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easy to deduce from (1.37) that at least one of the coordinates v0...m...0 corresponding 4 Quasiprojective Varieties 53 to the monomial um i the map is nonzero, and that, for example, on the open set vm0...0 = 0, u0 = vm0...0, ui = vm−1,0...1...0 for i ≥ 1 is a regular inverse of vm. Hence vm(Pn) is deﬁned by (1.37), and vm is an isomorphic embedding Pn → PN. The signiﬁcance of the Veronese embedding is that if F = ai0...inui0 0 · · · uin n is a form of degree m in the homogeneous coordinates of Pn and H ⊂ Pn is the hypersurface deﬁned by F = 0, then vm(H ) ⊂ vm(Pn) ⊂ PN is the intersection of vm(Pn) with the hyperplane of PN with equation ai0...invi0...in. Thus the Veronese embedding allows us to reduce the study of some problems concerning hypersurfaces of degree m to the case of hyperplanes. The mth Veronese image of the projective line vm(P1) ⊂ Pm is called the Veronese curve, the twisted m-ic curve, or the rational normal curve of degree m. 4.5 Exercises to Section 4 1 Prove that an afﬁne variety U is irreducible if and only if its projective closure U is irreducible. 0 its projective closure U in Pn. Prove 2 Associate with any afﬁne variety U ⊂ An that this deﬁnes a one-to-one correspondence between the afﬁne subvarieties of An 0 and the projective subvarieties of Pn with no components contained in the hyperplane S0 = 0. 3 Prove that the variety X = A2 \ (0, 0) is not isomorphic to an afﬁne variety. [Hint: Compute the ring k[X] of regular functions on X, and use the fact that if Y is an afﬁne variety, every proper ideal A k[Y ] deﬁ
nes a nonempty set.] 4 Prove that any quasiprojective variety is open in its projective closure. 5 Prove that every rational map ϕ : P1 → Pn is regular. 6 Prove that any regular map ϕ : P1 → An maps P1 to a point. 7 Deﬁne a birational map f from an irreducible quadric hypersurface X ⊂ P3 to the plane P2 by analogy with the stereographic projection of Example 1.22. At which points is f not regular? At which points is f −1 not regular? 8 In Exercise 7, ﬁnd the open subsets U ⊂ X and V ⊂ P2 that are isomorphic. 54 1 Basic Notions 9 Prove that the map y0 = x1x2, y1 = x0x2, y2 = x0x1 deﬁnes a birational map of P2 to itself. At which points are f and f −1 not regular? What are the open sets mapped isomorphically by f? (Compare Section 3.5, Chapter 4.) 10 Prove that the Veronese image vm(Pn) ⊂ PN is not contained in any linear subspace of PN. 11 Prove that the variety P2 \ X, where X is a plane conic, is afﬁne. [Hint: Use the Veronese embedding.] 5 Products and Maps of Quasiprojective Varieties 5.1 Products The deﬁnition of the product of afﬁne varieties (Example 1.5) was so natural as not to require any comment. For general quasiprojective varieties, things are somewhat more complicated. Because of this, we ﬁrst consider quasiprojective subvarieties of afﬁne spaces. If X ⊂ An and Y ⊂ Am are varieties of this type then X × Y = {(x, y) \| x ∈ X, y ∈ Y } is a quasiprojective variety in An × Am. Indeed, if X = X1 \ X0 and Y = Y1 \ Y0 where X1, X0 ⊂ An, and Y1, Y0 ⊂ Am are closed subvarieties, then writing X × Y = X1
× Y1 \ (X1 × Y0) ∪ (X0 × Y1) shows that X × Y is quasiprojective. This quasiprojective variety is the product of X and Y. At this point, we should check that if X and Y are replaced by isomorphic varieties then so is X × Y. This is easy to see. Suppose that ϕ : X → X ⊂ Ap and ψ : Y → Y ⊂ Aq are isomorphisms. Then deﬁned by (ϕ × ψ)(x, y) = (ϕ(x), ψ(y)) is a regular map, with regular inverse ϕ−1 × ψ −1. We return to quasiprojective varieties, and decide what properties we want the notion of product to have. Let X ⊂ Pn and Y ⊂ Pm be two quasiprojective varieties. Write X × Y for the set of pairs (x, y) with x ∈ X and y ∈ Y. We want to consider this set as a quasiprojective variety, and for this, we have to produce an embedding ϕ of X × Y into a projective space PN in such a way that the image ϕ(X × Y ) ⊂ PN is a quasiprojective subvariety. At the same time, it is reasonable to require that the deﬁnition is local, in the sense that for any points x ∈ X and y ∈ Y there exist afﬁne neighbourhoods X ⊃ U x and Y ⊃ V y such that ϕ(U × V ) is open in ϕ(X × Y ), and ϕ deﬁnes an isomorphism of the product of the afﬁne varieties U and V, whose deﬁnition we already know, to the subvariety ϕ(U × V ) ⊂ ϕ(X × Y ). It is easy to see that the local property of ϕ determines it uniquely; more precisely, if ψ : X × Y → PM is another embedding of the same kind, then ψ ◦ ϕ−1 deﬁnes an isomorphism between ϕ(X × Y ) and ψ(X × Y ). Indeed,
for this, it is enough to prove that for any x ∈ X and y ∈ Y, there exist neighbourhoods ϕ(X × Y ) ⊃ W1 ϕ(x, y) and ψ(X × Y ) ⊃ W2 ψ(x, y) such that ψ ◦ ϕ−1 : W1 → W2 is an 5 Products and Maps of Quasiprojective Varieties 55 isomorphism. Consider afﬁne neighbourhoods X ⊃ U x and Y ⊃ V y the existence of which is provided by the local property; passing if necessary to smaller afﬁne neighbourhoods, we can assume that U × V is isomorphic to both ϕ(U × V ) and ψ(U × V ). Then ϕ(U × V ) = W1 and ψ(U × V ) = W2 are the afﬁne neighbourhoods we need, since both are isomorphic to the product U × V of the afﬁne varieties U and V. We now proceed to construct an embedding ϕ with the required properties. For this, we can at once restrict to the case X = Pn, Y = Pm; for once an embedding ϕ : Pn × Pm → PN is constructed, it is easy to check that its restriction to X × Y ⊂ Pn × Pm has all the required properties. To construct the embedding ϕ, consider the projective space PN with homogeneous coordinates wij having two subscripts i = 0,..., n and j = 0,..., m; thus N = (n + 1)(m + 1) − 1. If x = (u0 : · · · : un) ∈ Pn and y = (v0 : · · · : vm) ∈ Pm then we set ϕ(x, y) = (wij ), with wij = uivj for 0 ≤ i ≤ n and 0 ≤ j ≤ m. (1.38) Multiplying the homogeneous coordinates of x or y by a common scalar obviously does not change the point ϕ(x, y) ∈ PN. To prove that ϕ(Pn × Pm) is a closed set of PN, we write out its deﬁning equations: wij wkl = w
kj wil for 0 ≤ i, k ≤ n and 0 ≤ j, l ≤ m. (1.39) Substituting the wij given by (1.38) shows at once that they satisfy (1.39). Conversely, if wij satisfy (1.39), and, say, w00 = 0, then setting k, l = 0 in (1.39) gives that (wij ) = ϕ(x, y), where x = (w00 : · · · : wn0) and y = (w00 : · · · : w0m). ⊂ Pn given by u0 = 0, Am 0 This argument proves at the same time that ϕ(x, y) determines x and y uniquely, that is, ϕ is an embedding Pn × Pm → PN with image the subvariety W ⊂ PN ⊂ Pm by deﬁned by (1.39). Consider the open sets An 0 ⊂ PN by w00 = 0, having inhomogeneous coordinates xi = ui/u0, v0 = 0, and AN 00 yj = vj /v0 and zij = wij /w00 respectively. Then obviously ϕ(An 0 ) = W ∩ 0 = W00. As we have just seen, on W00 we have zi0 = xi, z0j = yj and zij = AN 00 = W00 is xiyj = zi0z0j for i, j > 0. It follows from this that ϕ(Pn × Pm) ∩ AN 00 isomorphic to An+m with coordinates (x1,..., xn, y1,..., ym), and ϕ deﬁnes an → W00. This proves that ϕ satisﬁes the local requirement of isomorphism An 0 our construction. The embedding ϕ : Pn × Pm → PN with N = (n + 1)(m + 1) − 1 just constructed is called the Segre embedding, and the image Pn × Pm ⊂ PN the Segre variety. × Am 0 × Am Remark 1.3 The point (wij ) can be interpreted as an (n + 1) × (m + 1) matrix, and (1.39)
express the vanishing of the 2 × 2 minors: det wij wil wkj wkl = 0. 56 1 Basic Notions That is, they express the condition that the matrix (wij ) has rank 1, and (1.38) shows that such a matrix is a product of a 1 × (n + 1) column matrix and a (m + 1) × 1 row matrix. Thus ϕ(Pn × Pm) is a determinantal variety (see Example 1.26). Remark 1.4 The simplest case n = m = 1 has a simple geometric interpretation: in this case, (1.39) is the single equation w11w00 = w01w10, so that ϕ(P1 × P1) is just a nondegenerate quadric surface Q ⊂ P3. For α = (α0, α1) ∈ P1, the set ϕ(α × P1) is the line in P3 given by α1w00 = α0w10, α1w01 = α0w11. As α runs through P1, these lines give all the generators of one of the two families of lines of Q. Similarly the set ϕ(P1 × β) is a line of P3, and as β runs through P1, these lines give the generators of the other family. It is convenient, now that we have deﬁned the product X × Y of quasiprojective varieties using the embedding ϕ : Pn × Pm → PN, with N = (n + 1)(m + 1) − 1, to explain some ideas of algebraic geometry that are originally deﬁned in terms of Pn × Pm and of this embedding. Let us determine, for example, what are the subsets of Pn × Pm that are mapped by ϕ to algebraic subvarieties of PN ; these will then be the closed algebraic subvarieties of the product Pn × Pm. A subvariety V ⊂ PN is deﬁned by equations Fk(w00 : · · · : wnm) = 0, where the Fk are homogeneous polynomials in the wij. After making the substitution (1.38), we can write these in the coordinates ui and vj as equations Gk(u0 : · · · :
un; v0 : · · · : vm) = 0, where the Gk are homogeneous in each set of variables u0,..., un and v0,..., vm, and of the same degree in both. Conversely, it is easy to see that a polynomial with this bihomogeneity property can always be written as a polynomial in the products uivj. However, equations that are bihomogeneous in ui and vj always deﬁne an algebraic subvariety of Pn × Pm even if the degrees of homogeneity in the two sets of variables are different. For if G(u0 : · · · : un; v0 : · · · : vm) has degree r in ui and s in vj, and, say, r > s, then G = 0 is equivalent to the system of equations vr−s i G = 0 for i = 0,..., m, and we know that these deﬁne an algebraic variety. In what follows, we also need to answer the same question for the product Pn × Am. Suppose that Am = Am ⊂ Pm is given by v0 = 0. The equations of a 0 closed subset of Pn × Pm are Gk(u0 : · · · : un; v0 : · · · : vm) = 0. Suppose that Gk is homogeneous of degree rk in v0 : · · · : vm. Dividing the equation by vrk 0 and setting yj = vj /v0 gives equations gk(u0 : · · · : un; y1 : · · · : ym) = 0 that are homogeneous in the ui, and (in general) inhomogeneous in the yj. This proves the following result: Theorem 1.9 A subset X ⊂ Pn × Pm is a closed algebraic subvariety if and only if it is given by a system of equations Gk(u0 : · · · : un; v0 : · · · : vm) = 0 for k = 1,..., t, 5 Products and Maps of Quasiprojective Varieties 57 homogeneous separately in each set of variables ui and vj. Every closed algebraic subvariety of Pn × Am is given by a system of equations g
k(u0 : · · · : un; y1 : · · · : ym) = 0 for k = 1,..., t (1.40) that are homogeneous in (u0,..., un). Of course, the same kind of thing holds for a product of any number of spaces. For example, a subvariety of Pn1 × · · · × Pnk is given by a system of equations homogeneous in each of the k sets of variables. 5.2 The Image of a Projective Variety is Closed The image of an afﬁne variety under a regular map does not have to be a closed set; this is illustrated in Examples 1.13–1.14 for a map from an afﬁne variety to an afﬁne variety. For maps from an afﬁne variety to a projective variety it is even more obvious: an example is given by the embedding of An into Pn as the open subset An 0. In this respect, projective varieties are fundamentally different from afﬁne varieties. Theorem 1.10 The image of a projective variety under a regular map is closed. The proof uses a notion that will occur later. Let f : X → Y be a regular map between arbitrary quasiprojective varieties. The subset Γf of X × Y consisting of pairs (x, f (x)) is called the graph of f. Lemma 1.4 The graph of a regular map is closed in X × Y. Proof First of all, it is enough to assume that Y is projective space. Indeed, if Y ⊂ Pm then X × Y ⊂ X × Pm, and f deﬁnes a map f : X → Pm with Γf = Γf ⊂ X × Y ⊂ X × Pm. Thus set Y = Pm. Let ι be the identity map from Pm to itself. Consider the regular map (f, ι) : X × Pm → Pm × Pm given by (f, ι)(x, y) = (f (x), y). Obviously Γf is the inverse image under the regular map (f, ι) of the graph Γι of ι. We proved in Section 4.2 that the inverse image of a closed set under a regular map is
closed. Hence everything reduces to proving that Γι ⊂ Pm × Pm is closed. But Γι consists of points (x, y) ∈ Pm × Pm such that x = y. If x = (u0 : · · · : um) and y = (v0 : · · · : vm) then the condition is that (u0 : · · · : um) and (v0 : · · · : vm) are proportional; this condition can be expressed uivj = uj vi, that is, wij = wj i for i, j = 0,..., m. This proves that Γι is closed, and therefore the lemma. We return to the proof of the theorem. Let Γf be the graph of f, and p : X × Y → Y the second projection, deﬁned by p(x, y) = y. Obviously f (X) = p(Γf ). In view of Lemma 1.4, Theorem 1.10 follows from the following more general assertion. 58 1 Basic Notions Theorem 1.11 If X is a projective variety, and Y a quasiprojective variety, the second projection p : X × Y → Y takes closed sets to closed sets. Proof The proof of this theorem can be reduced to a simple particular case. First of all, if X ⊂ Pn is a closed subset then the theorem for X follows from the theorem for Pn: for X × Y is closed in Pn × Y, so that if Z is closed in X × Y, it is also closed in Pn × Y. Thus we can assume that X = Pn. Secondly, since closed is a local property, it is enough to cover Y by afﬁne open sets Ui and prove the theorem for each of these. Hence we can assume that Y is an afﬁne variety. Finally if Y ⊂ Am then Pn × Y is closed in Pn × Am, and hence it is enough to prove the theorem in the particular case X = Pn and Y = Am. Let’s see what the theorem means in this case. According to Theorem 1.9, any closed subvariety Z ⊂ Pn × Am is deﬁned by (1.40), that we write in the form gi(u
; y) = 0 for i = 1,..., t. Write p : Z → Am for the restriction of the second projection. Obviously the inverse image p−1(y0) of y0 ∈ Am consists of all nonzero solutions of the system gi(u, y0) = 0, and hence y0 ∈ p(Z) if and only if the system of equations gi(u; y0) = 0 has a nonzero solution in (u0,..., un). Thus Theorem 1.11 asserts that for any system of (1.40), the subset T of y0 ∈ Am for which gi(u; y0) = 0 has a nonzero solution is closed. Now in view of Lemma 1.1, gi(u; y0) = 0 has a nonzero solution if and only if g1(u, y0),..., gt (u, y0) ⊃ Is for all s = 1, 2,.... We now show that for given s ≥ 1, the set of points y0 ∈ Am for which (g1(u, y0),..., gt (u, y0)) ⊃ Is is a closed set Ts. Then T = Ts, and T is also closed. Write ki for the degree of the homogeneous polynomial gi(u, y) in the variables u0,..., un. Let {M α}α be the monomials of degree s in u0,..., un written out in some order. The condition (g1(u, y0),..., gt (u, y0)) ⊃ Is means that each monomial M α can be expressed in the form M α = t i=1 gi(u, y0)Fi,α(u). (1.41) Comparing the homogeneous components of degree s shows that there must also be an expression (1.41) for M α with deg Fi,α = s − ki, or Fi,α = 0 if ki > s. Let {N β }β be the monomials of degree s − ki written out in some order. We see that the i conditions (1.41) hold if and only if every monomial M α is a linear combination of the polynomials g
i(u, y0)N β i. This, in turn, is equivalent to the condition that the polynomials gi(u, y0)N β i span the entire vector space S of homogeneous polynomials of degree s in u0,..., un. Conversely, (g1(u, y0),..., gt (u, y0)) ⊃ Is means that gi(u, y0)N β i do not span S. To turn this condition into equations for Ts, write out the coefﬁcients of the M α appearing in all the polynomials gi(u, y0)N β i as a rectangular matrix {aαβ}, and set to zero all of its σ × σ minors, where σ = dim S. These minors are obviously polynomials in the coefﬁcients of the polynomials gi(u, y0), and 5 Products and Maps of Quasiprojective Varieties 59 are therefore polynomials in the coordinates of the point y0; they give the equations of the set Ts. Theorem 1.11 is proved, and with it Theorem 1.10. Remark One sees from the proof that Theorem 1.10 generalises to a wider class of maps f : X → Y between quasiprojective varieties, namely those that factor as a composite of a closed embedding ι : X → Pn × Y (that is, an isomorphism of X with a closed subvariety) and the projection p : Pn × Y → Y. Such maps are said to be proper. For example, if f : X → Y is a regular map of projective varieties then the restriction f : f −1(U ) → U to an open subset U ⊂ Y is proper. Obviously if f : X → Y is a proper map the inverse image f −1(y) of a point y ∈ Y is a projective variety. Corollary 1.1 If ϕ is a regular function on an irreducible projective variety then ϕ ∈ k, that is, ϕ is constant. Proof We can view ϕ as a map f : X → A1, and hence as a map f : X → P1. Since ϕ is a regular function, f is a regular
map, and hence so is f ; by Theorem 1.10 its image f (X) ⊂ P1 is closed. But since f itself is regular, f (X) = f (X), and therefore f (X) is a closed subset of P1 and is contained in A1, that is, it does not contain the point at inﬁnity x∞ ∈ P1. It follows from this that either f (X) = A1 or f (X) is a ﬁnite set S ⊂ A1 (see Example 1.3). The ﬁrst case is impossible, since f (X) is also supposed to be closed in P1, and A1 is not. Hence f (X) = S. If S f −1(αi), and t > 1 would consists of ﬁnitely many points α1,..., αt then X = contradict the irreducibility of X. Hence S consists of one point only, and so ϕ is constant. The corollary is proved. Corollary 1.1 and Theorem 1.7 provide an example of afﬁne and projective varieties having diametrically opposite properties. On an afﬁne variety there is a host of regular functions (they make up the whole coordinate ring k[X]), but on an irreducible projective variety, only the constants. The next result is a second example of afﬁne and projective varieties being opposites. Corollary 1.2 A regular map f : X → Y from an irreducible projective variety X to an afﬁne variety Y maps X to a point. Proof Suppose that Y ⊂ Am. Then f is given by m functions f (x) = (ϕ1(x),..., ϕm(x)). Each of the functions ϕi is constant by Corollary 1.1, that is ϕi = αi ∈ k. Hence f (X) = (α1,..., αm). The corollary is proved. We give another example of an application of Theorem 1.10. For this, we use the representation of forms of degree m in n + 1 variables by points of the projective space PN with N = νn,m = − 1, as in
Example 1.28. m+n m Proposition Points ξ ∈ PN corresponding to reducible homogeneous polynomials F form a closed set. 60 1 Basic Notions Remark 1.5 The proposition asserts that the condition for a homogeneous polynomial to be reducible can be written as polynomial conditions on its coefﬁcients. For curves of degree 2, that is, the case m = n = 2, this relation is well known 2 from coordinate geometry: if F = i=0 aij UiUj then F is irreducible if and only if det \|aij \| = 0. Remark 1.6 Passing to inhomogeneous coordinates, we see that in the vector space of all polynomial of degree ≤m, the reducible polynomials together with the polynomials of degree <m form a closed set. Proof Proceeding to the proof of the proposition, we write X ⊂ PN for the set of points ξ corresponding to reducible polynomials, and Xk for the set of points corresponding to polynomials F that split as a product of two polynomials of degrees k and m − k (for k = 1,..., m). Obviously X = Xk, and we need only prove that each Xk is closed. n+k k Consider the projective space Pνn,k and Pνn,m−k of forms of degree k and m − k, − 1 is as in Example 1.28. Multiplying polynomials of degree where νn,k = k and m − k deﬁnes a map f : Pνn,k × Pνn,m−k → PN, and it is easy to see that f is regular. Obviously Xk = f (Pνn,k × Pνn,m−k ). We saw in Section 5.1 that the product of two projective spaces is a projective variety, and hence Xk closed follows by Theorem 1.10. The proposition is proved. 5.3 Finite Maps The projection map introduced in Section 4.4 has an important property; in order to state this, we ﬁrst recall some notions from algebra. Let B be a ring, and A a subring containing the identity element 1B. We say that an element b ∈ B is integral over
A if it satisﬁes an equation bk + a1bk−1 + · · · + ak = 0 with ai ∈ A. B is integral over A if every element b ∈ B is integral over A. It is easy to prove (see for example Atiyah and Macdonald [8, Proposition 5.1 and Corollary 5.2 of Chapter 5]) that a ring B that is ﬁnitely generated as an A-algebra is integral over A if and only if it is ﬁnite as a module over A. Let X and Y be afﬁne varieties and f : X → Y a regular map such that f (X) is dense in Y. Then f ∗ deﬁnes an isomorphic inclusion k[Y ] → k[X]. We view k[Y ] as a subring of k[X] by means of f ∗. Deﬁnition 1.1 f is a ﬁnite map if k[X] is integral over k[Y ]. From the properties of integral rings recalled above it follows that the composite of two ﬁnite maps is again ﬁnite. A typical example of a map that is not ﬁnite is Example 1.13. 5 Products and Maps of Quasiprojective Varieties 61 Example 1.29 Let X be an afﬁne algebraic variety, G a ﬁnite group of automorphisms of X and Y = X/G the quotient space (see Example 1.21). Then the map ϕ : X → Y is ﬁnite. Indeed, the proof of Proposition A.6 shows that the generators ui of the algebra k[X] are integral over the algebra k[X]G = k[Y ]. It follows from this that k[X] is integral over k[Y ]. If f is a ﬁnite map then any point y ∈ Y has at most a ﬁnite number of inverse images. Indeed, suppose that X ⊂ An and let t1,..., tn be the coordinates of An viewed as functions on X. It is enough to prove that any coordinate ti takes only a ﬁnite number of values on the set f −1(y). By deﬁnition ti satis�
�es an equation + a1t k−1 + · · · + ak = 0 with ai ∈ k[Y ]. For y ∈ Y and x ∈ f −1(y), we get an t k i i equation ti(x)k + a1(y)ti(x)k−1 + · · · + ak(y) = 0, (1.42) which has only a ﬁnite number of roots. The meaning of the ﬁnite condition is that as y moves in Y, none of the roots of (1.42) tends to inﬁnity, since the coefﬁcient 1 of the leading term does not vanish on Y. Thus as y moves in Y, points of f −1(y) can merge together, but cannot disappear. We make this remark more precise in the following result. Theorem 1.12 A ﬁnite map is surjective. Proof Let X and Y be afﬁne varieties, f : X → Y a ﬁnite map, and y ∈ Y. Write my for the ideal of k[Y ] consisting of functions that take the value 0 at y. If t1,..., tn are the coordinate functions on Y and y = (α1,..., αn) then my = (t1 − α1,..., tn − αn). The equations of the variety f −1(y) then have the form f ∗(t1) = α1,..., f ∗(tn) = αn, and by the Nullstellensatz f −1(y) = ∅ if and only if the elements f ∗(ti) − αi generate the trivial ideal: f ∗(t1) − α1,..., f ∗(tn) − αn = k[X]. From now on we view k[Y ] as a subring of k[X], and do not distinguish between a function u ∈ k[Y ] and f ∗(u) ∈ k[X]. Then the above condition is of the form (t1 − α1,..., tn − αn) = k[X], that is, myk[X] = k[X]. Since k[X] is integral over
k[Y ] it follows that it is a ﬁnite k[Y ]-module; Theorem 1.12 follows from this and the following purely algebraic assertion: Lemma If a ring B is a ﬁnite A-module where A ⊂ B is a subring containing 1B, then for an ideal a of A, a A =⇒ aB B. See Proposition A.11, Corollary A.1 for the proof. This completes the proof of Theorem 1.12. Corollary A ﬁnite map takes closed sets to closed sets. 62 1 Basic Notions Proof It is enough to check this for an irreducible closed set Z ⊂ X. We apply Theorem 1.12 to the restriction of f to Z, that is f : Z → f (Z). This is clearly a ﬁnite map between afﬁne varieties, hence f (Z) = f (Z) by Theorem 1.12, that is, f (Z) is closed. The corollary is proved. Finiteness is a local property: Theorem 1.13 If f : X → Y is a regular map of afﬁne varieties, and every point x ∈ Y has an afﬁne neighbourhood U x such that V = f −1(U ) is afﬁne and f : V → U is ﬁnite, then f itself is ﬁnite. Proof Set k[X] = B, k[Y ] = A. Principal open sets were deﬁned in Section 4.2. We can take a neighbourhood U of any point of Y such that U is a principal open set and satisﬁes the assumption of the theorem (see Exercise 11 of Section 5.5). Let D(gα) be a family of such open sets, which we can take to be ﬁnite. Then Y = D(gα), that is, the ideal generated by the gα is the whole of A. In our case Vα = f −1(D(gα)) = D(f ∗(gα)) and k[D(gα)] = A[1/gα], k[Vα] = B[1/gα]. By assumption B[1/gα] has a ﬁn
ite basis ωi,α over A[1/gα]. We can assume that ωi,α ∈ B, since if the basis consisted of elements ωi,α/gmi α with ωi,α ∈ B then the elements ωi,α would also be a basis. We take the union of all the bases ωi,α and prove that they form a basis of B over A. An element b ∈ B has an expression b = i ai,α gnα α ωi,α for each α. Since the gnα α hα = 1. Hence α gnα α generate the unit ideal of A, there exist hα ∈ A such that b = b α gnα α hα = i α ai,αhαωi,α, which proves the theorem. Deﬁnition 1.2 A regular map f : X → Y of quasiprojective varieties is ﬁnite if any point y ∈ Y has an afﬁne neighbourhood V such that the set U = f −1V is afﬁne and f : U → V is a ﬁnite map between afﬁne varieties. Obviously, for a ﬁnite map f the set f −1(y) is ﬁnite for every y ∈ Y. It follows from Theorem 1.12 that any ﬁnite map is surjective. This property has important consequences, that relate to arbitrary maps. Theorem 1.14 If f : X → Y is a regular map and f (X) is dense in Y then f (X) contains an open set of Y. 5 Products and Maps of Quasiprojective Varieties 63 Proof The assertion of the theorem reduces at once to the case that both X and Y are irreducible and afﬁne, and we assume this in what follows. Then k[Y ] ⊂ k[X]. We write r for the transcendence degree of the ﬁeld extension k(X)/k(Y ), and choose r elements u1,..., ur ∈ k[X] that are algebraically independent over k(Y ). Then k[X] ⊃ k[Y ][u1,..., ur ] ⊃ k[Y ] and k[
Y ][u1,..., ur ] = k Y × Ar. This represents f as the composite f = g ◦ h of two maps h : X → Y × Ar and g : Y × Ar → Y, where g is simply the projection to the ﬁrst factor. Any element v ∈ k[X] is algebraic over k[Y × Ar ], hence there exists an element a ∈ k[Y × Ar ] such that av is integral over k[Y × Ar ]. Let v1,... vm be a system of generators of k[X], and a1,..., am ∈ k[Y × Ar ] elements such that each aivi is integral over k[Y × Ar ], and set F = a1 · · · am. Since all the functions ai are invertible on the principal open set D(F ) ⊂ Y × Ar, the functions vi on D(h∗(F )) ⊂ X are integral over k[Y × Ar ][1/F ], that is, the restricted map h∗(F ) h : D → D(F ) is ﬁnite. Thus h(D(h∗(F ))) = D(F ) by Theorem 1.12, so that D(F ) ⊂ h(X). It remains to prove that g(D(F )) contains an open set of Y. Suppose that F = F (y, T ) = Fα(y)T α, where T α are monomials in the variables T1,..., Tr, the coordinates of Ar. For points y ∈ Y at which not all Fα(y) = 0, there exist values Ti = τi for which F (y, τ ) = 0. Hence g(D(F )) ⊃ D(Fα). This proves Theorem 1.14. Theorem 1.14 shows one respect in which regular maps of algebraic varieties are simpler than continuous or differentiable maps. The famous example of an everywhere dense line in the torus T = R2/Z2, a map such as √ f : R1 → T given by f (x) = (x, 2x) mod Z2 is an example of a situation that cannot happen for algebraic varieties, by Theorem 1.14. Theorem 1.15 If X ⊂
Pn is a closed subvariety disjoint from a d-dimensional linear subspace E ⊂ Pn then the projection π : X → Pn−d−1 with centre E (see Example 1.27) deﬁnes a ﬁnite map X → π(X). Proof Let y0,..., yn−d−1 be homogeneous coordinates on Pn−d−1, and suppose that π is given by yj = Lj (x) for j = 0, where x ∈ X. Obviously Ui = π −1(An−d−1 ) ∩ X is given by the condition Li(x) = 0, and is an afﬁne open subset of X. We prove that π : Ui → An−d−1 ∩ π(X) is a ﬁnite map. Any function g ∈ k[Ui] is of the form g = Gi(x0,..., xn)/Lm i, where Gi is a form of degree m. i i 64 1 Basic Notions Consider the map π1 : X → Pn−d given by zj = Lm j (x) for j = 0 and zn−d = Gi(x), where z0,..., zn−d are homogeneous coordinates in Pn−d. This is a regular map, and its image π1(X) ⊂ Pn−d is closed by Theorem 1.10. Suppose that π1(X) is given by equations F1 = · · · = Fs = 0. Since X is disjoint from E, the forms Li for i = 0 have no common zeros on X. Hence the point 0 = (0 : · · · : 0 : 1) ∈ Pn−d is not the equations z0 = · · · = zn−d−1 = contained in π1(X), or in other words, F1 = · · · = Fs = 0 do not have solutions in Pn−d. By Lemma 1.1, it follows from this that (z0,..., zn−d−1, F1,..., Fs) ⊃ Ik for some k > 0. In particular, (z0,.
.., zn−d−1, F1,..., Fs) zk n−d. This means that we can write zk n−d = n−d−1 j =0 zj Hj + s j =1 Fj Pj, where Hj and Pj are polynomials. Writing H (q) for the homogeneous component of H of degree q, we deduce from this that Φ(z0,..., zn−d ) = zk n−d − zj H (k−1) j = 0 on π1(X). (1.43) The homogeneous polynomial Φ is of degree k and as a polynomial in zn−d it has leading coefﬁcient 1: Φ = zk n−d − k−1 j =0 Ak−j (z0,..., zn−d−1)zj n−d. (1.44) If we substitute in (1.43) the formulas deﬁning the map π1 : X → Pn−d, we get n−d−1, Gi) = 0 on X, with Φ of the form (1.44). Dividing this 0,..., Lm that Φ(Lm relation by Lmk i we get the required relation gk − k−1 j =0 0,..., 1,..., xm xm n−d−1 gj = 0, Ak−j where xr = yr /yi are coordinates on An−d−1 i. The theorem is proved. Using the Veronese embedding (Example 1.28) allows the following substantial generalisation of Theorem 1.15. Theorem 1.16 Suppose that F0,..., Fs are forms of degree m on Pn having no common zeros on a closed variety X ⊂ Pn. Then ϕ(x) = F0(x) : · · · : Fs(x) deﬁnes a ﬁnite map ϕ : X → ϕ(X). 5 Products and Maps of Quasiprojective Varieties 65 − 1) and Proof Let vm : Pn → PN be the Veronese embedding (with N = Li the linear
forms on PN corresponding to the forms Fi on Pn. Then obviously ϕ = π ◦ vm where π is the projection deﬁned by the linear forms L0,..., Ls. Since vm : X → vm(X) is an isomorphism, Theorem 1.16 follows from Theorem 1.15. n+m m 5.4 Noether Normalisation Consider an irreducible projective variety X ⊂ Pn distinct from the whole of Pn. Then there exists a point x ∈ Pn \ X, and the map ϕ obtained by projecting X away from x will be regular. The image ϕ(X) ⊂ Pn−1 is projective by Theorem 1.10, and the map ϕ : X → ϕ(X) is ﬁnite by Theorem 1.15. If ϕ(X) = Pn−1 then we can repeat the same argument for it. We ﬁnally arrive at a map X → Pm, which is ﬁnite, since it is a composite of ﬁnite maps. The result we have proved is called the Noether normalisation theorem: Theorem 1.17 For an irreducible projective variety X there exists a ﬁnite map ϕ : X → Pm to a projective space. The analogous result also holds for afﬁne varieties. To prove this, consider an afﬁne variety X ⊂ An. Embed An as an open An ⊂ Pn, and write X for the projective closure of X in Pn. Suppose that X = An. We choose a point at inﬁnity x ∈ Pn \ An with x /∈ X, and consider the projection ϕ : X → Pn−1 from this point. Here X will map to points in the ﬁnite part of Pn−1, that is, to points of An−1 = Pn−1 ∩ An. We can repeat this process as long as X = An, and as a result we arrive at a projection ϕ : X → Pm for which ϕ(X) = Am. This proves the following result. Theorem 1.18 For an irreducible afﬁne variety X there exists a ﬁnite
map ϕ : X → Am to an afﬁne space. Theorems 1.17–1.18 allow us to reduce the study of certain (very coarse) properties of projective and afﬁne varieties to the case of projective and afﬁne spaces. When m = 1 this point of view is due to Riemann, who considered algebraic curves as coverings of the Riemann sphere (P1 over the complex number ﬁeld C). Theorem 1.18 means that an integral domains A that is ﬁnitely generated over the ﬁeld k is integral over a subring isomorphic to a polynomial ring. This result can also easily be proved directly. 5.5 Exercises to Section 5 1 Prove that the Segre variety ϕ(Pn × Pm) ⊂ PN (where N = (n + 1)(m + 1) − 1) is not contained in any linear subspace strictly smaller than the whole of PN. 66 1 Basic Notions 2 Consider the two maps of varieties P1 × P1 → P1 given by p1(x, y) = x and p2(x, y) = y. Prove that p1(X) = p2(X) = P1 for any closed irreducible subset X ⊂ P1 × P1, unless X is of one of the following types: (a) a point (x0, y0) ∈ P1 × P1; (b) a line x0 × P1 for x0 ∈ P1 a ﬁxed point; (c) a line P1 × y0. 3 Verify Theorem 1.10, Corollary 1.1 directly for the case X = Pn. 4 Let X = A2 \ x where x is a point. Prove that X is not isomorphic to an afﬁne nor a projective variety (compare Exercise 3 of Section 4.5). 5 The same question as Exercise 4, for X = P2 \ x. 6 The same question as Exercise 4, for X = P1 × A1. 7 Is the map f : A1 → X ﬁnite, where X is given by y2 = x3, and f by f (t) = (t 2, t 3). 8 Let X �
�� Ar be a hypersurface of Ar and L a line of Ar through the origin. Let ϕL be the map projecting X parallel to L to an (r − 1)-dimensional subspace not containing L. Write S for the set of all lines L such that ϕL is not ﬁnite. Prove that S is an algebraic variety. [Hint: Prove that S = X ∩ Pr−1 ∞.] Find S if r = 2 and X is given by xy = 1. 9 Prove that any intersection of afﬁne open subsets is afﬁne. [Hint: Use Example 1.20.] 10 Prove that forms of degree m = kl in n + 1 variables that are lth powers of forms correspond to the points of a closed subset of PN, where N = − 1 = νn,m. n+m m 11 Let f : X → Y be a regular map of afﬁne varieties. Prove that the inverse image of a principal afﬁne open set is a principal afﬁne open set. 6 Dimension 6.1 Deﬁnition of Dimension In Section 2 we saw that closed algebraic subvarieties X ⊂ A2 are ﬁnite sets of points, algebraic plane curves, and A2 itself. This division into three cases corresponds to the intuitive notion of dimension, with varieties of dimension 0, 1 and 2. Here we give the deﬁnition of the dimension of an arbitrary algebraic variety. How could we arrive at this deﬁnition? First, of course, we take the dimension of Pn and An to be n. Secondly, if there exists a ﬁnite map X → Y then it is natural to suppose that X and Y have the same dimension. Since by Noether normalisation 6 Dimension 67 (Theorems 1.17–1.18), any projective or afﬁne variety X has a ﬁnite map to some Pm or Am, it is natural to take m as the deﬁnition of the dimension of X. However, the question then arises as to whether this is well deﬁned: might there not exist two ﬁnite maps f : X → Am and g : X → An with m = n? Suppose that
X is irreducible. Then the ﬁniteness of a regular map f : X → Am implies that the rational function ﬁeld k(X) is a ﬁnite extension of the ﬁeld f ∗(k(Am)), which is in turn isomorphic to k(t1,..., tm). Hence k(X) has transcendence degree m over k; this gives a characterisation of the number m independent of the choice of the ﬁnite map f : X → Am. This gives some motivation for the deﬁnition of dimension. Deﬁnition The dimension of an irreducible quasiprojective variety X is the transcendence degree of the function ﬁeld k(X); it is denoted by dim X. The dimension of a reducible variety is the maximum of the dimension of its irreducible components. If Y ⊂ X is a closed subvariety of X then the number dim X − dim Y is called the codimension of Y in X, and written codim Y or codimX Y. Algebraic varieties of dimension 1 and 2 are called curves and surfaces.3 Note that if X is an irreducible variety and U ⊂ X is open then k(U ) = k(X), and hence dim U = dim X. Example 1.30 dim An = dim Pn = n, because the ﬁeld k(An) is the ﬁeld of rational functions in n variables. Since dimension is by deﬁnition invariant under birational equivalence, we see that An and Am are not birational if n = m. Example 1.31 An irreducible plane curve is 1-dimensional, as we saw in Section 1.3. Example 1.32 If X consists of a single point then obviously dim X = 0, and thus the same holds if X is a ﬁnite set. Conversely, if dim X = 0 then X is a ﬁnite set. It is enough to prove this for an irreducible afﬁne variety X. Let X ⊂ An, and write t1,..., tn for the coordinates on An as functions on X, that is, as elements of k[X]. By assumption the ti are algebra
ic over k, and can hence only take ﬁnitely many values. It follows from this that X is ﬁnite. Example 1.33 We prove that if X and Y are irreducible varieties then dim(X × Y ) = dim X + dim Y. We need only consider the case that X ⊂ AN and Y ⊂ AM are afﬁne varieties. Suppose that dim X = n, dim Y = m, and let t1,..., tN and u1,..., uM be coordinates of AN and AM considered as functions on X and Y respectively, such that 3n-dimensional varieties are often called n-folds, for example 3-folds, 4-folds (or threefolds, fourfolds). 68 1 Basic Notions t1,..., tn are algebraically independent in k(X) and u1,..., um in k(Y ). By deﬁnition k[X ×Y ] is generated by the elements t1,..., tN, u1,..., uM, and under the current assumptions all of these are algebraically dependent on t1,..., tn, u1,..., um. Hence it is enough to prove that these elements are algebraically independent. Suppose that there is a relation F (T, U ) = F (T1,..., Tn, U1,..., Um) = 0 on X × Y. Then for any point x ∈ X we have F (x, U1,..., Um) = 0 on Y. Since u1,..., um are algebraically independent in k(Y ), every coefﬁcient ai(x) of the polynomial F (x, U ) is zero; this means that the corresponding polynomial ai(T1,..., Tn) is 0 on X. Now we use the fact that t1,..., tn are algebraically independent in k(X) and deduce from this that ai(T1,..., Tn) = 0, and hence F (T, U ) is identically 0. Example 1.34 The Grassmannian Grass(r, n) (see Example 1.24)
is covered by = 0 isomorphic to the afﬁne space Ar(n−r). Thus dim Grass(r, n) = open sets pi1...ir r(n − r). It also follows from this that Grass(r, n) is rational. Theorem 1.19 If X ⊂ Y then dim X ≤ dim Y. If Y is irreducible and X ⊂ Y is a closed subvariety with dim X = dim Y then X = Y. Proof It is enough to prove the assertions for X and Y irreducible afﬁne varieties. Suppose X ⊂ Y ⊂ AN with dim Y = n. Then any n + 1 of the coordinate functions t1,..., tN are algebraically dependent as elements of k[Y ], that is, are connected by a relation F (ti1,..., tin+1) = 0 on Y. A fortiori this holds on X. This means that the transcendence degree of k(X) is at most n, so that dim X ≤ dim Y. Now suppose that dim X = dim Y = n. Then some n of the coordinates t1,..., tN are algebraically independent on X; suppose that these are t1,..., tn. Then a fortiori they are algebraically independent on Y. Let u ∈ k[Y ] with u = 0 on Y. Then u on Y is algebraically dependent on t1,..., tn, that is, there is a polynomial a(t, U ) ∈ k[t1,..., tn][U ] such that the relation a0(t1,..., tn)uk + · · · + ak(t1,..., tn) = 0 (1.45) holds on Y. We can choose a(t, U ) to be irreducible, and then ak(t1,..., tn) = 0 on Y. Relation (1.45) holds a fortiori on X. Suppose that u = 0 on X. Then (1.45) implies that ak(t1,..., tn) = 0 on X. Since by assumption t1,..., tn are independent on X, it follows that ak(t1
,..., tn) = 0 on the whole of AN. This contradicts ak(t1,..., tn) = 0 on Y. Thus if u = 0 on X then also u = 0 on Y, and therefore X = Y. The theorem is proved. We have seen that an irreducible algebraic plane curve is 1-dimensional. The following result is a generalisation. Theorem 1.20 Every irreducible component of a hypersurface in An or Pn has codimension 1. Proof It is enough to consider the case of a hypersurface in An. Suppose that a variety X ⊂ An is given by an equation F (T ) = 0. The factorisation F = F1... Fk 6 Dimension 69 of F into irreducible factors corresponds to an expression X = X1 ∪ · · · ∪ Xk, where Xi is deﬁned by Fi = 0. It is obviously sufﬁcient to prove the theorem for each variety Xi. Let us prove that Xi is irreducible: if Xi were reducible, there would exist polynomials G and H such that GH = 0 on Xi but G, H = 0 on Xi. From the Nullstellensatz it follows that Fi \| (GH )l for some l > 0. Since Fi is irreducible it follows from this that Fi \| G or Fi \| H, and this contradicts G = 0, H = 0 on Xi. Suppose that the variable Tn actually appears in the polynomial Fi(T ), and prove that the coordinates t1,..., tn−1 are algebraically independent on Xi. Indeed, a relation G(t1,..., tn−1) = 0 on Xi would imply that Fi \| Gl for some l > 0, which is impossible since G does not involve Tn. Thus dim Xi ≥ n − 1; since X = An, it follows from Theorem 1.19 that dim Xi = n − 1. Theorem 1.20 is proved. Theorem 1.21 Let X ⊂ An be a variety, and suppose that all the components of X have dimension n − 1. Then X is a hypersurface and the ideal AX is principal. Proof We only need consider the case that X is irreducible. Since X = An (because dim X
= n − 1), there exists a nonzero polynomial F which is zero on X. Since X is irreducible, some irreducible factor H of F is also zero on X. Write Y ⊂ An for the hypersurface deﬁned by H = 0; we saw in the proof of Theorem 1.20 that Y is irreducible. Then X ⊂ Y, so that X = Y by Theorem 1.19. If G ∈ AX then by the Nullstellensatz H \| Gl for some l > 0, and then G ∈ (H ) by the irreducibility of H, that is AX = (H ). Theorem 1.21 is proved. The following analogue of Theorem 1.21 is proved similarly: Theorem 1.21 Let X ⊂ Pn1 × · · · × Pnk be a variety, and suppose that all the components of X have dimension n1 + · · · + nk − 1. Then X is deﬁned by one equation that is homogeneous in each of the k sets of variables. Proof We need only replace the unique factorisation of polynomials used in the proof of Theorem 1.21 by the unique factorisation of polynomials that are homogeneous in each of the k groups of variables into irreducible polynomials of the same type. This comes from the fact that if F (x0,..., xn1, y0,..., yn2,..., u0,..., unk ) } },..., {u0,..., unk is homogeneous in each of the k sets of variables {x0,..., xn1 and F factorises as F = G · H, then G and H have the same homogeneity property. Theorem 1.21 is proved. 6.2 Dimension of Intersection with a Hypersurface If we try to study varieties deﬁned by more than one equation, we come up at once against the question of the dimension of intersection of a variety with a hypersurface. We study this question ﬁrst for projective varieties. If X is closed in PN and a form 70 1 Basic Notions F is not zero on X then we write XF for the closed subvariety
of X deﬁned by F = 0. For any projective variety X ⊂ PN we can ﬁnd a form G(U0,..., UN ) of any speciﬁed degree m which does not vanish on any components Xi of X. For this, it is enough to choose one point xi ∈ Xi in each irreducible component of X, and ﬁnd a linear form L not vanishing on any of these; then we can take G = Lm to be the appropriate power of L. Suppose that X ⊂ PN is closed, and that a form F is not zero on any component of X. By Theorem 1.19 we have dim XF < dim X. Set XF = X(1) and apply the same argument to X(1), ﬁnding a form F1 with deg F1 = deg F not vanishing on any component of X(1). We get a chain of varieties X(i) and forms Fi such that X = X(0) ⊃ X(1) ⊃ · · ·, with X(i+1) = X(i) Fi and F0 = F. (1.46) By Theorem 1.19, dim X(i+1) < dim X(i). Hence if dim X = n, then X(n+1) is empty. In other words, the forms F0 = F, F1,..., Fn have no common zeros on X. Suppose now that X is irreducible. Consider the map ϕ : X → Pn given by ϕ(x) = F0(x) : · · · : Fn(x). (1.47) This map satisﬁes the assumptions of Theorem 1.16, and by this theorem the map X → ϕ(X) is ﬁnite. But if X → Y is a ﬁnite map then, as we have seen, dim X = dim Y. Hence dim ϕ(X) = dim X = n, and since ϕ(X) ⊂ Pn is closed by Theorem 1.10, we get ϕ(X) = Pn by Theorem 1.19. Suppose now that dim X(1) = dim XF < n − 1. Then in (1.
46), already X(n) is empty. In other words, the forms F0,..., Fn−1 have no common zeros on X. This means that the point (0 : · · · : 0 : 1) is not contained in ϕ(X), which contradicts ϕ(X) = Pn. Thus we have proved the following result. Theorem 1.22 If a form F is not 0 on an irreducible projective variety X then dim XF = dim X − 1. Recall that this means that XF contains one or more irreducible components of dimension dim X − 1. Corollary 1.3 A projective variety X contains subvarieties of any dimension s < dimX. Corollary 1.4 (Inductive deﬁnition of dimension) If X is an irreducible projective variety then dim X = 1 + sup dim Y, where Y runs through all proper subvarieties of X. Corollary 1.5 The dimension of a projective variety X can be deﬁned as the maximal integer n for which there exists a strictly decreasing chain Y0 Y1 · · · Yn ∅ of length n of irreducible subvarieties Yi ⊂ X. 6 Dimension 71 Corollary 1.6 The dimension n of a projective variety X ⊂ PN can be deﬁned as N − s − 1, where s is the maximum dimension of a linear subspace of PN disjoint from X. Proof Let E ⊂ PN be a linear subspace of dimension s. If s ≥ N − n then E can be deﬁned by ≤n equations, and successive application of Theorem 1.22 proves that dim(X ∩ E) ≥ 0, and hence X ∩ E is nonempty (the dimension of the empty set is −1!). Setting m = 1 in the construction of the chain (1.46) gives n + 1 linear forms L0,..., Ln with no common zeros on X. If E is the linear subspace deﬁned by these, then dim E = N − n − 1 and X ∩ E is empty. Corollary 1.6 is proved. Corollary 1.7 The variety of common zeros of r forms F1,..., Fr on an ndimensional project
ive variety has dimension ≥n − r. The proof is by r − 1 applications of Theorem 1.22. Corollary 1.7 provides a rather strong existence theorem. Proposition If r ≤ n then r forms have a common zero on an n-dimensional projective variety. For example, in the case X = Pn, this says that n homogeneous equations in n + 1 variables have a nonzero solution. This existence theorem allows us to make a number of important deductions. Corollary 1.8 Any two curves of P2 intersect. This is clear, since a curve is given by a single homogeneous equation. However, there exist nonintersecting curves on a nonsingular quadric surface Q ⊂ P3, for example the lines of one family of generators. Therefore P2 and Q are not isomorphic. Since they are birational (Example 1.22) we get an example of two varieties that are birational but not isomorphic. This example will appear again later (Sections 4.1 and 4.5, Chapter 2, Example 4.11 of Section 2.3, Chapter 4). Corollary 1.9 Theorem 1.21 fails already for the curves on a nonsingular quadric surface Q: there exist curves C ⊂ Q that cannot be deﬁned by setting to zero a single form on P3. Indeed, if we assume that each of the disjoint curves C1 and C2 which we found on Q is deﬁned by one equation F1 = 0 and F2 = 0, we get a contradiction to Corollary 1.7, according to which the system of equations G = F1 = F2 = 0 have a common solution (where G is the equation of Q). Corollary 1.10 Any curve of degree ≥3 has an inﬂexion point. Proof We have seen in Section 1.6 that the inﬂexion points of an algebraic plane curve with equation F = 0 is deﬁned by H (F ) = 0, where H (F ) is the Hessian 72 1 Basic Notions form of F. If F has degree n then H (F ) has degree 3(n − 2). Therefore for n ≥ 3 the system of equations F = H (F ) = 0 has a nonzero solution; that is, the curve F = 0 has an inﬂexion
point. Corollary 1.10 is proved. The simplest case is when n = 3. We see that every cubic curve in P2 has an inﬂexion point. Choose a coordinate system (ξ0, ξ1, ξ2) so that the inﬂexion point is (0, 0, 1), and the inﬂexional tangent is the line ξ1 = 0. Setting u = ξ0/ξ2, v = ξ1/ξ2, we see easily that our assumption is equivalent to saying that the equation ϕ(u, v) of the curve has no constant term, or term in u or u2. Changing to coordinates x = ξ0/ξ1, y = ξ2/ξ1, so that the inﬂexion point is at inﬁnity, we ﬁnd that the equation of our cubic has no term in y3, y2x or yx2, that is, it is of the form ay2 + (bx + c)y + g(x) = 0, where g is a polynomial of degree ≤3. If a = 0 then the inﬂexion point is singular. If a = 0 we can assume that a = 1. Assuming that char k = 2, we can complete the square by setting y1 = y + (1/2)(bx + c) and reduce the equation = g1(x), where g1(x) has degree ≤3, and = 3 if the cubic curve is to the form y2 1 nonsingular. Thus the equation of a nonsingular cubic has Weierstrass normal form in some coordinate system. In Section 1.4 we proved only the weaker statement that a cubic is birational to a curve with equation in Weierstrass normal form. Corollary 1.11 (Tsen’s theorem) Let F (x1,..., xn) be a form in n variables of degree m < n whose coefﬁcients are polynomials in one variable t. Then the equation F (x1,..., xn) = 0 has a solution in polynomials xi = pi(t). l Proof We look for xi of the form xi = j =0 uij t j with
unknown coefﬁcients uij. Substituting these expressions in the equation F (x1,..., xn) = 0, we get a polynomial in t all of whose coefﬁcients must be set to 0. If the maximum of the degrees of the coefﬁcients of a polynomial F equals k then the number of equations is at most ml + k + 1. The number of indeterminates is n(l + 1). Since by assumption n > m, for l sufﬁciently large, the number of unknowns is greater than the number of equations, and hence the system has a nonzero solution. Example 1.35 An important particular case of Tsen’s theorem is when n = 3 and F is a quadratic form. It can be given the following geometric interpretation: suppose that a surface X ⊂ P2 × A1 is deﬁned by the equation q(x0 : x1 : x2; t) = 2 i,j =0 aij (t)xixj with aij (t) ∈ k[t], where (x0 : x1 : x2) are coordinates in P2 and t a coordinate on A1. The ﬁbres of the map X → A1 are the conics q(x0 : x1 : x2; a) = 0 for a ∈ A1, and the surface is called a conic bundle or pencil of conics. Tsen’s theorem proves that a pencil of conics has a section, that is, there exists a regular map ϕ : A1 → X such that ϕ(a) is a point of the ﬁbre over a for every a ∈ A1. Another interpretation of this result is as follows. Consider our surface X as 2 i,j =0 aij xixj = 0 in P2 over the the conic C with equation q(x0 : x1 : x2; t) = 6 Dimension 73 algebraically nonclosed ﬁeld K = k(t). Obviously K(C) = k(X). Then C has a point with coordinates in K. We assume that the curve C is irreducible for a general point t ∈ A1, that is, that det \|aij (t)\| is not identically 0; we
say that the pencil of conics is nondegenerate in this case. In Section 1.2 we saw that the conic is then rational, with the birational map to P1 deﬁned over K = k(t). In other words, the ﬁeld K(C) is isomorphic over K to the ﬁeld K(x), and since K(C) = k(X) it follows that k(X) is isomorphic to K(x) = k(t, x). We have proved the next result. Corollary 1.12 A nondegenerate pencil of conics over A1 is a rational surface. Theorem 1.23 Under the assumptions of Theorem 1.22, every component of XF has dimension dim X − 1. Proof Consider the ﬁnite map ϕ : X → Pn (with n = dim X) constructed in the proof of Theorem 1.22, and let An ⊂ Pn for i = 0,..., n be the afﬁne open sets i covering Pn. Then using the Veronese embedding with m = deg F, it is easy to see that ϕ−1(An i ) = Ui are afﬁne open sets of X. It is obviously enough to prove that each component of the afﬁne variety XF ∩ Ui has dimension n − 1 for each i. From now on our arguments apply to some ﬁxed Ui, which we denote by U. Obviously XF ∩ U = V (f ), where f = F /Fi, that is, XF coincides on U with the set of zeros of the regular function f ∈ k[U ]. We constructed above a ﬁnite map ϕ : U → An, given by n regular functions f1,..., fn, with f = f1. To prove that each component of V (f ) has dimension n − 1, we only need to prove that it has dimension ≥n − 1. We prove that the functions f2,..., fn are algebraically independent on each component. Let P ∈ k[T2,..., Tn]. To prove that R = P (f2,..., fn) does not vanish on any component of V (f ) it is enough to
prove that for Q ∈ k[U ], RQ = 0 on V (f ) =⇒ Q = 0 on V (f ). Indeed, if V (f ) = U (1) ∪ · · · ∪ U (t) is an irredundant decomposition into irreducible components, and R = 0 on U (1), then take Q to be any function that vanishes on U (2) ∪ · · · ∪ U (t) but not on U (1). Then RQ = 0 on V (f ) but Q = 0 on V (f ). By the Nullstellensatz our assertion can be restated as follows: if f \| (RQ)l for some l > 0 then f \| Qk for some k > 0. Thus Theorem 1.23 follows from the following purely algebraic fact: Lemma Set B = k[T1,..., Tn], and let A ⊃ B be an integral domain that is integral over B; write x = T1, and let y = P (T2,..., Tn) = 0. Then for any u ∈ A, x \| (yu)l in A for some l > 0 =⇒ x \| uk for some k > 0. Proof of the Lemma The only property of x and y that we use is that they are relatively prime in the UFD k[T1,..., Tn]. Note that we can replace yl by z and ul by 74 1 Basic Notions v, and then it is enough to prove that if x and z are relatively prime in k[T1,..., Tn] then x \| zv in A implies that x \| vk for some k > 0. Thus the lemma asserts that the property of polynomials x, z ∈ B being relatively prime is in a certain sense preserved on passing to a ring A that is integral over B. Write K for the ﬁeld of fractions of B. If t ∈ A is integral over B then it is algebraic over K. Let F (T ) ∈ K[T ] be the minimal polynomial of t over K, that is, the polynomial of least degree with leading coefﬁcient 1 such that F (t) = 0. Division with remainder shows that any polyn
omial G(T ) ∈ K[T ] with G(t) = 0 is divisible by F (T ) in K[T ]. Now from this it follows that t is integral over B if and only if F [T ] ∈ B[T ]. Indeed, if t is integral and G(t) = 0 for G ∈ B[T ] with leading coefﬁcient 1, then G(T ) = F (T )H (T ) in K[T ]. But B = k[T1,..., Tn] is a UFD, so a simple application of Gauss’ lemma shows that F (T ), H (T ) ∈ B[T ]. It is now easy to complete the proof of the lemma. Suppose that zv = xw with v, w ∈ A and let F (T ) = T k + b1T k−1 + · · · + bk be the minimal polynomial of w. Since w is integral over B, the coefﬁcients bi of F satisfy bi ∈ B. It is easy to see that the minimal polynomial G(T ) of v = xw/z is given by (x/z)kF (zT /x). Therefore T k−1 + · · · + xkbk zk G(T ) = T k + xb1 z vk−1 + · · · + xkbk zk and vk + xb1 z = 0., (1.48) Since v is integral over B, also xibi/zi ∈ B, and because x and z are relatively prime it follows that zi \| bi. It then follows from (1.48) that x \| vk. The lemma is proved, and with it Theorem 1.23. Corollary 1.13 If X ⊂ PN is an irreducible quasiprojective variety and F a form that is not identically 0 on X, then every (nonempty) component of XF has codimension 1. (XF = ∅ is of course possible for quasiprojective varieties.) Proof By deﬁnition X is open in some closed subset X ⊂ PN. Since X is irreducible, so is X, and hence dim X = dim X. By The
orem 1.23, (X)F = Yi with dim Yi = dim X − 1. But it is easy to see that XF = (X)F ∩ X; it fol(Yi ∩ X), and Yi ∩ X is either empty or is open in Yi, so that lows that XF = dim(Yi ∩ X) = dim X − 1. This proves Corollary 1.13. The particular case of this lemma that usually turns up is when X ⊂ An is an 0 given by u0 = 0, and write m = deg F 0 ; then XF = V (f ). In other words, XF is just the set of zeros of afﬁne variety. Let An ⊂ Pn be the subset An and f = F /um some regular function f ∈ k[X]. Corollary 1.14 Let X ⊂ PN be an irreducible n-dimensional quasiprojective variety, and Y ⊂ X the set of zeros of m forms on X. Then every (nonempty) component of Y has dimension ≥n − m. 6 Dimension 75 Proof The proof is by an obvious induction on m. In the case of an afﬁne variety X we can again say that Y is the set of zeros of m regular functions on X. If X is projective and m ≤ n then by the proposition after Theorem 1.22, Corollary 1.7 we can assert that Y = ∅. Corollary 1.14 is proved. Theorem 1.24 Let X, Y ⊂ PN be irreducible quasiprojective varieties with dim X = n and dim Y = m. Then any (nonempty) component Z of X ∩ Y has dim Z ≥ n + m − N. Moreover, if X and Y are projective and n + m ≥ N then X ∩ Y = ∅. Proof The theorem is obviously local in nature, and we therefore only need to prove it in the case of afﬁne varieties. Suppose that X, Y ⊂ AN. Write Δ ⊂ AN × AN = A2N for the diagonal (see Example 1.20). Then X ∩ Y is isomorphic to (X × Y ) ∩ Δ ⊂ A2N. The theorem follows from Corollary 1
.14, since Δ ⊂ A2N is deﬁned by N equations. For the ﬁnal sentence, apply the ﬁrst part to the afﬁne cone over X and Y. The theorem is proved. Theorem 1.24 can be stated in a more symmetric form, in which it generalises at once to the intersection of any number of subvarieties: codimX r i=1 Yi ≤ r i=1 codimX Yi. (1.49) 6.3 The Theorem on the Dimension of Fibres For a given regular map f : X → Y of quasiprojective varieties, and y ∈ Y, the set f −1(y) is called the ﬁbre of f over y. It is obviously a closed subvariety of X. The idea behind the terminology is that f ﬁbres X as the disjoint union of the ﬁbres over the different points y ∈ f (X). Theorem 1.25 Let f : X → Y be a regular map between irreducible varieties. Suppose that f is surjective: f (X) = Y, and that dim X = n, dim Y = m. Then m ≤ n, and (i) dim F ≥ n − m for any y ∈ Y and for any component F of the ﬁbre f −1(y); (ii) there exists a nonempty open subset U ⊂ Y such that dim f −1(y) = n − m for y ∈ U. Proof of (i) This property is obviously local over Y, and it is enough to prove it after replacing Y by any open set U ⊂ Y with U y and X by f −1(U ). Hence we can assume that Y is afﬁne. Suppose that Y ⊂ AN. In the chain of subvarieties of Y given by (1.46), Y (m) is a ﬁnite set Y (m) = Y ∩ Z, where Z is deﬁned by m 76 1 Basic Notions equations and y ∈ Z. The open set U can be chosen such that Z ∩ Y ∩ U = {y}, and so we can assume that Z ∩ Y = {y}. The subspace Z is deﬁned by m equations
g1 = · · · = gm = 0. Thus in Y the system of equations g1 = · · · = gm = 0 deﬁnes the point y. This means that in X the system of equations f ∗(g1) = · · · = f ∗(gm) = 0 deﬁnes the subvariety f −1(y). Assertion (i) now follows from Theorem 1.23, Corollary 1.14 (the afﬁne case). Proof of (ii) We can replace Y by an afﬁne open subset W and X by an open afﬁne set V ⊂ f −1(W ). Since V is dense in f −1(W ) and f is surjective, f (V ) is dense in W. Hence f deﬁnes an inclusion f ∗ : k[W ] → k[V ]. From now on we take k[W ] ⊂ k[V ], therefore k(W ) ⊂ k(V ). Write k[W ] = k[w1,..., wM ] and k[V ] = k[v1,..., vN ]. Since dim W = m and dim V = n, the ﬁeld k(V ) has transcendence degree n − m over k(W ). Suppose that v1,..., vn−m are algebraically independent over k(W ), and the remaining vi algebraic over k(W )[v1,..., vn−m], with relations Fi(vi; v1,..., vn−m; w1,..., wM ) = 0 for i = n − m + 1,..., N. Write vi for the function vi restricted to f −1(y) ∩ V. Then f −1(y) ∩ V k = k[v1,..., vN ]. (1.50) We now view Fi as a polynomial in vi, v1,..., vn−m, with coefﬁcients functions of w1,..., wM, and deﬁne Yi to be the subvariety of W given by the vanishing of the leading term of Fi. Set E = Yi and
U = W \ E. Obviously U is open and nonempty. By construction of E, if y ∈ U then none of the polynomials Fi(Ti; T1,..., Tn−m; w1(y),..., wM (y)) is identically zero, and therefore all the vi are algebraically dependent on v1,..., vn−m. Together with formula (1.50) this proves that dim f −1(y) ≤ n − m, so that (ii) of the proposition follows from (i). The theorem is proved. It is easy to give examples where (ii) does not hold for every y ∈ Y ; (see for example Exercise 6 of Section 2.4, and the end of Section 6.4). That is, the dimension of ﬁbres may jump up. Corollary The sets Yk = {y ∈ Y \| dim f −1(y) ≥ k} are closed in Y. Proof By Theorem 1.25, Yn−m = Y, and there exists a closed subset Y Y such that Yk ⊂ Y if k > n − m. If Zi are the irreducible components of Y and fi : f −1(Zi) → Zi the restrictions of f, then dim Zi < dim Y, and we can prove the corollary by induction on dim Y. The corollary is proved. Theorem 1.25 implies a criterion for a variety to be irreducible which is often useful. Theorem 1.26 Let f : X → Y be a regular map between projective varieties, with f (X) = Y. Suppose that Y is irreducible, and that all the ﬁbres f −1(y) for y ∈ Y are irreducible and of the same dimension. Then X is irreducible. 6 Dimension 77 Proof Let X = f (Xi) is closed. Since Y = Xi be an irreducible decomposition. By Theorem 1.10, each f (Xi) and Y is irreducible, Y = f (Xi) for some i. Set dim f −1(y) = n. For each i such that Y = f (Xi), by Theorem 1.25, (ii), (y)) = ni there exists a dense open set Ui �
�� Y and an integer ni such that dim(f for all y ∈ Ui. Extend the deﬁnition of Ui to i such that f (Xi) = Y by setting Ui. Then since f −1(y) is irreducible, we must have Ui = Y \f (Xi). Consider y ∈ f −1(y) ⊂ Xi for some i, say i = 0. Write f0 : X0 → Y for the restriction of f. Then f −1(y) ⊂ f (y) and n = n0. (y); but the opposite inclusion is trivial, so that f −1(y) = f −1 0 −1 0 −1 i Now since f0 is surjective, we know that f every y ∈ Y, and it has dimension ≥n0 by Theorem 1.25, (i), so that f f −1(y). Therefore X0 = X. The theorem is proved. −1 0 (y) ⊂ f −1(y) is nonempty for (y) = −1 0 A very special case of Theorem 1.26 is the irreducibility of a product of irre- ducible projective varieties; see Theorem 1.6. 6.4 Lines on Surfaces It is only natural, after the effort spent on the proof of Theorems 1.22–1.24 on the dimension of intersections, to look for some applications of these results. As an example, we now treat a simple question on lines on surfaces in P3. As a general rule, the notion of dimension is useful in cases when we need to give rigorous meaning to a statement that some set depends on a given number of parameters. For this, we must identify the set with some algebraic variety, and apply the notion of dimension we have introduced. For example, we have seen in Example 1.28 that hypersurfaces of Pn, deﬁned by equations of degree m, are in one-to-one correspondence with points of a projective space PN, where N = νn,m = n + m m − 1. We proceed to subvarieties that are not hypersurfaces, the simplest of which are lines in P3. In Example 1.24, we saw that lines l ⊂ P3 are in one-to-one correspondence with points of
the quadric hypersurface of Π ⊂ P5 deﬁned by p01p23 − p02p13 + p03p12 = 0. Obviously dim Π = 4. To study lines lying on surfaces, the following result is important. Lemma The conditions that the line l with Plücker coordinates pij be contained in the surface X with equation F = 0 are algebraic relations between the pij and the coefﬁcients of F, homogeneous in both the pij and the coefﬁcients of F. Proof We can write a parametric representation of l it terms of its Plücker coordinates: let x and y be a basis of a plane L ⊂ V, with dim L = 2, dim V = 4. Then it 78 1 Basic Notions is easy to check that as f runs through the space of all linear forms on V, the set of vectors of the form xf (y) − yf (x) (1.51) coincides with L. If f has coordinates (α0, α1, α2, α3), that is, if f (x) = αixi, then the vector (1.51) has coordinates zi = j αj pij, where pij = xiyj − xj yi. Hence if l is the line with Plücker coordinates pij, the points of l are the points with coordinates j αj pij for j = 0,..., 3. On substituting these expressions into the equation F (u0, u1, u2, u3) = 0 and equating to zero the coefﬁcients of all the monomials in αi, we get the condition that l ⊂ X, as a set of algebraic relations between the coefﬁcients of F and the Plücker coordinates pij. The lemma is proved. m+3 3 We proceed to the question we are interested in, the lines lying on surfaces in P3. For given m, consider the projective space PN with N = ν3,m = − 1, whose points parametrise surfaces in P3 of degree m, that is, given by a homogeneous equation of degree m. Write Γm ⊂ PN × Π for the set of pairs (ξ, η) ∈ PN × Π
such that the line l corresponding to η ∈ Π is contained in the surface X corresponding to ξ ∈ PN. By the lemma, Γm is a projective variety. Let us determine the dimension of Γm. For this, consider the projection maps ϕ : PN × Π → PN and ψ : PN × Π → Π given by ϕ(ξ, η) = ξ and ψ(ξ, η) = η. Obviously ϕ and ψ are regular maps. From now on, we only consider their restrictions to Γm. Note that ψ(Γm) = Π. This simply means that for every line l there is at least one surface of degree m passing through l, possibly reducible. We determine the dimension of the ﬁbres ψ −1(η) of ψ. By a projective transformation we can assume that the line corresponding to η is given by u0 = u1 = 0. Points ξ ∈ PN such that (ξ, η) ∈ ψ −1(η) ⊂ Γm correspond to surfaces of degree m passing through this line. Such a surface is given by F = 0, where F = u0G + u1H, with G and H arbitrary forms of degree m − 1. The set of such forms is of course a linear subspace of PN whose dimension is easy to calculate. It is equal to μ = m(m + 1)(m + 5) 6 − 1. (1.52) Thus dim ψ −1(η) = m(m + 1)(m + 5) 6 − 1 = N − (m + 1). It follows from Theorem 1.26 that Γm is irreducible. Applying Theorem 1.25 we get that dim Γm = dim ψ(Γm) + dim ψ −1(η) = m(m + 1)(m + 5. (1.53) 6 Dimension 79 Consider now the other projection ϕ : Γm → PN. Its image is a closed subset of PN, by Theorem 1.10. Obviously dim ϕ(Γm) ≤ Γm. Thus if dim Γm < N then ϕ(Γm) = PN
, or in other words, not every surface of degree m contains a line. By (1.53), the inequality dim Γm < N reduces to m > 3. We have obtained the following result. Theorem 1.27 For any m > 3, there exist surfaces of degree m that do not contain any lines. Moreover, such surfaces correspond to an open set of PN. Thus there exist nontrivial algebraic relations between the coefﬁcients of a form F (u0, u1, u2, u3) of degree m > 3 that are necessary and sufﬁcient for the surface given by F = 0 to contain a line. Of the remaining cases m = 1, 2, 3, the case m = 1 is trivial. We consider the case m = 2, although we already know the answer from 3-dimensional coordinate geometry. When m = 2 we have N = 9 and dim Γm = 10. It follows from Theorem 1.25 that dim ϕ−1(ξ ) ≥ 1. This is the well-known fact that any quadric surface contains inﬁnitely many lines. We remark in passing, and without details of the proof, that this already provides an example of the phenomenon mentioned in Section 6.3 of the dimension of ﬁbres jumping up: if the quadric surface corresponding to a point ξ is irreducible then dim ϕ−1(ξ ) = 1, whereas if it splits as a pair of planes then of course dim ϕ−1(ξ ) = 2. Now consider the case m = 3. In this case, dim Γm = N = 19. It is easy to construct a cubic surface X ⊂ P3 which contains only a ﬁnite number of lines. For example, if X is given in inhomogeneous coordinates by T1T2T3 = 1, (1.54) then X does not have a single line contained in A3. Indeed, if we write the equation of an afﬁne line in the form Ti = ait + bi for i = 1, 2, 3 and substitute in (1.54), we get a contradiction; whereas the intersection of X with the plane at inﬁnity contains 3 lines. Thus there exists a point of P19 for which ϕ−1(ξ ) is nonempty and dim �
�−1(ξ ) = 0. By Theorem 1.25, this is only possible if dim ϕ(Γ3) = 19. Using Theorem 1.19, we see that ϕ(Γ3) = P19. We have proved the following result. Theorem 1.28 Every cubic surface contains at least one line. There exists an open subset U of the space P19 parametrising all cubic surfaces such that a surface corresponding to a point of U contains only ﬁnitely many lines. Cubic surfaces that contain inﬁnitely many lines do exist, for example cubic cones. Thus again the dimension of ﬁbres can jump up. We will see later that most cubic surfaces contain only ﬁnitely many lines, and we will determine the number of these. 80 1 Basic Notions 6.5 Exercises to Section 6 1 Let L ⊂ Pn be an (n − 1)-dimensional linear subspace, X ⊂ L an irreducible closed variety and y a point in Pn \ L. Join y to all points x ∈ X by lines, and denote by Y the set of points lying on all these lines, that is, the cone over X with vertex y. Prove that Y is an irreducible projective variety and dim Y = dim X + 1. 2 Let X ⊂ A3 be the reducible curve whose components are the 3 coordinate axes. Prove that the ideal AX cannot be generated by 2 elements. 3 Let X ⊂ P2 be the reducible 0-dimensional variety consisting of 3 points not lying on a line. Prove that the ideal AX cannot be generated by 2 elements. 4 Prove that any ﬁnite set S ⊂ A2 can be deﬁned by two equations. [Hint: Choose the coordinates x, y in A2 in such a way that all points of S have different x coor(x − αi) = 0, dinates; then show how to deﬁne S by the two equations y = f (x), where f (x) is a polynomial.] 5 Prove that any ﬁnite set of points S ⊂ P2 can be deﬁned by two equations. 6 Let X ⊂ A3 be an algebraic curve, and x, y, z coordinates
in A3; suppose that X does not contain a line parallel to the z-axis. Prove that there exists a nonzero polynomial f (x, y) vanishing at all points of X. Prove that all such polynomials form a principal ideal (g(x, y)), and that the curve g(x, y) = 0 in A2 is the closure of the projection of X onto the (x, y)-plane parallel to the z-axis. 7 We use the notation of Exercise 6. Suppose that h(x, y, z) = g0(x, y)zn + · · · + gn(x, y) is the irreducible polynomial of smallest positive degree in z contained in the ideal AX. Prove that if f ∈ AX has degree m as a polynomial in z, then we can = hU + v(x, y), where v(x, y) is divisible by g(x, y). Deduce that the write fgm 0 equation h = g = 0 deﬁnes a reducible curve consisting of X together with a ﬁnite number of lines parallel to the x-axis, deﬁned by g0(x, y) = g(x, y) = 0. 8 Use Exercises 6–7 to prove that any curve X ⊂ A3 can be deﬁned by 3 equations. 9 By analogy with Exercises 6–8, prove that any curve X ⊂ P3 can be deﬁned by 3 equations. 10 Let F0(x0,..., xn),..., Fn(x0,..., xn) be forms of degree m0,..., mn and consider the system of n + 1 equations in n + 1 variables F0(x) = · · · = Fn(x) = 0. Write Γ for the subset of Pνn,mi × Pn (where νn,m = n+m m n i=0 (F0,..., Fn, x) \| F0(x) = · · · = Fn(x) = 0 Γ = − 1) deﬁned by. 6 Dimension 81 Pνn,mi and ψ : Γ → Pn, prove By considering the two projection
maps ϕ : Γ → − 1. Deduce from this that there exists a polynothat dim Γ = dim ϕ(Γ ) = mial R = R(F0,..., Fn) in the coefﬁcients of the forms F0,..., Fn such that R = 0 is a necessary and sufﬁcient condition for the system of n + 1 equations in n + 1 variables to have a nonzero solution. What is the polynomial R if the forms F0,..., Fn are linear? i νn,mi i 11 Prove that the Plücker hypersurface Π ⊂ P5 contains two systems of 2-dimensional linear subspaces. A plane of the ﬁrst system is deﬁned by a point ξ ∈ P3 and consists of all points of Π corresponding to lines l ⊂ P3 through ξ. A plane of the second system is deﬁned by a plane Ξ ⊂ P3 and consists of all points of Π corresponding to lines l ⊂ P3 contained in Ξ. There are no other planes contained in Π. 12 Let F (x0, x1, x2, x3) be an arbitrary form of degree 4. Prove that there exists a polynomial Φ in the coefﬁcients of F such that Φ(F ) = 0 is a necessary and sufﬁcient condition for the surface F = 0 to contain a line. 13 Let Q ⊂ P3 be an irreducible quadric surface and ΛX ⊂ Π the set of points on the Plücker hypersurface Π ⊂ P5 corresponding to lines contained in Q. Prove that ΛX consists of two disjoint conics. Chapter 2 Local Properties 1 Singular and Nonsingular Points 1.1 The Local Ring of a Point This chapter investigates local properties of points of algebraic varieties, that is, properties of points x ∈ X that remain unchanged if X is replaced by any neighbourhood of x. Since any point has an afﬁne neighbourhood, in the study of local properties of points we can restrict ourselves to afﬁne varieties. The basic local invariant of a point x of a variety is its local ring Ox
, the ring consisting of all functions, each of which is regular in some neighbourhood of x. This deﬁnition requires a little care, however, since each function is regular in a different neighbourhood. If X is irreducible, Ox is the subring of the function ﬁeld k(X) consisting of all functions f ∈ k(X) that are regular at x. Recalling the deﬁnition of k(X) as the ﬁeld of fractions of the coordinate ring k[X] we see that Ox consists of fractions f/g with f, g ∈ k[X] and g(x) = 0. This construction becomes clearer if we focus on its general and purely algebraic nature. It can be applied to an arbitrary commutative ring A and prime ideal p of A. In this generality there is a new difﬁculty caused by possible zerodivisors in A. Consider the set of pairs (f, g) with f, g ∈ A and g /∈ p; we identify pairs ac- cording to the rule (f, g) = f, g ⇐⇒ ∃ h ∈ A \ p such that h fg − gf = 0. (2.1) Algebraic operations are deﬁned on this set as follows: (f, g) + (f, g) f, g f, g fg + gf, gg ff, gg., = = (2.2) (2.3) It is easy to check that in this way we get a ring. It is called the local ring of A at the prime ideal p, and denoted by Ap. I.R. Shafarevich, Basic Algebraic Geometry 1, DOI 10.1007/978-3-642-37956-7_2, © Springer-Verlag Berlin Heidelberg 2013 83 84 2 Local Properties The map ϕ : A → Ap given by ϕ(h) = (h, 1) is a homomorphism. The elements ϕ(g) with g /∈ p are invertible in Ap, and any element u ∈ Ap can be written u = ϕ(f )/ϕ(g) with g /∈ p; we sometimes use the somewhat imprecise notation u = f
/g. The elements of the form ϕ(f )/ϕ(g) with f ∈ p and g /∈ p form an ideal m ⊂ Ap; moreover every element u ∈ Ap with u /∈ m has an inverse. Therefore m contains every other ideal of Ap. We arrive at one of the fundamental notions of commutative algebra: a ring O is a local ring if it has an ideal m ⊂ O with m = O such that m contains every other ideal of O. Lemma If A is a Noetherian ring then so is every local ring Ap. Proof Indeed, for any ideal a ⊂ Ap, set a = ϕ−1(a). This is an ideal of A, and so by assumption has a ﬁnite basis, a = (f1,..., fr ). If u ∈ a then u = ϕ(f )/ϕ(g) with f, g ∈ A and g /∈ p. By the identiﬁcation rule (2.1), it follows that there exists h ∈ A \ p such that hf ∈ a, and since 1/ϕ(hg) ∈ Ap, we get u ∈ ϕ( a )Ap = (ϕ(f1),..., ϕ(fr )). Hence a = (ϕ(f1),..., ϕ(fr )), and so has a ﬁnite basis. The lemma is proved. If A = k[X] is the afﬁne coordinate ring of an afﬁne variety X and p = mx the maximal ideal of a point x ∈ X then Ap is called the local ring of x, and denoted by OX,x or Ox. It is Noetherian by the lemma. For each pair (f, g) deﬁning an element of Ox the function f/g is regular in the neighbourhood D(g) of x. The rule (2.1) means that in Ox we identify functions f/g and f /g that are equal in some neighbourhood of x (in the present case D(hgg)). Thus we can also deﬁne Ox as the ring whose elements are regular functions in different neighbourhoods of x, with the identiﬁcation rule just given. The de�
��nition is already independent of the choice of some afﬁne neighbourhood U of x. We choose, in particular, the variety V so that all its irreducible components pass through x. Then a function f that is 0 on some neighbourhood U ⊂ V of x will be 0 on the whole of V. Hence the homomorphism ϕ : k[V ] → Ox is an inclusion, and we identify k[V ] with a subring of Ox. In this situation, we can get rid of the factor h in the identiﬁcation rule (2.1). In other words, Ox consists of functions on V without any identiﬁcation, and all functions ϕx ∈ Ox are of the form f/g, with f, g ∈ k[V ] and g(x) = 0. A similar construction is applicable to any irreducible subvariety Y of an afﬁne variety X. Here we need to set A = k[X] and p = aY. In this case, the local ring Ap is called the local ring of X at the irreducible subvariety Y (or along Y ), and denoted OX,Y or OY. If X is irreducible then OY ⊂ k(X) is the ring consisting of all rational functions that are regular at some point of Y (and hence regular on a dense open subset of Y ). The maximal ideal mY ⊂ OY consists of functions vanishing along Y, and the residue ﬁeld OY /mY = k(Y ) is the function ﬁeld of Y. The passage to the case of an irreducible closed subvariety Y of an arbitrary quasiprojective variety X is just as obvious as when Y was a point. The local ring 1 Singular and Nonsingular Points 85 OY is deﬁned in this case as the local ring of the subvariety Y ∩ V, where V ⊂ X is any open afﬁne variety such that Y ∩ V = ∅. The local ring is independent of the choice of V.4 1.2 The Tangent Space We will deﬁne the tangent space to an afﬁne variety X at a point x as the set of all lines through x tangent to
X. To deﬁne tangency of a line L ⊂ AN to a variety X ⊂ AN, suppose that the coordinate system in AN is chosen so that x = (0,..., 0) = 0. Then L = {ta \| t ∈ k}, where a = 0 is a ﬁxed point. To study the intersection of X with L, suppose that X is given by a system of equations F1 = · · · = Fm = 0 with AX = (F1,..., Fm). The set X ∩ L is then given by the equations F1(ta) = · · · = Fm(ta) = 0. Since we are now dealing with polynomials in one variable t, their common roots are the roots of their highest common factor. Suppose that f (t) = hcf F1(ta),..., Fm(ta) = c (t − αi)ki. (2.4) The values t = αi correspond to the points of intersection of L with X. Note that in (2.4), a root t = αi has an associated multiplicity ki, that is naturally interpreted as the multiplicity of intersection of L with X. In particular, since L ∩ X 0, one of the roots of f (t) in (2.4) is t = 0. We arrive at the following deﬁnition. Deﬁnition 2.1 The intersection multiplicity of a line L with a variety X at 0 is the multiplicity of t = 0 as a root of the polynomial f (t) = hcf(F1(ta),..., Fm(ta)). Thus the intersection multiplicity is the biggest power of t dividing all the Fi(ta). It is ≥1 by deﬁnition, since 0 ∈ L ∩ X. If the Fi(ta) are identically 0 then the intersection multiplicity is considered to be +∞. Obviously, f (t) = hcf{F (ta) \| F ∈ Ax}, and hence the multiplicity of intersec- tion is independent of the choice of the generators Fi of AX. 4Quite generally, OX,Y is a subring of the direct product of function ﬁelds k(Xi ) of the irreducible components Xi
of X that meet Y, that is, OX,Y ⊂ k(Xi ). Y ∩Xi =∅ We can also view it as a quotient of the local ring OAn,Y of rational functions on the ambient space regular on a dense open set of Y, modulo the ideal aXOAn,Y of functions vanishing on X. In discussing rational maps and rational functions as in Chapter 1, a point to grasp is that rational functions are deﬁned as fractions, and the locus where they are regular is determined subsequently; otherwise you have to worry about when two functions or maps with different domains are equal (for example, is the function z/z with a removable singularity equal to 1?). 86 2 Local Properties Deﬁnition 2.2 A line L is tangent to X at 0 if it has intersection multiplicity ≥2 with X at 0. We now write out the conditions for L to be tangent to X. Since X 0, each of the polynomials Fi(T ) has constant term 0. For i = 1,..., m, we write Li for the linear term, so that Fi = Li + Gi, where Gi has only terms of degree ≥2. Then Fi(at) = tLi(a) + Gi(ta), and Gi(ta) is divisible by t 2. Therefore Fi(at) is divisible by t 2 if and only if Li(a) = 0. Thus the condition for tangency is L1(a) = · · · = Lm(a) = 0. (2.5) Deﬁnition 2.3 The geometric locus of points on lines tangent to X at x is called the tangent space to X at x. It is denoted by Θx, or by ΘX,x if we need to specify which variety is intended. Thus (2.5) are the equations of the tangent space. They show that Θx is a linear subspace of AN. Example 2.1 The tangent space to An at any point is just An itself. Example 2.2 Let X ⊂ An be a hypersurface and AX = (F ). If X 0 and F = L + G (in the above notation) then Θ0 is deﬁned by the single equation L(T1,..., Tn) =
0. Hence if L = 0 then dim Θ0 = n − 1 and if L = 0 then Θ0 = An, so that dim Θ0 = n. Obviously L = ∂F ∂xi (0)xi, so that for n = 2 the deﬁnition coincides with that given in (1.13). Example 2.3 The tangent space at (0, 0) to the curve y(y − x2) = 0 in A2 is the whole of A2. (Although both its components have the same tangent line y = 0.) 1.3 Intrinsic Nature of the Tangent Space Deﬁnition 2.3 was given in terms of the deﬁning equations of a subvariety X ⊂ AN. Hence it is not obvious that under an isomorphism f : X → Y the tangent spaces at x and at f (x) are isomorphic (that is, have the same dimension). We now prove this; for this, we reformulate the notion of tangent space so that it depends only on the coordinate ring k[X]. We recall some deﬁnitions. If F (T1,..., TN ) is a polynomial and x = (x1,..., xN ) a point, then F has a Taylor series expansion F (T ) = F (x) + F (1)(T ) + · · · + F (k)(T ), 1 Singular and Nonsingular Points 87 where F (i) are homogeneous polynomials of degree i in the variables Tj − xj. The linear form F (1) is the differential of F at x, and is denoted by dF or dxF ; we have dxF = N i=1 ∂F ∂Ti (x)(Ti − xi). It follows from the deﬁnition that dx(F + G) = dxF + dxG, dx(F G) = F (x)dxG + G(x)dxF. (2.6) Using this notation, we can write the (2.5) of the tangent space to X at x ∈ X in the form or dxF1 = · · · = dxFm = 0, N i=1 ∂Fj ∂Ti (x)(Ti − xi
) = 0 for j = 1,..., m, (2.7) (2.8) where AX = (F1,..., Fm). Suppose that g ∈ k[X] is deﬁned as the restriction to X of a polynomial G. If we set dxg = dxG then the answer depends on the choice of the polynomial G; more precisely, it would only be deﬁned up to adding a terms dxF with F ∈ AX. Since AX = (F1,..., Fm), we have F = A1F1 + · · · + AmFm, and by (2.6) and the fact that Fi(x) = 0, we get that dxF = A1(x)dxF1 + · · · + Am(x)dxFm. Using (2.7) we see that all the linear forms dxF for F ∈ AX are 0 on Θx, and hence, if we write dxg for the restriction to Θx of the linear form dxG, that is, dxg = dxG\|Θx, (2.9) we get a map that sends any function g ∈ k[X] into a well-deﬁned linear form dxg on Θx. Deﬁnition The linear function dxg deﬁned by (2.9) is called the differential of g at x. Obviously, dx(f + g) = dxf + dxg, dx(fg) = f (x)dxg + g(x)dxf. (2.10) We thus have a homomorphism dx : k[X] → Θ ∗ x is the space of linear forms on Θx. Since dxα = 0 for α ∈ k, we can replace the study of this map by that of dx : mx → Θ ∗ x, where mx = {f ∈ k[X] \| f (x) = 0}. Obviously mx is an ideal of k[X]. x, where Θ ∗ Theorem 2.1 The map dx deﬁnes an isomorphism of the vector spaces mx/m2 x and Θ ∗ x. 88 2 Local Properties x and ker dx = m2 Proof We need to show that im
dx = Θ ∗ x. The ﬁrst of these is obvious. Any linear form ϕ on Θx is induced by some linear function f on AN, and dxf = ϕ. To prove the second assertion, suppose that x = (0,..., 0) and that g ∈ mx satisﬁes dxg = 0. Suppose that g is induced by a polynomial G ∈ k[T1,..., TN ]. By assumption the linear form dxG is 0 on Θx, and hence is a linear combination of (2.7) deﬁning this subspace, that is, dxG = λ1dxF1 + · · · + λmdxFm. Set G1 = G − λ1F1 − · · · − λmFm. We see that G1 does not have any terms of degree 0 or 1 in T1,..., TN, and therefore G1 ∈ (T1,..., TN )2. Furthermore, G1\|X = G\|X = g, and hence g ∈ (t1,..., tN )2, where ti = Ti \|X. Since obviously mx = (t1,..., tN ), this proves the theorem. As is well known, if L is a vector space and M = L∗ is the vector space of all linear forms on L then L can be identiﬁed with the vector space of all linear forms on M, that is, L = M ∗. Applying this in our case gives the following. Corollary 2.1 The tangent space Θx at a point x is isomorphic to the vector space of all linear forms on mx/m2 x. x is called the cotangent space to X at x. The vector space mx/m2 From this, we make a deduction concerning the behaviour of tangent spaces under a regular map f : X → Y between varieties. Suppose that x ∈ X and y = f (x). Then f deﬁnes a map f ∗ : k[Y ] → k[X], and obviously f ∗(my) ⊂ mx and f ∗(m2 x. Linear functions, like any functions, are contravariant (map
in the opposite direction) and since by Corollary 2.1 the tangent spaces ΘX,x and ΘY,y are isomorphic to the vector space of y respectively. We get a map ΘX,x → ΘY,y. This linear forms on mx/m2 x and my/m2 map is called the differential of f and denoted by dxf. x ; thus f induces a map f ∗ : my/m2 → mx/m2 y) ⊂ m2 It is easy to check that if g : Y → Z is another regular map and z = g(y) then the differential d(g ◦ f ) : ΘX,x → ΘZ,z of the composite map is given by d(g ◦ f ) = dg ◦ df. If f is the identity map X → X then for any point x ∈ X the differential dxf is also the identity map on the tangent space at any point. These observations imply the following result. y Corollary 2.2 Under an isomorphism of varieties, the tangent spaces at corresponding points are isomorphic. In particular the dimension of the tangent space at a point is invariant under isomorphism. Theorem 2.2 The tangent space ΘX,x is a local invariant of a point x of a variety X. Namely, ΘX,x is the dual vector space of the vector space mx/m2 x, where mx is the maximal ideal of the local ring Ox of x. Proof We show how to determine Θx in terms of the local ring Ox of x. Recall that the differential of a rational function F /G, where F, G ∈ k[T1,..., Tn], at a point 1 Singular and Nonsingular Points where G(x) = 0, is given by 89 dx(F /G) = G(x)dxF − F (x)dxG G2(x). We can view a function f ∈ Ox as the restriction to X of a rational function F /G, and deﬁne the differential as dxf = dx(F /G)\|Θx. All the arguments given before Theorem 2.1 and during its proof go through as before, and we see that dx ∼→ Θ ∗ deﬁ
nes an isomorphism dx : mx/m2 x, where now mx is the maximal ideal x {f ∈ Ox \| f (x) = 0} of the local ring Ox. This proves Theorem 2.2. We deﬁne the tangent space Θx at a point x of any quasiprojective variety X x)∗, where mx is the maximal ideal of the local ring Ox of x. By Theo- as (mx/m2 rem 2.2, Θx is then also the tangent space at x to any afﬁne neighbourhood of x. The tangent space is thus deﬁned as an abstract vector space, not realised as a subspace of any ambient space. However, if X is afﬁne and X ⊂ AN then the embedding i : X → AN deﬁnes an embedding di : ΘX,x → ΘAN,x. Since ΘAN,x may be identiﬁed with AN, we can view ΘX,x as embedded in AN, thus returning to the deﬁnition given in Section 1.2. If X ⊂ PN is a projective variety and x ∈ X with x ∈ AN i, then ΘX,x is an afﬁne i. The closure of ΘX,x in PN does not depend on the choice of linear subspace of AN the afﬁne piece AN i. Despite the ambiguity in using the same term for two different objects, the closure Θ X,x ⊂ PN is sometimes also called the (projective) tangent space to X at x. The usual veriﬁcation shows that Θ X,x ⊂ PN is deﬁned by the equations N i=0 ∂Fα ∂ξi (x)ξi = 0, where {Fα} is a homogeneous basis for the ideal of X. The intrinsic nature of the tangent space provides answers to certain questions on embedding varieties in afﬁne spaces. For example, if x ∈ X is a point such that dim Θx = N, then X is not isomorphic to any subvariety of an afﬁne space An with ∼→ Y ⊂ An would
take Θx isomorphically to the n < N. An isomorphism f : X subspace Θf (x) ⊂ An. From this, for any n > 1, one can construct an example of a curve X ⊂ An not isomorphic to any curve Y ⊂ Am with m < n. Namely, take X to be the image of A1 under the map x1 = t n, x2 = t n+1,..., xn = t 2n−1. (2.11) It is enough to prove that the tangent space to X as x = (0,..., 0) is the whole of An. This means that all polynomials F ∈ AX do not contain linear terms in T1,..., Tn. Let F ∈ AX and write F = n i=1 aiTi + G with G ∈ (T1,..., Tn)2. 90 2 Local Properties Substituting (2.11) in F, we get the following identity in t t n, t n+1,..., t 2n−1 ait n+i−1 + G ≡ 0. n i=1 But if any ai = 0 this is impossible, since the terms ait n+i−1 have degree ≤2n − 1, and terms coming from G(t n,..., t 2n−1) have degree ≥2n, so that they cannot cancel out. It follows from the proof just given that no neighbourhood of x in the curve X is isomorphic to a quasiprojective variety in Am with m < n. We now consider some examples of tangent spaces. We start by giving an interpretation of the tangent space to a point q ∈ P(V ) of the projective space corresponding to a vector space V. The tangent space ΘV,v to V at v can naturally be identiﬁed with V, since mv/m2 v is identiﬁed with the vector space of linear forms on V, that is, V ∗. The map π : V \ 0 → P(V ) given by π(ξ0,..., ξn) = (ξ0 : · · · : ξn) has differential dvπ :
ΘV,v = V → ΘP(V ),π(v). If ξ0 = 0 at v then in coordinates xi = ξi/ξ0 a linear form ϕ ∈ ΘV,v goes over into the function ψ = (dvπ)(ϕ) on mπ(v)/m2 π(v) for which ψ(xi) = ϕdv(ξi/ξ0) = ξiϕ(ξ0) − ξ0ϕ(ξi) ξ 2 0. It follows that the image of dvπ is the whole of ΘP(V ),π(v), and the kernel consists of the vectors (η0,..., ηn) satisfying ξiη0 = ξ0ηi, that is, proportional to (ξ0,..., ξn). Thus for ξ ∈ P(V ) we have ΘP(V ),ξ ∼= V / lξ, (2.12) where lξ = π −1(ξ ) is the line in V corresponding to a point ξ ∈ P(V ). From this, we can say that if X ⊂ P(V ) is a projective variety deﬁned by a system of homogeneous equations, and X ⊂ V the afﬁne cone over X, deﬁned in V by the ∼= ΘX,x/ lx, where x = π(x), and lx is as in (2.12). We same equations, then ΘX,x apply this interpretation of the tangent space to a projective variety to the algebraic varieties considered in Examples 1.24–1.26. Example 2.4 (The Grassmannian) We consider here only X = Grass(2, n). It is de2 V ). Differentiating these equations, ﬁned by the equations x2 = x ∧ x = 0 in P( we get that the tangent space to the afﬁne cone X ⊂ 2 V 2 V at x consists of y ∈ such that x ∧ y = 0. (2.13) 2 V is the point corresponding to a 2-plane L ⊂ V, so Suppose that
x ∈ 2 L = kx, and let f ∈ Hom(L, V /L). Then it is easy to check that for any that basis e1, e2 of L, the bivector y = e1 ∧ f (e2) − e2 ∧ f (e1) is uniquely determined 2 V /kx, is independent of the choice of a basis in L up to a scalar multiple, in 2 L ⊂ 1 Singular and Nonsingular Points 91 and satisﬁes (2.13). Moreover, any solution to (2.13) is obtained in this way. Thus for any 2-plane L ⊂ V, we see that ΘGrass(2,V ),L ∼= Hom(L, V /L). (2.14) We will show in Example 6.24 of Section 4.1, Chapter 6, that a similar relation holds for any Grass(r, V ), and give an interpretation. Remark Our starting point for deducing (2.14) was that Grass(2, V ) is given by the system of equations x ∧ x = 0. But in order to apply the deﬁnition of the tangent space given in Section 1.2, we need to know that these equations not only deﬁne X = Grass(2, V ) set-theoretically, by also generate the ideal AX. At present, we can only assert that if we write out the equations x ∧ x = 0 as F1 = · · · = Fm = 0 2 V ), the space deﬁned by then, after restricting to some afﬁne piece of P( (∂Fi/∂Tj )(x)(Ti − xi) = 0 is isomorphic to Hom(L, V /L). From this it is already not hard to deduce that AX = (F1,..., Fm), and hence the relation (2.14) holds without any reservations (see Exercise 15 of Section 3.3). Example 2.5 (Variety of associative algebras) Differentiating the associativity relation (1.28), we see that the tangent space to the variety of associative algebras is deﬁned by the equations ij xm αl lk il xl αm j k j k
xm il + αm lkxl ij (2.15) + αl. = l l = ηm Suppose that xm ij f (x, y) with x, y ∈ A given by f (ei, ej ) = take the form ij satisfy these equations. Consider the bilinear function ij em. The relations (2.15) then m ηm xf (y, z) + f (x, yz) = f (xy, z) + f (x, y)z for all x, y, z ∈ A. Functions of this type are called 2-cocycles on A. Thus the tangent space to the variety of algebras at a point corresponding to an algebra A is isomorphic to the space of 2-cocycles on A. Remark As in Example 2.4, we started from the relations (1.28), that deﬁne the variety of associative algebras only in the set-theoretic sense. Whether the left-hand sides of these equations generate the ideal of the variety seems not to be known; it is known that this fails for Lie algebras, and it is plausible that it also fails for associative algebras. Thus the space of 2-cocycles on A is only equal to the tangent space at A to the variety of algebras for those dimensions n for which the (1.28) generate the ideal of the variety of algebras, or for A for which these equations generate the ideal locally. However, the associativity relations (1.28) are so natural that any information deduced from them should have some kind of meaning. In particular, for a discussion of the space of 2-cocycles, see Section 3.4, Chapter 5, and Example 6.25 of Section 4.1, Chapter 6. 92 2 Local Properties Example 2.6 (Variety of quadrics) Let V be the vector space of symmetric n × n matrixes A = (xij ), with xij = xj i, and consider the variety Δ ⊂ P(V ) given by det A = 0 for A ∈ V. It is easy to see that det A is an irreducible polynomial, so that Δ is an irreducible hypersurface. The tangent space to the af�
��ne cone Δ at a matrix A consists of matrixes B ∈ V such that ((d/dt) det(A + tB))\|t=0 = 0. Since d dt det(A + tB)\|t=0 = det A1 + · · · + det An, where Ai is the matrix obtained by replacing the ith row of A by that of B, this expression is 0 if rank A < n − 1. For these points ΘΔ,A = P(V ). Suppose that A has rank n − 1. Transformations A → t CAC with C a nondegenerate matrix obviously deﬁne automorphisms of Δ. We can use such a transformation to put the quadratic form f corresponding to A in the form x2 n−1. Thus we 1 can assume that f = x2 n−1, and then the same argument shows that 1 ((d/dt) det(A + tB))\|t=0 equals the entry bnn of B. Hence at such points, the tangent space ΘΔ,A can be identiﬁed with the subspace of matrixes B ∈ V with bnn = 0, that is, the space of quadrics passing through the vertex of the singular quadric f = 0. + · · · + x2 + · · · + x2 1.4 Singular Points We now explain what can be said concerning the dimension of tangent spaces of an irreducible quasiprojective variety X. Our result will be local in nature, so that we restrict ourselves to considering afﬁne varieties. Let X ⊂ AN be an irreducible variety. Consider the subset Θ of the direct product AN × X consisting of pairs (a, x) with a ∈ AN and x ∈ X such that a ∈ Θx. Equation (2.7) shows that Θ is closed in AN × X. Write π : Θ → X for the second projection, π(a, x) = x. Obviously π(Θ) = X, and π −1(x) = {(a, x) \| a ∈ Θx}. Thus Θ is ﬁbred over X with the tangent spaces Θx at different points of X as ﬁbres; Θ → X is
called the tangent ﬁbre space to X. We apply to Θ the results on the dimensions of ﬁbres of a map Theorem 1.25 and its corollary; then we see that there exists a number s such that dim Θx ≥ s for every x ∈ X, and the points y ∈ X for which dim Θy > s form a closed proper subvariety Y X, that is, a variety of smaller dimension. Deﬁnition Let X be an irreducible variety and set s = minx∈X dim Θx. We say that a point x ∈ X is nonsingular5 if dim Θx = s; we also say that X is nonsingular at x. A variety X is nonsingular if it is nonsingular at every x ∈ X. If dim Θx > s then x is a singular point of X. 5The term smooth is used interchangeably with nonsingular in the current literature. The ﬁrst English edition of this book used the archaic term simple, which goes back to Zariski, and is a literal translation of the Russian, but is not in current use. 1 Singular and Nonsingular Points 93 As we have just seen, nonsingular points of X form an open nonempty subvariety, and singular points a closed proper subvariety. Consider the example of a hypersurface (Example 2.2), which contains as the particular case n = 2 the case of algebraic plane curves considered in Section 1.5, Chapter 1. If AX = (F ) then the equation of the tangent space at x is n i=1 ∂F ∂Ti (x)(Ti − xi) = 0. We now prove that in this case s = min dim Θx = n − 1. This is obviously equivalent to saying that the ∂F /∂Ti are not all identically 0 on X. In characteristic 0 this would mean that F is constant, and in characteristic p > 0 that all the indeterminates only appear in F in powers that are multiples of p. But then, as in Section 1.5, Chapter 1, since the ﬁeld k is algebraically closed, it would follow that F = F p 1, and this contradicts AX = (F ). Thus in our example, nonsingular points x ∈
X have dim Θx = dim X = n − 1. We now prove that the same holds for an arbitrary irreducible variety, and that the general case reduces to that of a hypersurface. Theorem 2.3 The dimension of the tangent space at a nonsingular point equals the dimension of the variety. Proof In view of the deﬁnition of nonsingular point, the theorem asserts that dim Θx ≥ dim X for every point x of an irreducible variety X, and that the set of points x with dim Θx = dim X is open and nonempty. This is obviously a local assertion, and we need only consider the case of an afﬁne variety. We have seen that there exists an s such that dim Θx ≥ s for every x ∈ X, and the set of points x with dim Θx = s is open and nonempty. It only remains to prove that s = dim X. We now use Theorem 1.8, which asserts that X is birational to a hypersurface Y. Let ϕ : X → Y be the birational map of Theorem 1.8. By Proposition of Section 4.3, Chapter 1, there exist nonempty open sets U ⊂ X and V ⊂ Y such that ϕ ∼→ V. By the remarks made before the statement deﬁnes an isomorphism ϕ : U of the theorem, the set W of nonsingular points of the variety Y is open, and dim Θy = dim Y = dim X for all y ∈ W. The set W ∩ V is also open and nonempty, and hence ϕ−1(W ∩ V ) ⊂ U is also open and nonempty. Since the dimension of the tangent space is invariant under isomorphism, dim Θx = dim X for x ∈ ϕ−1(W ∩V ). The theorem is proved. Consider now reducible varieties. Already the inequality dim Θx ≥ dim X fails for them. For example, if X = X1 ∪ X2 with dim X1 = 1 and dim X2 = 2, and if x ∈ X1 \ X2 is a nonsingular point of X1 then dim Θx = 1, whereas dim X = 2. This is only to be expected, since a component of X not
passing through x contributes to the dimension dim X, but does not affect Θx. Hence it is natural to introduce the following notion. The dimension of X at a point x, denoted by dimx X, is the maximum of the dimensions of the irreducible components of X through x. Obviously dim X = maxx∈X dimx X. 94 2 Local Properties Deﬁnition A point x of an afﬁne variety is nonsingular if dim Θx = dimx X. It follows from Theorem 2.3 that dim Θx ≥ dimx X for any point x ∈ X. Indeed, if Xi for i = 1,..., s are the irreducible components of X passing through x, and ⊂ Θx, so Θ i that x is the tangent space to Xi at this point then dim Θ i x ≥ dim Xi and Θ i x dim Θx ≥ max i dim Θ i x ≥ max i dim Xi = dimx X. It follows from Theorem 2.3 exactly as before that the singular points are con- tained in a subvariety of X of smaller dimension. The passage to an arbitrary quasiprojective variety is obvious: a point x ∈ X is nonsingular if it is nonsingular on an afﬁne neighbourhood U x. This is equivalent to dim Θx = dimx X. A variety is nonsingular or smooth if all its points are nonsingular. + · · · + x2 r Examples of singular points of algebraic plane curves appeared in Section 1.5, Chapter 1. We now consider a quadric Q ⊂ Pn. In a suitable coordinate system, = 0 for some r ≤ n (here we are assuming that Q has the equation x2 0 char k = 2). The singular points of Q are given by x0 = · · · = xr = 0, and if r = n, there are none. If r < n then the singular points form a (n−r −1)-dimensional linear subspace L ⊂ Pn. Intersecting Q with the r-dimensional subspace xr+1 = · · · = xn = 0 gives a nonsingular quadric S ⊂ Pr. For any point q = (α0 : · · · : αn) ∈ Q the
points s = (α0 : · · · : αr ) ∈ S, because the equation of Q does not involve the last n − r coordinates xr+1,..., xn. If s is ﬁxed, the points q ∈ Q with arbitrary αr+1,..., αn form the (n − r)-dimensional linear subspace spanned by s and L. These subspaces sweep out Q. For this reason, we say that Q is a cone with vertex the linear subspace L and base the nonsingular quadric S. Example 1.34 showed that dim Grass(r, n) = r(n − r), and that Grass(r, n) is nonsingular and rational. In exactly the same way, in the space of quadrics, by Example 2.6, the open set of the determinantal hypersurface Δ consisting of quadrics of rank n − 1 is nonsingular. In the case of the variety of associative algebras (Example 2.5), the situation is more complicated; there are both nonsingular and singular points, that is, “nonsingular” and “singular” algebras. 1.5 The Tangent Cone The simplest invariant measuring how far a singular point is from being nonsingular is the dimension of its tangent space. However, there is a much ﬁner invariant, the tangent cone to X at x. We do not need this notion in what follows, and therefore leave the detailed working out of the following arguments to the reader as a (very easy) exercise. Let X be an irreducible afﬁne variety. The tangent cone to X at x ∈ X consists of lines through x that we deﬁne as the analogue of limiting positions of secants in differential geometry. 1 Singular and Nonsingular Points 95 Suppose that X ⊂ AN with x = (0,..., 0), and that we make AN into a vector space using the choice of x as origin. In AN +1 ∼= AN × A1, consider the set X of pairs (a, t) with a ∈ AN and t ∈ A1 such that at ∈ X. Obviously X is closed in AN +1. We have, as usual, the two projections ϕ : X →
A1 and ψ : X → AN. One sees easily that X is reducible (if X = AN ), and consists of two components: X = X1 ∪ X2, where X2 = {(a, 0) \| a ∈ AN } and X1 is the closure in X of ϕ−1(A1 \ 0). Write ϕ1 and ψ1 for the restriction to X1 of the projections ϕ and ψ. The set ψ1(X1) is the closure of the set of points on all secants of X through x. The set Tx = ψ1(ϕ −1 1 (0)) is called the tangent cone to X at x. It is easy to write out the equations of the tangent cone. The equations of X are of the form F (at) = 0 for all F ∈ AX. Suppose that F = Fk + Fk+1 + · · · + Fl, where Fj is a form of degree j and Fk = 0. Then F (at) = t kFk(a) + t k+1Fk+1(a) + · · · + t lFl(a). Since F (0) = 0, we automatically have k ≥ 1, and the equation of the component X2 inside X is t = 0. It is easy to see that the equations of Tx are Fk = 0 for all F ∈ AX. The form Fk is the leading form of F. Thus Tx is deﬁned by setting to 0 the leading forms of all polynomials F ∈ AX. Since Tx is deﬁned by homogeneous polynomials, it is a cone with vertex x. It is easy to see that Tx ⊂ Θx, and that Tx = Θx if x is a nonsingular point. We consider the example of an algebraic plane curve X ⊂ A2. If AX = (F (x, y)) and Fk is the leading form of F then Tx has equation Fk(x, y) = 0. Since Fk is a form in two variables, and k is algebraically closed, Fk splits as a product of linear forms, Fk(x, y) = (αix + βiy)li. Hence in this case Tx breaks up into several lines αix
+ βiy = 0. These lines are called the tangent lines to X at x, and li their multiplicities. If k > 1 then Θx = A2. The number k is called the multiplicity of the singular point x. When k = 2 or 3 we say that x ∈ X is a double point or triple point. For example if F = x2 − y2 + x3 and x = (0, 0) then Tx consists of the two lines x + y = 0 and x − y = 0; if F = x2y − y3 + x4 and x = (0, 0) then Tx consists of 3 lines y = 0, x + y = 0 and x − y = 0. If F = y2 − x3 and x = (0, 0) then y = 0 is a tangent line with multiplicity 2. In exactly the same way as the ﬁrst deﬁnition of tangent space given in Section 1.2, the above deﬁnition of tangent cone uses a notion that is not invariant under isomorphism. However, one can prove that the tangent cone Tx is invariant under isomorphism, and is a local invariant of x ∈ X. 1.6 Exercises to Section 1 1 Prove that the local ring OX,x of a point x of an irreducible variety X is the union in k(X) of all the rings k[U ] for U a neighbourhood of x. 2 The map ϕ(t) = (t 2, t 3) deﬁnes a birational map from the line A1 to the curve y2 = x3. What are the rational functions in t that correspond to the functions in the local ring Ox of the point (0, 0)? 96 2 Local Properties 3 The same question for the birational map from A1 to the curve of (1.2). 4 Prove that the local ring Ox of the curve xy = 0 at (0, 0) is isomorphic to the subring O ⊂ O1 ⊕ O2, where O1 and O2 are copies of the local ring of 0 in A1, consisting of functions f1, f2 with f1 ∈ O1 and f2 ∈ O2 such that f1(0) = f2(0). 5
Determine the local ring at (0, 0, 0) of the curve consisting of the three coordinate axes in A3. 6 Determine the local ring at (0, 0) of the curve xy(x − y) = 0. Prove that this curve is not isomorphic to that of Exercise 5. 7 Prove that if x ∈ X and y ∈ Y are nonsingular points then (x, y) ∈ X × Y is nonsingular. 8 Prove that if X = X1 ∪ X2 and x ∈ X1 ∩ X2, then the tangent spaces ΘX,x, ΘX1,x and ΘX2,x satisfy Does equality always hold? ΘX1,x + ΘX2,x ⊂ ΘX,x. 9 Prove that a hypersurface of degree 2 with a singular point is a cone. 10 Prove that if a hypersurface X of degree 3 has two singular points then the line joining them is contained in X. 11 Prove that if a plane curve of degree 3 has three singular points then it breaks up as a union of 3 lines. 12 Prove that the singular points of the hypersurface X ⊂ Pn deﬁned by F (x0,..., xn) = 0 are determined by the system of equations F (x0,..., xn) = 0, and ∂F ∂xi (x0,..., xn) = 0 for i = 0,..., n. If deg F is not divisible by the characteristic of the ﬁeld, then the ﬁrst equation follows from the others. 13 Prove that if a hypersurface X ⊂ Pn contains a linear subspace L of dimension r ≥ n/2 then X is singular. [Hint: Choose the coordinate system so that L is given by xr+1 = · · · = xn = 0, write out the equation of X and look for singular points contained in L.] 14 For what values of a does the curve x3 0 a singular point? What are its singular points then? Is it reducible? + x3 2 + x3 1 + a(x0 + x1 + x2)3 = 0 have 1 Singular and Nonsingular Points 97 15 Deter
mine the singular points of the Steiner surface in P3: 1 x2 x2 2 + x2 2 x2 0 + x2 0 x2 1 − x0x1x2x3 = 0. 16 For what values of a does the surface x4 0 singular points, and what are these points? + x4 1 + x4 2 + x4 3 − ax1x2x3x4 have 17 Let PN with N = νn,m be as in Example 1.28. Prove that over a ﬁeld of characteristic 0, the points of the space PN for which the corresponding hypersurfaces has a singular point form a hypersurface in PN. [Hint: Use the result of Exercise 10 of Section 6.5, Chapter 1.] 18 Let F (x0, x1, x2) = 0 be the equation of an irreducible curve X ⊂ P2 over a ﬁeld of characteristic 0. Consider the rational map ϕ : X → P2 given by the formulas ui = ∂F /∂xi(x0, x1, x2) for i = 0, 1, 2. Prove (a) ϕ(X) is a point if and only if X is a line; (b) if X is not a line, then ϕ is regular at x ∈ X if and only if x is nonsingular. The image ϕ(X) is called the dual curve of X. 19 Prove that if X is a conic then so is ϕ(X). 20 Find the dual curve of x3 0 + x3 1 + x3 2 = 0. 21 Prove that if X ⊂ Pn is a nonsingular hypersurface and not a hyperplane, then as x runs through X, the tangent hyperplanes Θx form a hypersurface in the dual space Pn∗. 22 Let ϕ be the regular map of a variety X ⊂ An consisting of the linear projection to some subspace. Determine the map dϕ on the linear subspaces Θx for x ∈ X. 23 Let x be a point of a variety X and mx ⊂ Ox the local ring at x and its maximal ideal. Prove that for every integer t > 0, the module mt is a ﬁnite dimensional vector space
over k. x/mt+1 x 24 Let A be a ring and M an A-module. A derivation of A into M is a map d : A → M satisfying the property d(ab) = a db + b da for all a, b ∈ A. If A and M are vector spaces over a ﬁeld k, then we require in addition that dα = 0 for α ∈ k. Check that the multiplication map d → cd for c ∈ A turns the set of all derivations from A to M into an A-module. It is denoted by Der(A, M). If A and M are vector spaces over k, we write Derk(A, M). For A = k[X] the coordinate ring of an afﬁne variety X, if we view k as the residue ﬁeld at a point x ∈ X, which is an A-module by the rule f · α = f (x) · α, then prove that Derk(A, k) ∼= Θx. 98 2 Local Properties 2 Power Series Expansions 2.1 Local Parameters at a Point We study a nonsingular point x of a variety X, with dimx X = n. Deﬁnition Functions u1,..., un ∈ Ox are local parameters at x if each ui ∈ mx, and the images of u1,..., un form a basis of the vector space mx/m2 x. x, we see that u1,..., un ∈ Ox is a In view of the isomorphism dx : mx/m2 x system of local parameters if and only if the n linear forms dxu1,..., dxun on Θx are linearly independent. Since dim Θx = n, this in turn is equivalent to saying that the system of equations → Θ ∗ dxu1 = · · · = dxun = 0 (2.16) has 0 as its only solution in Θx. i We can replace X by an afﬁne neighbourhood X of x on which the u1,..., un are regular functions; set A = AX. Now write X i for the hypersurface in X deﬁned by. Let Ui be a polynomial that deﬁnes the function u
i on X. ui = 0, and set Ai = AX Then Ai ⊃ (A, Ui), and by deﬁnition of the tangent space it follows that Θi ⊂ Li, where Θi is the tangent space to X i at x and Li ⊂ Θx is deﬁned by dxUi = 0. From the assumption that the system (2.16) has 0 as its only solution in Θx it follows that Li = Θx, that is, dim Li = n − 1, and from the theorem on dimension of intersection and the inequality dim Θi ≥ dim X i it follows that dim Θi ≥ n − 1. Hence dim Θi = n − 1, and it follows that x is a nonsingular point of X i. In some neighbourhood of x, the intersection of the varieties X X i with dim Y > 0 passed through x, the tangent space to Y at x would be contained in all the Θi, and this again contradicts the assumption that (2.16) has 0 as its only solution. i is exactly x: for if some component Y of We have thus proved the following assertion. Theorem 2.4 If u1,..., un are local parameters at x such that the ui are regular on X, and Xi = V (ui), then x is a nonsingular point on each of the Xi and Θi = 0, where Θi is the tangent space to Xi at x. Here we meet a general property of subvarieties that will appear frequently in what follows. Deﬁnition Subvarieties Y1,..., Yr of a nonsingular variety X are transversal at a point x ∈ Yi if codimΘX,x ΘYi,x = r i=1 r i=1 codimX Yi. (2.17) 2 Power Series Expansions Figure 7 Transversal curves on a surface 99 For example, two curves on a nonsingular surface are transversal at a point of intersection if they are both nonsingular and their tangent lines are different (Figure 7). Using the inequality (1.49) for the subspaces ΘYi,x ⊂ ΘX,x, and the inequalities codim ΘYi,x ≤
codimX Yi we see that (2.17) implies the equality dim ΘYi,x = dim Yi, so that each Yi is nonsingular at x, and the equality codimΘX,x ΘYi,x = r i=1 r i=1 codim ΘYi,x, so that the vector subspaces ΘYi,x ⊂ ΘX,x are transversal, in the sense that their intersection is as small as possible for their dimensions. From the inclusion r i=1 ΘYi,x ⊃ ΘY,x, where Y = Yi, we deduce in the same way that Y is non- singular at x. Thus Theorem 2.4 asserts that the subvarieties V (ui) are transversal. Let X be an afﬁne neighbourhood of x in which = x = (0,..., 0). If X ⊂ AN and ti are coordinates in AN, then x is deﬁned by t1 = · · · = tN = 0, and X i is deﬁned in X by u1 = · · · = un = 0. By the Nullstellensatz it follows that (t1,..., tN )k ⊂ (u1,..., un) for some k > 0, where (t1,..., tN ) and (u1,..., un) denote ideals of k[X]. A fortiori the same holds for the ideals (t1,..., tN ) and (u1,..., un) in Ox. Note that (t1,..., tN ) = mx, so that mk ⊂ (u1,..., un). In fact x a more precise statement holds. X i Theorem 2.5 Local parameters at x generate the maximal ideal mx of Ox. Proof This is an immediate consequence of Nakayama’s lemma (Proposition A.11) applied to the maximal ideal mx as an Ox -module. By Lemma of Section 1.1, mx is a ﬁnite Ox -module. Since local parameters generate mx/m2 x, they generate mx by Nakayama’s lemma. The theorem
is proved. Example Let X be a nonsingular afﬁne variety and G a ﬁnite group of automorphisms of X, as in Example 1.21. Suppose that G acts freely on X, that is, g(x) = x implies that g = e for any g ∈ G and x ∈ X, where e is the identity map. We prove that under these assumptions the quotient variety X/G is again nonsingular. Let f : X → Y = X/G be the quotient map constructed in Example 1.21, and 100 2 Local Properties set n = dim X = dim Y. Choose x ∈ X, set y = f (x) ∈ Y and let u1,..., un be local parameters at x with ui ∈ k[X]. Then u1,..., un generate mx. For each ui, we construct a function ui ∈ k[X] such that ui ≡ ui mod m2 g(x) for all g ∈ G with g = e. For this, we need only multiply ui by the square of an element h ∈ k[X] with h(x) = 1 and h(g(x)) = 0 for all g = e. x and ui ∈ m2 Set vi = S(ui), where the averaging operator S is as in Proposition A.6 and Example 1.21. Since g∗ui ∈ m2 x, and hence v1,..., vn are local parameters at x. But vi ∈ k[Y ] and vi(y) = 0. Let us prove that my = (v1,..., vn). Let h ∈ my ∩ k[Y ]. Then f ∗(h) ∈ mx and f ∗(h) = hivi. Applying the operator S to this, in view of S(f ∗(h)) = f ∗(h) and S(vi) = vi, we get that f ∗(h) = ≤ n, and it follows that y is nonsingular. S(hi)vi. Thus dim my/m2 y x for g = e, we have vi ≡ 1/\|G\| · ui mod m2 It is important to note that nonsingularity of a point x is characterised by
a purely algebraic property of the local ring Ox. By deﬁnition x ∈ X is nonsingular if and = dimx X. The left-hand side of the equality is deﬁned for any only if dimk mx/m2 x Noetherian local ring O. The right-hand side can also be expressed as a property of the local ring Ox. Namely, by Theorem 1.23, Corollary 1.3, the dimension of X at x can be deﬁned as the smallest r for which there exist r functions u1,..., ur ∈ mx such that, in some neighbourhood of x, the set deﬁned by u1 = · · · = ur = 0 consists of x only. By the Nullstellensatz, this property is equivalent to (u1,..., ur ) ⊃ mk x for some k > 0. For an arbitrary Noetherian local ring O with maximal ideal m, the smallest number r for which there exist r functions u1,..., ur ∈ m such that (u1,..., ur ) ⊃ mk for some k > 0 is called the dimension of O and denoted by dim O. By Nakayama’s lemma, the ideal m itself is generated by n elements, where n = dimO/m(m/m2). Hence dim O ≤ dimO/m m/m2. If dim O = dimO/m(m/m2) then the local ring O is said to be regular. We see that x is a nonsingular point if and only if the local ring Ox is regular. This is the algebraic meaning of nonsingularity of a point. 2.2 Power Series Expansions The idea of associating power series with elements of the local ring Ox is based on the following arguments. For any function f ∈ Ox, set f (x) = α0 and f1 = f − α0; then f1 ∈ mx. Let u1,..., un be a system of local parameters at x. By deﬁnition, u1,..., un generate the whole of the vector space mx/m2 x. Thus there n exist α1,..., αn ∈ k such that f1 − i=1
αiui = x. Set f2 = f1 − n f − α0 − gj hj with gj, hj ∈ mx. i=1 αiui. Since f2 ∈ m2 As above, there exist βj i, γj i ∈ k such that n i=1 αiui ∈ m2 x, we can write f2 = gj − n i=1 βj iui ∈ m2 x and hj − n i=1 γj iui ∈ m2 x. 2 Power Series Expansions 101 i βj iui)( αlkuluk ∈ m3 j ( x, Now set and therefore f3 = f − α0 − x. Continuing in the same way, we can obviously ﬁnd forms Fi ∈ k[T1,..., Tn] of degree deg Fi = i such that f − n l,k=1 αlkuluk. Then f2 − i γj iui) = n αlkuluk ∈ m3 i=1 αiui − i=0 Fi(u1,..., un) ∈ mk+1 k x. Deﬁnition 2.4 The formal power series ring in variables (T1,..., Tn) = T is the ring whose elements are inﬁnite expressions of the form Φ = F0 + F1 + F2 + · · ·, (2.18) where Fi ∈ k[T ] is a form of degree i, and the ring operations are deﬁned by the rules: if Ψ = G0 + G1 + G2 + · · · then Φ + Ψ = (F0 + G0) + (F1 + G1) + (F2 + G2) + · · ·, and ΦΨ = H0 + H1 + H2 + · · ·, where Hi = Gj Fl. j +l=i The formal power series ring is denoted by k[[T ]]. It contains k as the power series with Fi = 0 for i > 0. If i is the ﬁrst index for which Fi = 0 then Fi is called the leading term of (2.18). The leading term of a product is equal to the product of the leading
terms, so that k[[T ]] has no zerodivisors. The arguments discussed above allow us to associate a power series Φ = F0 + F1 + F2 + · · · with a function f ∈ Ox. We arrive at the following deﬁnition. Deﬁnition 2.5 A formal power series Φ is called the Taylor series of a function f ∈ Ox if for every k ≥ 0 we have f − Sk(u1,..., un) ∈ mk+1 x, with Sk = k i=0 Fi. (2.19) Example Let X = A1 with coordinate t, and let x be the point t = 0. Then mx = m=0 αmt m with any rational function (t), and one can associate a power series f (t) = P (t)/Q(t) with Q(0) = 0 such that ∞ that is, P (t) Q(t) − k m=0 αmt m ≡ 0 mod t k+1, P (t) − Q(t) k m=0 αmt m ≡ 0 mod t k+1. 102 2 Local Properties This is the usual procedure for ﬁnding the coefﬁcients of a power series of a rational function by the method of unknown coefﬁcients. For example, 1 1 − t = ∞ m=0 t m, because 1 1 − t − k m=0 t m = t k+1 1 − t ≡ 0 mod t k+1. The correspondence f → Φ depends in an essential way on the choice of the system of local parameters u1,..., un. The arguments we have just given prove the following assertion. Theorem 2.6 Every function f has at least one Taylor series. Up to now we have used in essence not that x is nonsingular, but only that u1,..., un generate mx/m2 x. Now we make use of the nonsingularity of x. Theorem 2.7 If x is nonsingular, then a function has a unique Taylor series. Proof It is obviously enough to prove that any Taylor series of the function f = 0 is equal to 0. By (2.19), this is equivalent to the assertion that if u1,..., un are local parameters of a nons
ingular point x, then for a form Fk(T1,..., Tn) of degree k, Fk(u1,..., un) ∈ mk+1 x =⇒ Fk(T1,..., Tn) = 0. (2.20) Suppose that this is not the case. By means of a nondegenerate linear transformation, we can arrange that the coefﬁcient of T k n in Fk is nonzero. Indeed, this coefﬁcient equals Fk(0,..., 0, 1), and if Fk(α1,..., αn) = 0 (and such α1,..., αn certainly exist, given that Fk = 0), then we just have to carry out a linear transformation taking the vector (α1,..., αn) to (0,..., 0, 1). Thus we can assume that + G1(T1,..., Tn−1)T k−1 + · · · + Gk(T1,..., Tn−1), Fk(T1,..., Tn) = αT k n n where α = 0 and Gi is a form of degree i in T1,..., Tn−1. By Theorem 2.5, it follows easily that any element of mk+1 can be written as a form of degree k in u1,..., un with coefﬁcients in mx. Hence the left-hand side of (2.20) can be expressed in the form x + G1(u1,..., un−1)uk−1 αuk n = μuk n + H1(u1,..., un−1)uk−1 n n + · · · + Gk(u1,..., un−1) + · · · + Hk(u1,..., un−1), (2.21) where μ ∈ mx and Hi are forms of degree i. It follows that (α − μ)uk ∈ n (u1,..., un−1). Since α = 0, it follows that α − μ /∈ mx and (α − μ)−
1 ∈ Ox, ∈ (u1,..., un−1). We see that V (un) ⊃ V (u1) ∩ · · · ∩ V (un−1). It foland hence uk n lows that Θn ⊃ Θ1 ∩ · · · ∩ Θn−1, where Θi is the tangent space to V (ui) at x, and hence Θ1 ∩ · · · ∩ Θn = Θ1 ∩ · · · ∩ Θn−1. Therefore dim Θ1 ∩ · · · ∩ Θn ≥ 1, and this contradicts Theorem 2.4. The theorem is proved. 2 Power Series Expansions 103 Thus we have a uniquely determined map τ : Ox → k[[T ]] that takes each function to its Taylor series. A simple veriﬁcation based on the deﬁnition (2.19) of τ shows that it is a homomorphism. We leave this veriﬁcation to the reader. What is the kernel of τ? If τ (f ) = 0 for a function f ∈ Ox, then by (2.19) this means that f ∈ mk+1 x. Thus we are talking about functions that are analogues of the functions in analysis with every derivative at some point equal to 0. In our case such a function must be equal to 0. This follows from Propositions A.12 and Lemma of Section 1.1. for all k. In other words, f ∈ k=0 mk ∞ x As a corollary we get the following result. Theorem 2.8 A function f ∈ Ox is uniquely determined by any of its Taylor series. In other words, τ is an isomorphic inclusion of the local ring Ox into the formal power series ring k[[T ]]. Recall that in this section we have nowhere used that the variety X is irreducible. Conversely, Theorem 2.8 allows us to make certain deductions concerning irreducibility. Theorem 2.9 If x is a nonsingular point of X then there is a unique component of X passing through x. Proof We replace X by an afﬁne neighbourhood U of x contained in X = X \ Zi, where Zi are the components of X not passing through x. Then k
[U ] ⊂ Ox. By Theorem 2.8, Ox is isomorphic to a subring of the formal power series ring k[[T ]]. Since k[[T ]] has no zerodivisors, the same holds for k[U ], which is isomorphic to a subring of k[[T ]]. Hence U is irreducible, as asserted in the theorem. Corollary The set of singular points of an algebraic variety X is closed. Proof Let X = Xi be a decomposition into irreducible components. It follows from Theorem 2.9 that the set of singular points of X is the union of the sets Xi ∩ Xj for i = j and the sets of singular points of Xi. As a union of a ﬁnite number of closed sets, it is closed. → Ox/mn ∈ Ox/mn x If x is a singular point, the best we can do is to send an element f ∈ Ox into the sequence of residue classes ξn = f + mn x. This sequence has the folx lowing compatibility property: if θn+1 : Ox/mn+1 x is the quotient map then θn+1(ξn+1) = ξn. The set of all such compatible sequences {ξn} under componentwise addition and multiplication forms a ring Ox, called the completion of Ox. We have just deﬁned a homomorphism τ : Ox → Ox by τ (f ) = {ξn}, where ξn = f + mn x. The same argument as in the case of a nonsingular point x shows that τ is an inclusion. The ring Ox is local, with maximal ideal M consisting of all compatible sequences {ξn} with ξn ∈ mx. It can be shown that applying the same construction again to Ox gives nothing new, that is, (Ox)= Ox, and τ in this case is an isomorphism. If x is nonsingular, then Ox is just the formal power series ∈ Ox/mn 104 2 Local Properties ring. In the general case, Ox is an important characteristic of a singular point. If for x ∈ X and y ∈ Y the completed local rings Ox and Oy are isomorphic, we say that the varieties X and Y are formally analytically equivalent in neighbourhoods of these points. Since for a nons
ingular point x of an n-dimensional variety the local ring Ox is isomorphic to that of a point x ∈ An, all nonsingular points of all varieties of the same dimension have formally analytically equivalent neighbourhoods. Compare Exercises 8–16 of Section 3.3. 2.3 Varieties over the Reals and the Complexes Suppose that k = R or C. We prove that in this case, the formal Taylor series of a function f ∈ Ox converges for small values of T1,..., Tn. Let X ⊂ AN be a variety, with AX = (F1,..., Fm), and suppose that dimx X = n. If x ∈ X is a nonsingular point then the matrix ∂Fi ∂Tj (x) i=1...m j =1...N has rank N − n. Suppose that the minor det ∂Fi ∂Tj (x) = 0, i=1...N−n j =n+1...N (2.22) and that x is the origin. Then the restrictions t1,..., tn to X of the ﬁrst n coordinates form a system of local parameters on X at x. Write X for the union of all components of the variety deﬁned by F1 = · · · = FN −n = 0 (2.23) that pass through x. By (2.22), the dimension of the tangent space Θ to X at x equals n, and by the theorem on dimension of intersections, dimx X ≥ n. Since dim Θ ≥ dimx X, then dimx X = n and x is a nonsingular point of X. From this it follows by Theorem 2.9 that X is irreducible. Obviously X ⊃ X, so that dim X = dim X implies that X = X. We see that X can be deﬁned in some neighbourhood of x by the N − n equations (2.23), and that these satisfy (2.22). By the implicit function theorem (see for example Goursat [32, §§187–190, Chapter IX, Vol. 1], or Fleming [27, Section 4.6]), there exists a system of power series Φ1,..., ΦN −n in n variables T1,..., Tn
and an ε > 0 such that Φj (T1,..., Tn) converges for all Ti with \|Ti\| < ε, and T1,..., Tn, Φ1(T ),..., ΦN −n(T ) Fi = 0; (2.24) moreover, the coefﬁcients of the power series Φ1,..., ΦN −n are uniquely determined by the relation (2.24). 2 Power Series Expansions 105 However, assuming that t1,..., tn are chosen as local parameters, the formal power series τ (Tn+1),..., τ (TN ) also satisfy (2.24), and hence must coincide with Φ1,..., ΦN −n, and it therefore follows that τ (Tn+1),..., τ (TN ) converge if \|Tj \| < ε for j = 1,..., n. Any function f ∈ Ox can be written in the form f = P /Q, where P = P (T1,..., Tn) and Q = Q(T1,..., Tn) are polynomials and Q(x) = 0; and then τ (f ) = P (τ (T1),..., τ (Tn)) Q(τ (T1),..., τ (Tn)). The convergence of τ (f ) then follows from standard theorems on convergence of power series. In the same way, one can show that if u1,..., un is another system of local pa- rameters then det ∂τ (ui) ∂Tj (0,..., 0) i=1...n j =1...n is nonzero, the Taylor series of t1,..., tn with respect to the system of local parameters u1,..., un are obtained by inverting the series τ (ui) = Φi(T1,..., Tn) for i = 1,..., n, and hence they also have positive radius of convergence. Therefore for f ∈ Ox, with respect to any choice of the system of local parameters, the series τ (f ) has positive radius of convergence.
The implicit function theorem asserts not only that the convergent series Φ1,..., ΦN −n exist, but also that there exists η > 0 such that any point (t1,..., tN ) ∈ X with \|ti\| < η for i = 1,..., N is given by the form tn+i = Φi(t1,..., tn) for i = 1,..., N − n. It follows that (t1,..., tN ) → (t1,..., tn) is a homeomorphism of the set {(t1,..., tN ) ∈ X \|ti\| < η} to a domain of n-dimensional space. Since in our case k = R or C, projective space PN over k is a topological space. An algebraic variety X ⊂ PN is also a topological space. In the respective cases k = R or C, this topology on X is called the real or complex topology of X. This topology and notions deduced from it should not be confused with the terms of topological nature such as closed set, neighbourhood, open set, closure, etc. used up to now. The preceding arguments show that for an n-dimensional variety X, any nonsingular point has a neighbourhood in the real topology that is homeomorphic to a domain of Rn. Hence if every point of X is nonsingular then X is an n-dimensional manifold in the sense of topology. If k = C then a nonsingular point x ∈ X has a neighbourhood in the complex topology that is homeomorphic to a domain in n-dimensional complex space Cn, and hence to a domain in R2n. Therefore if all points of X are nonsingular, X is a 2n-dimensional manifold. It is easy to prove that PN over k = R or C is compact in the real or complex topology. Thus if X is projective, it is compact. If k = C, the converse also holds: a quasiprojective variety X that is compact in its complex topology is projective. See Exercise 2 of Section 2.6, Chapter 7. In conclusion we note that everything we have said in this section (excluding the preceding paragraph) carries over word-for-
word to the case that k is a p-adic number ﬁeld. 106 2 Local Properties 2.4 Exercises to Section 2 1 Prove that for an n-dimensional variety X, the set of points where n given functions fail to form a system of local parameters is closed. 2 Prove that a polynomial f ∈ k[T ] = k[A1] is a local parameter at the point T = α if and only if α is a simple root of f. 3 Prove that a formal power series Φ = F0 + F1 + · · · has an inverse in k[[T ]] if and only if F0 = 0. 4 Let T be an indeterminate. Consider the ring k((T )) of expressions of the form α−nT −n + α−n+1T −n+1 + · · · + α0 + α1T + · · ·. Prove that k((T )) is a ﬁeld, and is isomorphic to the ﬁeld of fractions of k[[T ]]. (It is called the ﬁeld of formal Laurent series.) 5 Let S ⊂ A2 be the circle given by X2 + Y 2 = 1, over a ﬁeld k of characteristic 0. Prove that X is a local parameter at x = (0, 1), and that the Taylor series expansion of Y is given by τ (Y ) = (−1)n (1/2)(1/2 − 1) · · · (1/2 − n + 1) X2n. n! ∞ n=0 6 Prove that if x is a singular point then any function f ∈ Ox has an inﬁnite number of different Taylor series. 7 Let X = A1 and x ∈ X. Prove that τ (Ox) is not the whole of k[[T ]]. 3 Properties of Nonsingular Points 3.1 Codimension 1 Subvarieties The theory of local rings allows us to prove an important property of nonsingular varieties analogous to Theorem 1.21. The question under discussion is that of deﬁning a codimension 1 subvariety Y ⊂ X by means of a single equation. For a singular variety, this property fails in general; (compare Corollary 1.9). We prove, however, that
it holds locally on a nonsingular variety. To state the result, we introduce the following deﬁnition. Deﬁnition Functions f1,..., fm ∈ Ox are local equations of a subvariety Y ⊂ X in a neighbourhood of x if there exists an afﬁne neighbourhood X of x such that f1,..., fm ∈ k[X] and aY = (f1,..., fm) in k[X], where Y = Y ∩ X. 3 Properties of Nonsingular Points 107 It is convenient to restate this condition in terms of the local ring Ox of x. For this, consider the ideal aY,x ⊂ Ox consisting of functions f ∈ Ox that are equal to 0 on Y in some neighbourhood of x. For an afﬁne variety X, we obviously have aY,x = f = u/v u, v ∈ k[X] with u ∈ aY and v(x) = 0, and if all components of Y pass through x, then aY = aY,x ∩ k[X]. Lemma Functions f1,..., fm ∈ Ox are local equations of Y in a neighbourhood of x if and only if aY,x = (f1,..., fm). Proof If aY = (f1,..., fm) in k[X] then obviously also aY,x = (f1,..., fm) in Ox. Conversely, suppose that aY,x = (f1,..., fm) with fi ∈ Ox, and let aY = (g1,..., gs) with gi ∈ k[X]. For i = 1,..., s, since gi ∈ aY,x, we can write gi = m j =1 hij fj with hij ∈ Ox. (2.25) The functions fi and hij are all regular in some principal open neighbourhood U of x. Suppose that U = X \ V (g) with g ∈ k[X]. Then k[U ] consists of elements of the form u/gl with u ∈ k[X] and l ≥ 0. Then by (2.25), inside k[U ] we
have (g1,..., gs) = aY k[U ] ⊂ (f1,..., fm). We prove that aY k[U ] = aY, where now Y = Y ∩ U. From this, it then follows that aY ⊂ (f1,..., fm), and since fi ∈ aY, this implies the assertion of the lemma. It remains to prove that aY k[U ] = aY. The inclusion aY k[U ] ⊂ aY is obvious. Let v ∈ aY. Then v = u/gl with u ∈ k[X], and hence u = vgl; hence u ∈ aY, and since 1/gl ∈ k[U ], we get v = u/gl ∈ aY k[U ]. The lemma is proved. Our aim is to prove the following result. Theorem 2.10 An irreducible subvariety Y ⊂ X of codimension 1 has a local equation in a neighbourhood of any nonsingular point x ∈ X. The proof follows exactly the steps of the proof of Theorem 1.21. There, however, we used the fact that k[T ] is a UFD. Here the part of k[T ] is played by the local ring Ox. It has the analogous property. Theorem 2.11 The local ring Ox of a nonsingular point is a UFD. The proof of Theorem 2.11 is based on ﬁrst establishing that the power series ring k[[T ]] is a UFD. This is a fairly elementary fact, similar to the corresponding result for polynomial rings. We indicate only the main steps of the proof. An entirely elementary proof (not depending on the remainder of the book) can be found in Zariski and Samuel [81, Theorem 6 of Section 1, Chapter VII, Vol. 2]. 108 2 Local Properties We say that a power series Φ(T1,..., Tn) is regular with respect to the variable n with cm = 0. Tn if its initial form is of degree m, say, and contains the term cmT m A linear transformation of the variables T1,..., Tn obviously induces an automorphism of k[[T ]]. We can, in particular, carry out a
linear transformation so that any given nonzero power series Φ becomes regular with respect to Tn. Lemma 2.1 (Weierstrass preparation theorem) Suppose that a power series Φ ∈ k[[T ]] is regular with respect to Tn and has initial form of degree m; then there exists a power series U ∈ k[[T ]] with nonzero constant term such that the series ΦU is a polynomial in Tn over k[[T1,..., Tn−1]], that is, ΦU = T m n + R1T m−1 n + · · · + Rm, with Ri = Ri(T1,..., Tn−1) ∈ k[[T1,..., Tn−1]] for i = 1,..., m. Proof See Zariski and Samuel [81, Theorem 5 of Section 1, Chapter VII, Vol. 2]. Lemma 2.2 The formal power series ring k[[T ]] is a UFD. Lemma 2.1 allows us to prove this assertion by induction on the number of variables T1,..., Tn by reducing it to the analogous statement for polynomials in Tn with coefﬁcients in k[[T1,..., Tn−1]]. The proof is carried out in detail in Zariski and Samuel [81, Theorem 6 of Section 1, Chapter VII, Vol. 2]. Proof of Theorem 2.11 We write Ox for the ring of formal power series, and view Ox as a subring Ox ⊂ Ox (this is possible by Theorem 2.8). Write mx for the ideal of Ox consisting of formal power series with constant term 0. Then mk x is the ideal of formal power series having no terms of degree <k. By deﬁnition of the inclusion Ox → Ox (see (2.19)), it follows that mk x. Thus the assumptions of Proposition A.14 are satisﬁed, and this guarantees that Ox a UFD (Lemma 2.2) implies Ox a UFD. The theorem is proved. ∩ Ox = mk x Proof of Theorem 2.10 As we have already said, the proof of Theorem 2.10 is exactly the same as that of Theorem 1.21. Since the assertion is
local in nature, we can assume that X is afﬁne. Let f ∈ Ox be any function that vanishes on Y. Factorise f into prime factors in Ox. By the irreducibility of Y, one factor must also vanish on Y. We denote this by g, and prove that it is a local equation of Y. Replacing X by a smaller afﬁne neighbourhood of x, we can assume that g is regular on X. Since V (g) ⊃ Y, and both are codimension 1 subvarieties, we have V (g) = Y ∪ Y. If Y x there exist functions h and h such that hh = 0 on V (g), but neither h nor h are 0 on V (g). Therefore, g divides (hh)r in k[X] for some r, and so a fortiori g \| (hh)r in Ox. Since Ox is a UFD it then follows that g divides either h or h in Ox. Then either h or h vanishes on V (g) in some neighbourhood of x, and hence, after passing to a smaller neighbourhood, on the whole of V (g). This contradicts the assumption Y x. Thus Y x, and again replacing X by a sufﬁciently small 3 Properties of Nonsingular Points 109 afﬁne neighbourhood of x, we can assume that V (g) = Y. If now u vanishes on Y then g divides us in k[X] for some s > 0, and hence a fortiori in Ox. It follows that g divides u in Ox. Thus aY,x = (g) and the theorem is proved. Theorem 2.10 has many applications. Here is the ﬁrst of these (compare Theo- rem 1.2). Theorem 2.12 If X is a nonsingular variety and ϕ : X → Pn a rational map to projective space, then the set of points at which ϕ is not regular has codimension ≥2. Proof Recall that the set of points at which a rational map ϕ is not regular (the locus of indeterminacy of ϕ) is a closed set. The assertion of the theorem is local in nature, and it is enough to prove it in a neighbourhood of a nonsingular point x ∈ X. We can write ϕ in the form ϕ
= (f0 : · · · : fn) with fi ∈ k(X), and without changing ϕ, we can multiply the fi through by a common factor in such a way that all the fi ∈ Ox, and they have no common factor in Ox. Now ϕ fails to be regular only at points at which f0 = · · · = fn = 0. But no codimension 1 subvariety Y can be contained in the locus deﬁned by these equations; indeed, by Theorem 2.10, aY,x = (g) and all the fi would have g as a common factor in Ox, which contradicts the assumption. The theorem is proved. Corollary 2.3 Any rational map of a nonsingular curve to projective space is regular. Corollary 2.4 If two nonsingular projective curves are birational then they are isomorphic. Let k = C be the complex number ﬁeld. It follows from Corollary 2.4 that if two curves X and X are birational then the set of points of X and X with their complex topologies are homeomorphic. Indeed, in this case, regular functions, hence also regular maps, are deﬁned by convergent power series, and are hence continuous. The same holds for the set of real points of curves X and X deﬁned by equations with real coefﬁcients, and such that there exists a birational map ϕ : X → X deﬁned over R, that is, deﬁned by rational functions with real coefﬁcients. This sometimes allows us to deduce easily that two curves are not birational over R. For example the curve X deﬁned by y2 = x3 − x consists of two connected components (Figure 8). Therefore X is not rational over R, since P1 is homeomorphic to the circle, and has only one connected component. Using similar ideas, we can prove that the curve X given by y2 = x3 − x is also irrational over C. For this, we need to compare the topological spaces of complex points of X and of P1 with their complex topologies, and prove that they are not homeomorphic. Indeed, the ﬁrst is homeomorphic to the torus and the second to the sphere. This is a particular case of results proved in Section 3.
3, Chapter 7. Figure 9 shows what the real points of X look like as a subset of its complex points. 110 Figure 8 y2 = x3 − x over R 2 Local Properties Figure 9 y2 = x3 − x over C 3.2 Nonsingular Subvarieties Theorem 2.10 does not generalise to subvarieties Y ⊂ X of codimension greater than 1; compare, for example, Exercise 2 of Section 6.5, Chapter 1. But a similar statement does hold for a subvariety that is nonsingular at x. We prove a slightly more precise fact. We start with an auxiliary assertion. Theorem 2.13 Let X be an afﬁne variety, x ∈ X a nonsingular point, and suppose that u1,..., un are regular functions on X that form a system of local parameters at x. Then for m ≤ n, the subvariety Y deﬁned by u1 = · · · = um = 0 is nonsingular at x, we have aY = (u1,..., um) in some afﬁne neighbourhood of x, and um+1,..., un form a system of local parameters on Y at x. Proof The proof is by induction on m. For m = 1, Theorem 2.10 shows that aY = (f ) in some afﬁne neighbourhood of x. Suppose that u1 = f v; then dxu1 = v(x)dxf. Now dxu1 = 0, since u1 is an element of a system of local parameters at x. Thus v(x) = 0, so that aY = (u1) in some smaller open set. Since dxu1 = 0 it follows that x is a nonsingular point of Y. The tangent space ΘY,x to Y at x is obviously obtained from ΘX,x by imposY,x, that is, ing the condition dxu1 = 0. Therefore dxu2,..., dxun is a basis of Θ ∗ u2,..., un is a system of local parameters on Y at x. 3 Properties of Nonsingular Points 111 In the general case, we let X ⊂ X be the subvariety deﬁned by u1 = 0. Then Y �
� X is deﬁned in X by the equations u2 = · · · = um = 0, and we can use induction. The theorem is proved. Now we prove that any subvariety Y ⊂ X that is nonsingular at x is given by the procedure described in Theorem 2.13 in some neighbourhood of x. Theorem 2.14 Let X be a variety, Y ⊂ X a subvariety, and suppose that x ∈ Y is a nonsingular point of both X and Y. Then we can choose a system of local parameters u1,..., un on X at x and an afﬁne neighbourhood U of x such that aY = (u1,..., um) in U. In the special case X = An and k = R or C, a similar fact has already been proved in Section 2.3. → mY,x/m2 Proof The inclusion of the tangent spaces ΘY,x → ΘX,x corresponds to a surjective map of the dual spaces ϕ : mX,x/m2 Y,x, deﬁned by restricting X,x functions from X to Y. We can choose a basis u1,..., un of mX,x/m2 X,x such that u1,..., um ∈ aY and such that the restrictions to Y of um+1,..., un form a basis of mY,x/m2 Y,x. Consider an afﬁne neighbourhood of x in which all the ui are regular, and in this consider the subvariety Y deﬁned by u1 = · · · = um = 0. By construction, Y ⊃ Y. We prove that Y = Y, so that the assertion will follow by Theorem 2.13. By Theorem 2.13, Y is nonsingular at x, and hence is irreducible in a neighbourhood of x by Theorem 2.9. It follows by Theorem 2.13 that dim Y = n − m. It is clear by construction that dim ΘY,x = n − m. Hence Y = Y, and since aY = (u1,..., um) by Theorem 2.13, also aY = (u1,..., um
) in some neighbour- hood of x. The theorem is proved. 3.3 Exercises to Section 3 1 Prove that if t is a local parameter of a nonsingular point of an algebraic curve then any function f ∈ Ox can be uniquely written in the form f = t nu with n ≥ 0 and u an invertible element of Ox. Use this to deduce Theorem 2.11 for curves. 2 Prove the converse of Theorem 2.4: if codimension 1 subvarieties D1,..., Dn intersect transversally at x and u1,..., un are their local equations in a neighbourhood of x then u1,..., un form a system of local parameters at x. 3 Is Theorem 2.12, Corollary 2.4 true without the nonsingularity assumption? What about Theorem 2.12 itself? 4 Prove that a point x of an algebraic curve X is nonsingular if and only if x has a local equation on X. 112 2 Local Properties 5 X ⊂ A3 is the cone given by x2 + y2 − z2. Prove that the generator L deﬁned by the equations x = 0, y = z does not have a local equation in any neighbourhood of (0, 0, 0). 6 Let ϕ : P2 → P2 be the rational map deﬁned by ϕ(x0 : x1 : x2) = (x1x2 : x0x2 : x0x1). Consider the point x = (1 : 0 : 0) and a curve C ⊂ P2 that is nonsingular at x. By Theorem 2.12, the map ϕ restricted to C is regular at x, and therefore maps x to some point that we denote by ϕC(x). Prove that ϕC1 (x) = ϕC2 (x) if and only if C1 and C2 touch at x, that is, ΘC1,x = ΘC2,x. 7 Prove that if ϕ = f/g is a rational function, f and g are regular at a nonsingular point x and the power series τ (f ) is divisible by τ (g) then ϕ is regular at x. [Hint: Use the arguments of Proposition A.14.]
8 We use the following assertion in subsequent exercises. Let X ⊂ An be an afﬁne variety and x ∈ X. Suppose that aX = (f1,..., fm). Prove that Ox = k[[T1,..., Tn]]/aX, where aX = τ (f1),..., τ (fm), and τ (fi) is the Taylor series of fi as in Section 2.2. [Hint: Use the results of Atiyah and Macdonald [8, Chapter 10].] 9 Prove that a formal analytic equivalence of An with itself (that is, a formal analytic automorphism) in a neighbourhood of 0 is given by power series Φ1,..., Φn with no constant terms such that the determinant formed by the linear terms is nonzero. 10 Prove that two plane curves with equations F = 0 and G = 0 passing through the origin 0 ∈ A2 are formally analytically equivalent in a neighbourhood of 0 if and only if there exists a formal analytic automorphism of A2 given by power series Φ1, Φ2 such that F (Φ1, Φ2) = GU, where U is a power series with nonzero constant term. 11 Prove that any nonsingular algebraic curve having the origin 0 as a double point with distinct tangents is formally analytically equivalent in a neighbourhood of 0 to the curve xy = 0. [Hint: Use Exercise 10. Look for Φ1, Φ2 modulo higher and higher powers of the ideal (x, y).] 12 Classify double points of algebraic plane curves up to formal analytic equivalence over a ﬁeld k of characteristic 0. 13 Let X be the hypersurface in An with equation F = F2(T ) + F3(T ) + · · · + Fk(T ) = 0, where F2(T ) is a quadratic form of maximal rank n. Prove that X is 4 The Structure of Birational Maps 113 formally analytically equivalent in a neighbourhood of 0 to the quadratic cone. + 14 Construct an inﬁnite number of nonsingular projective curves, with no two isomorphic over R. 15 Suppose that a nonsingular irreducible afﬁne n-dimensional variety X �
�� An is given by equations F1 = · · · = Fm = 0, and that for every x = (x1,..., xn) ∈ X the (∂Fi/∂Tj )(x)(Tj − xj ) = 0 is n-dimensional. Prove that then space deﬁned by aX = (F1,..., Fm). 16 Deduce from Exercise 15 that if the Plücker equations x ∧ x = 0 of the Grassmannian Grass(2, r) are written out as F1 = · · · = Fm = 0 then F1,..., Fm generate the ideal of Grass(2, r). (Compare Example 1.24 and the remark after Example 2.4.) 4 The Structure of Birational Maps 4.1 Blowup in Projective Space We proved in the preceding section that a birational map between nonsingular projective curves is an isomorphism (Theorem 2.12, Corollary 2.4). This is no longer true for higher dimensional varieties. For example, stereographic projection establishes a birational equivalence between a nonsingular quadric surface Q ⊂ P3 and the projective plane P2, but this is not a regular map (see Exercise 7 of Section 4.5, Chapter 1, and compare Proposition of Section 6.2, Chapter 1 and Corollary 1.7). This section deﬁnes and studies the simplest and most typical case of a birational map that is not an isomorphism, the blowup. We consider the two projective spaces Pn with homogeneous coordinates x0,..., xn and Pn−1 with homogeneous coordinates y1,..., yn. For points x = (x0 : · · · : xn) ∈ Pn and y = (y1 : · · · : yn) ∈ Pn−1, we denote the point (x, y) ∈ Pn × Pn−1 of the product also by (x0 : · · · : xn; y1 : · · · : yn). Consider the closed subvariety Π ⊂ Pn × Pn−1 deﬁned by the equations xiyj = xj yi for i, j = 1,..., n.
(2.26) Deﬁnition 2.6 The map σ : Π → Pn deﬁned by restricting the ﬁrst projection Pn × Pn−1 → Pn is called the blowup6 of Pn centred at ξ. 6This notion appears in the literature under many other names: σ -process, monoidal transformation, dilation, quadratic transformation, etc. 114 2 Local Properties Write ξ for the point ξ = (1 : 0 : · · · : 0) ∈ Pn. If (x0 : · · · : xn) = ξ then (2.26) imply that (y1 : · · · : yn) = (x1 : · · · : xn), so that the map σ −1 : Pn \ ξ → Π deﬁned by (x0 : · · · : xn), (x1 : · · · : xn) (x0 : · · · : xn) → (2.27) is the inverse of σ. However, if (x0 : · · · : xn) = ξ then (2.26) are satisﬁed by any values of the yi. Thus σ −1(ξ ) = ξ × Pn−1, and σ deﬁnes an isomorphism between Pn \ ξ and Π \ (ξ × Pn−1). The point ξ is called the centre of the blowup σ. Let us describe the structure of Π in a neighbourhood of points of the form (ξ ; y1,..., yn). We have yi = 0 for some i, so that the chosen point is contained in the open set Ui of Π deﬁned by x0 = 0, yi = 0; in Ui we can even assume that x0 = 1, yi = 1. Then (2.26) take the form xj = yj xi for j = 1,..., n with j = i. It follows that Ui is isomorphic to afﬁne space An with coordinates y1,..., yi, xi,..., yn. We see in particular that Π is nonsingular,
and thus by Theorem 2.9 is irreducible in a neighbourhood of every point. We will see presently that Π is actually irreducible. For this, to get a clearer idea of the effect of the blowup, we consider σ over some line L through ξ. Suppose that L is given by xj = αj xi for some i and j = 1,..., n with j = i. On L the map (2.27) takes the form σ −1(x0 : · · · : xn) = (x0 : · · · : xn; α1 : · · · : 1 : · · · : αn), with αi = 1 in the ith place. We see thus that σ −1 is regular on L and maps it to a curve σ −1(L) ⊂ Π that intersects ξ × Pn−1 in the point (ξ ; α1 : · · · : 1 : · · · : αn). We can interpret this result as follows: σ −1 is not regular at ξ, but considering it on L we get a regular map σ −1 : L → Π. We can use this to extend the deﬁnition of σ −1 on L over the point ξ ; over R or C, this means that we deﬁne σ −1(x) for x ∈ L \ ξ and deﬁne σ −1(ξ ) by letting x tend to ξ along the direction of L. However, the result depends on the choice of L, since passing to the limit x → ξ depends on the direction along which x approaches ξ. Choosing different lines L we get all possible points of ξ × Pn−1. Thus, although σ −1 is not regular at ξ, on resolving the indeterminacy arising from this we don’t get arbitrary points of Π, but only points of ξ × Pn−1. Bearing this picture in mind one says that σ −1 blows up ξ to ξ × Pn−1. Note that at the same time we have proved the irreducibility of Π. Indeed, ξ × Pn−1 Π \ ξ × Pn−1. ∪ Π = Since Π
\ (ξ × Pn−1) is isomorphic to Pn \ ξ it is irreducible, hence so is its closure Π \ (ξ × Pn−1). Thus we need only show that ξ × Pn−1 ξ × Pn−1 ⊂ Π \. But obviously σ −1(L) ⊂ Π \ (ξ × Pn−1), so that also σ −1(L) ∩ ξ × Pn−1 ⊂ Π \ ξ × Pn−1. 4 The Structure of Birational Maps 115 Figure 10 The blowup σ : Π → P2 But we have just seen that for suitable choice of L the left-hand side here is an arbitrary point of ξ × Pn−1. For n = 2 we have an intuitive picture of the map σ : Π → P2 and its effect on the lines L: σ −1(L) intersects the line ξ × P1 in a point that moves as L rotates around ξ. Thus Π looks like one twist of a helix (Figure 10). 4.2 Local Blowup For an arbitrary quasiprojective variety X and a nonsingular point ξ ∈ X, we now construct a variety Y and a map σ : Y → X analogous to that constructed in Section 4.1 for X = Pn and ξ = (1 : 0 : · · · : 0). We begin with an auxiliary construction. Let X be a quasiprojective variety and ξ ∈ X a nonsingular point, and suppose that u1,..., un are functions that are regular everywhere on X and such that (a) the equations u1 = · · · = un = 0 have the single solution ξ in X; and (b) u1,..., un form a local system of parameters on X at ξ. Consider the product X × Pn−1 and the subvariety Y ⊂ X × Pn−1 consisting of points (x; t1 : · · · : tn) with x ∈ X and (t1 : · · · : tn) ∈ Pn−1, such that ui(x)tj = uj (x)ti for i, j = 1,.
.., n. The regular map σ : Y → X obtained as the restriction to Y of the ﬁrst projection X × Pn−1 → X is called the local blowup of X with centre in ξ. Note that in general this construction does not apply to the case that X is projective, since we require the existence of nonconstant everywhere regular functions 116 2 Local Properties u1,..., un on X. Thus the new notion does not include the previous notion of blowup in the case X = Pn. The two are related as follows: write X ⊂ Pn for the afﬁne subset deﬁned by x0 = 0, and set Y = σ −1(X). Then the map σ : Y → X induced on Y by the blowup Π → Pn is a local blowup. The following properties proved in Section 4.1 for the blowup of Pn are proved in exactly the same way for a local blowup. The map σ : Y → X is regular and deﬁnes an isomorphism Y \ ξ × Pn−1 ∼→ X \ ξ. At a point y ∈ σ −1(ξ ), we have ti = 0 for some i, and we can set sj = tj /ti for j = i. Then the equations of Y take the form uj = uisj for j = 1,..., n with j = i. We see from this that the maximal ideal of y is given by my = = u1 − u1(y),..., un − un(y), s1 − s1(y),..., sn − sn(y) s1 − s1(y),..., ui − ui(y),..., sn − sn(y). Hence dim ΘY,y ≤ n, and since dim σ −1(X \ ξ ) = n, the variety Y is nonsingular at every point y ∈ σ −1(X \ ξ ). Since ξ × Pn−1 Y = ∪ σ −1(X \ ξ ), Y is either irreducible, equal to the closure σ −1(X \ ξ ) of the set σ −1(X \ ξ ), or
has a second component isomorphic to Pn−1. In the second case, the two components would have to intersect: for otherwise σ −1(X \ ξ ) would be closed in X × Pn−1, but then by Theorem 1.11 also its image X \ ξ ⊂ X would be closed. But a point of intersection of the two components is singular, and this contradicts Theorem 2.9. Thus Y is irreducible and nonsingular and s1 − s1(y),..., ui − ui(y),..., sn − sn(y) are local parameters at a point y ∈ σ −1(ξ ) at which ti = 0. A local blowup is obviously a proper map (see the discussion after the proof of Theorem 1.11). We now prove a property that can reasonably be called the independence of the local blowup on the choice of the system of local parameters u1,..., un. Lemma Let v1,..., vn be another system of functions on X satisfying the above conditions (a) and (b) and σ : Y → X the local blowup constructed as above in terms of v1,..., vn. Then there exist an isomorphism ϕ : Y → Y such that the diagram ϕ−→ Y Y σ σ X is commutative. 4 The Structure of Birational Maps 117 Proof We have Y ⊂ X × Pn−1, where the homogeneous coordinates in Pn−1 are t 1,..., t n. In the open sets Y \ σ −1(ξ ) and Y \ σ −1(ξ ), we set x; v1(x) : · · · : vn(x), x; u1(x) : · · · : un(x) ϕ(x; t1 : · · · : tn) = : · · · : t x; t = n 1 ψ. (2.28) It follows from property (a) of the ui that ϕ and ψ are regular maps and ψ : Y \ σ −1(ξ ) → Y. ϕ : Y \ σ −1(ξ ) → Y We now consider the open set of Y in which ti = 0 and
set sj = tj /ti. Since vk(ξ ) = 0, and u1,..., un is a basis of the ideal mξ, we have vk = n j =1 hkj uj with hkj ∈ Oξ. (2.29) Since in our open set uj = uisj, it follows that vk = ui n j =1 σ ∗(hkj )sj = uigk, where gk = n j =1 σ ∗(hkj )sj. (2.30) We set ϕ(x; t1 : · · · : tn) = (x; g1 : · · · : gn). Obviously this map coincides with (2.28) wherever both are deﬁned, since there gk = vk/ui. Let us check that ϕ is regular. For this, we must prove that g1,..., gn are not simultaneously 0 at any point η ∈ σ −1(ξ ). Suppose that g1(η) = · · · = gn(η) = 0. Since not all the sj (η) = 0 (because si = 1), it follows from (2.30) that det \|hkj (ξ )\| = 0. But vk ≡ ξ and it would follow from this that the vk are linearly dependent in mξ /m2 ξ, whereas they form a system of local coordinates at ξ. Thus we have deﬁned a global map ϕ : Y → Y, and similarly a map ψ : Y → Y. It is enough to prove that these are mutually inverse on the open sets where the formulas (2.28) hold; and there it is obvious. The lemma is proved. hkj (ξ )uj modulo m2 4.3 Behaviour of a Subvariety Under a Blowup Let X ⊂ PN be a quasiprojective subvariety, and σ : Π → PN the blowup deﬁned in Section 4.1. We investigate the inverse image σ −1(X) of the subvariety X, which is, of course, a quasiprojective subvariety of Π. Theorem 2.15 Suppose that X

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