers. For n = 1 we have k[t]/(t) = k and Spec k is
a point while Spec k[ε] is a point with a tangent vector. This can be seen by looking at
maps out of Spec k and Spec k[ε]. If X is a variety over k, then a map Spec k → X is the
same as picking a point and a map Spec k[ε] → X is picking a point and a tangent vector.
Note that the map on the underlying topological spaces is a homeomorphism, which holds
in general for infinitesimal thickenings.
Definition 2.2. A map of schemes Y → X is formally étale if any commutative diagram
of solid lines where Z → Z 0 is an infinitesimal thickening , can be completed with a unique
dashed line:
Z
Y
Z0
X
We call such a map a lift of Z 0 → X.
BEZOUT’S THEOREM
3
Example 2.3. The map in example 2.1 is formally étale. We will check that for any
diagram of the form
C
G
C[x, x−1 ]
H
C[ε]
F
C[x, x−1 ]
that H exists and is unique. Note G sends x to λn for some λ ∈ C× , so by commutativity
F (x) = λn + aε
H(x) = λ + bε
1
for some a, b ∈ C. Commutativity of the diagram also forces b = (a/n) n . This is a well
defined map and is unique. Note that this shows that k × → k × given by x 7→ xn is étale
as long as the characteristic of k does not divide n.
This definition categorizes étale maps as those for which infinitesimal motion lifts
uniquely, which is similar to the criterion for local diffeomorphisms in differential geometry. Take Z → Z 0 to be the map Spec k → Spec k[ε] from example 2.2. If f : Y → X
is étale we consider the map on Zariski tangent spaces Dy f : Ty Y → Tf (y) X. Interpreting
maps from Spec k[ε] as picking out a point and a tangent vector, the existence of a lift
shows Dy f is surjective, while uniqueness ensures that Dy is injective.
2.2. Proper maps. The algebraic analog of unique path lifting is the valuative criterion
of properness.
Definition 2.3. A map of schemes Y → X is said to be valuatively proper if for any
discrete valuation ring R with field of fractions K and any commutative diagram of solid
lines, there exists a unique dashed line completing the diagram:
Spec K
Y
Spec R
X
Again, such a map is called a lift of Spec R → X.
Example 2.4. The map Pnk → Spec k is proper, closed embeddings are proper.
Proposition 2.1. Proper maps are stable under pullback, meaning if Y → X is proper
and Z → X any map then Y ×X Z → Z is proper.
Proof. Consider the diagram
Spec K
Z ×X Y
Y
Spec R
Z
X
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CHRISTIAN KLEVDAL
The dashed arrow exists and is unique since Y → X is proper, and the dotted arrow exists
and is unique by the universal property of the fiber product. The dotted arrow is the
required lift.
Considering the maps Pn → Spec k and Pm → Spec k we get the following corollary:
Corollary 2.1. Projection maps π : Pn × Pm → Pn are proper.
3. Transverse Intersections
Recall the earlier set up, where X is the space whose point are triples (f, g, p) where
f, g are of degree d, e respectively and p ∈ V (f ) ∩ V (g), and π : X → PN is the projection
map. The following proposition relates the intersection theory of V (f ) and V (g) with
properties of the map π.
Proposition 3.1. Curves V (f ) and V (g) intersect transversally if and only if the map
π : X → PN is formally étale at (f, g).
Proof. The intersection V (f ) ∩ V (g) = X(f,g) is transverse ⇐⇒ it is dimension 0 and
reduced ⇐⇒ no infinitesimal motion is possible in the fiber ⇐⇒ π is étale at (f, g). Proposition 3.2. The set U ⊆ PN over which π is étale is nonempty and open. Consequently π −1 U → U is proper étale morphism.
Proof. The fact that π −1 U → U is proper follows from the stability of proper maps under
base change.
To show U is open, we need to use a different criterion for étale maps, in particular
that (nice enought) maps of schemes are étale if they are flat and the sheaf of relative
differentials vanish. Generic flatness says that the locus V ⊂ PN over which π −1 V is flat
is open and dense. The support Z ⊂ X of ΩX/PN is closed so π(Z) is closed since π is
proper. Thus U = V ∩ (PN \ Z) which is open.
4. Proof of Bézout’s Theorem
We write V = π −1 U for U as above. Then π : V → U is both proper and étale. Suppose
a1 , . . . , ad ∈ k and b1 , . . . , be ∈ k are sets of distinct points and let
f = (x − a1 ) · · · (x − ad ),
g = (y − b1 ) · · · (y − be ).
The pair (f, g) represent a point in U and the fiber X(f,g) = Spec k[x, y]/(f, g). Note
MaxSpec k[x, y]/(f, g) = {(x − ai , y − bj ) | 1 ≤ i ≤ d, 1 ≤ j ≤ e}
so |X(f,g) | = de. The number of intersection points |X(f,g) | is the number of k-rational
points (maps Spec k → X(f,g) , denoted by X(f,g) (k)), since these correspond to intersection
points. The only part left is to show that for any points q1 = (f1 , g1 ), q2 = (f2 , g2 ) ∈ U
that Xq1 (k) = Xq2 (k).
BEZOUT’S THEOREM
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First we find a hyperplane H ⊆ PN so that q1 , q2 6∈ H. Then PN \ H ' AN . Consider
the closed linear embedding
γ : A1 → AN
t 7→ (1 − t)q1 + tq2 .
Our strategy will be to show that |Xγ(0) | = |XL | = |Xγ(1) | where L is the generic point
of imγ. The inverse image γ −1 U will be the complement of V (h) ⊆ Spec k[t] for some
h ∈ k[t]. Since γ(0), γ(1) ∈ U it follows that h(0) 6= 0 6= h(1), and from this we get that
the image of h under the quotient maps k[t] → k[t]/(tn ) is a unit. Hence there is a well
defined morphism k[t, h−1 ] → lim k[t]/(tn ) = k[[t]]. Thus we have defined a map
←−
Spec k[[t]] → γ −1 U ⊆ A1 → U
and we know that Spec k[[t]] = {(0), (t)} sends the generic point (0) to L. Thus we have
defined a map
Start with a k rational point Spec k → Xγ(0) . From this, we get the following diagram
π −1 U
Spec k
···
Spec k[t]/(tn )
···
Spec k[[t]]
U
The dashed lines can be filled in uniquely since π −1 U → U is étale and Spec k →
Spec k[t]/(tn ) is an infinitesimal thickening. The dotted line comes from the universal
property of inverse limits. Restricting the dotted line to the generic point, we obtain a
morphism Spec k(t) → π −1 (U ) which is a point in XL , i.e. we have constructed a map
Ψ : Xγ(0) (k) → XL (k(t)). It is injective because the map Spec k[[t]] → π −1 U is the unique
extension of the map Spec k → π −1 U that we started with. We will show that Ψ is surjective. Starting with a point of XL , i.e. a lift of L, is the same thing as giving a commutative
diagram of solid lines
Spec k((t))
π −1 U
Spec k[[t]]
U
The dashed line exists and is unique by the valuative criterion of properness, and using
the composition Spec k → Spec k[[t]] we get a point in Xγ(0) that maps to the given point
in XL . Therefore Ψ : Xγ(0) (k) → XL (k(t)) is bijective. The exact same argument exhibits
a bijection Xγ(1) (k) → XL (k(t)) hence we conclude that Xγ(0) (k) = Xγ(1) (k).
5. Deformation Theory
A key part of the proof of Bézout’s theorem is to show that the solutions (number of
points in a fiber) are constant under small perturbations. Deformation theory studies how
spaces behave under these small perturbations.
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CHRISTIAN KLEVDAL
Definition 5.1. If X0 is a variety over k, an infinitesimal deformation of X0 is a scheme
X, flat over Spec A and a commutative diagram
X0
X
Spec k
Spec A
such that A is a local artinian k-algebra and X0 ' X ×Spec A Spec k.
In our case, A was taken to be k[t]/(tn ).