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] d denote 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, so the parameter space of curves of degree d should be (C (d+2 2 ) {0})/C = P (d+2 2 ) 1. Letting N = ( ) ( d+2 2 + e+2 ) 2 2, points of P N are pairs (f, g) where f is a polynomial of degree d and g a polynomial of degree e. Consider the projection map: π : P N P 2 P N Points of P N P 2 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)} P N P 2 is a closed subvariety, and π restricts to the projection map X P N. The fiber X (f,g) over a point (f, g) P N 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 P N 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, x X there is a bijection p 1 (x) p 1 (x ). This can be constructed by taking a path γ with γ(0) = x 1
2 CHRISTIAN KLEVDAL and γ(1) = x, and a point y p 1 (x) is sent to γ(1) where γ 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 x n 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 D p f : T p M T f(p) N is an isomorphism. Definition 2.1. A map of schemes Z Z is an infinitesimal thickening if it is a closed embedding and the sheaf of ideals I Z/Z is nilpotent. Example 2.2. The important examples of an infinitesimal thickening are Spec k[t]/(t n ) Spec k[t]/(t n+1 ) induced by the quotient k[t]/(t n+1 ) k[t]/(t n ). We write k[ε] = k[t]/(t 2 ) 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 is an infinitesimal thickening, can be completed with a unique dashed line: Z Y Z X We call such a map a lift of Z X.
BEZOUT S THEOREM 3 Example 2.3. The map in example 2.1 is formally étale. diagram of the form C G C[x, x 1 ] H We will check that for any F C[ε] C[x, x 1 ] that H exists and is unique. Note G sends x to λ n for some λ C, so by commutativity F (x) = λ n + aε H(x) = λ + bε for some a, b C. Commutativity of the diagram also forces b = (a/n) 1 n. This is a well defined map and is unique. Note that this shows that k k given by x x n 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 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 D y f : T y Y T f(y) X. Interpreting maps from Spec k[ε] as picking out a point and a tangent vector, the existence of a lift shows D y f is surjective, while uniqueness ensures that D y 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 P n k 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
4 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 P n Spec k and P m Spec k we get the following corollary: Corollary 2.1. Projection maps π : P n P m P n 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 P N 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 P N 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 P N 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 P N over which π 1 V is flat is open and dense. The support Z X of Ω X/P N is closed so π(z) is closed since π is proper. Thus U = V (P N \ 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 a 1,..., a d k and b 1,..., b e k are sets of distinct points and let f = (x a 1 ) (x a d ), g = (y b 1 ) (y b e ). 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 a i, y b j ) 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 q 1 = (f 1, g 1 ), q 2 = (f 2, g 2 ) U that X q1 (k) = X q2 (k).
BEZOUT S THEOREM 5 First we find a hyperplane H P N so that q 1, q 2 H. Then P N \ H A N. Consider the closed linear embedding γ : A 1 A N t (1 t)q 1 + tq 2. Our strategy will be to show that X γ(0) = X L = X γ(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) 0 h(1), and from this we get that the image of h under the quotient maps k[t] k[t]/(t n ) is a unit. Hence there is a well defined morphism k[t, h 1 ] lim k[t]/(t n ) = k[[t]]. Thus we have defined a map Spec k[[t]] γ 1 U A 1 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 X γ(0). From this, we get the following diagram π 1 U Spec k Spec k[t]/(t n ) 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]/(t n ) 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 X L, i.e. we have constructed a map Ψ: X γ(0) (k) X L (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 X L, 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 X γ(0) that maps to the given point in X L. Therefore Ψ: X γ(0) (k) X L (k(t)) is bijective. The exact same argument exhibits a bijection X γ(1) (k) X L (k(t)) hence we conclude that X γ(0) (k) = X γ(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.
6 CHRISTIAN KLEVDAL Definition 5.1. If X 0 is a variety over k, an infinitesimal deformation of X 0 is a scheme X, flat over Spec A and a commutative diagram X 0 X Spec k Spec A such that A is a local artinian k-algebra and X 0 X Spec A Spec k. In our case, A was taken to be k[t]/(t n ).