18.782 Arithmetic Geometry, Lecture Note 15

18.782 Introduction to Arithmetic Geometry Lecture #15 As usual, k is a perfect field and k¯ is a fixed algebraic closure of k. Recall that an affine ¯ (resp. Pn = Pn (k)). ¯ (resp. projective) variety is an irreducible alebraic set in An = An (k) 15.1 Rational maps of affine varieties Before defining rational maps we want to nail down two points on which we we were intentional vague in the last lecture. We defined a morphism φ : X → Y of varieties X ⊆ Am and Y ⊆ An as a “map defined by a tuple of polynomials (φ1 , . . . , φn ).” This definition is vague in two ways. First, is a morphism a map between two sets X and Y , or is it a tuple of polynomials? We shall adopt the second view; we still get a function by evaluating the polynomials at points in X. Given that we regard φ as a tuple (φ1 , . . . , φn ), the next question is in which ring do ¯ 1 , . . . , xm ] or k[X]. ¯ The function they its components φi lie? Are they elements of k[x ¯ define is the same in either case, but we shall regard the φi as elements of k[X]. This ˆ means that in order to evaluate φi at a point P ∈ X, we need to lift φi = φi + I(X) to ¯ 1 , . . . , xm ] and then compute φˆi (P ). Of course it does not matter a representative φˆi ∈ k[x which representative φˆi we pick; we define φi (P ) to be the value φˆi (P ) for any/every choice ˆ and thereby define φ(P ) for P ∈ X (but note that φ(P ) is not defined for any P 6∈ X). of φ, One advantage of this approach is that there is then a one-to-one correspondence between morphisms and the functions they define. If φ = (φ1 , . . . , φn ) and ψ = (ψ1 , . . . , ψn ) define ¯ 1 , . . . , xm ] contains the same function from X to Y then each of the polynomials φˆi −ψˆi in k[x X in its zero locus and therefore lies in the ideal I(X). This implies, by definition, that in ¯ ¯ i , . . . , xm ]/I(X) we have φi = ψi for 1 ≤ i ≤ m and therefore φ = ψ. k[X] = k[x ¯ ¯ We now want to extend these ideas to the function field k(X). Elements of k(X) have ¯ 6 0, and are called rational functions (or even the form r = f /g, with f, g ∈ k[X] and g = ¯ just functions), on X, even though they are formally elements of the fraction field of k[X] ¯ indeed, this is precisely the issue we and typically do not define a function from X to k; must now address. It seems natural to say that for a point P ∈ X we should define r(P ) to be f (P )/g(P ) whenever g(P ) = 6 0 and call it undefined otherwise. But there is a problem with this approach: the representation r = f /g is not necessarily unique.1 We also have r = p/q ¯ whenever pg = f q holds in k[X] (recall that this is part of the definition of a fraction field, it is a set of equivalence classes of fractions). The values f (P )/g(P ) and p(P )/q(P ) are necessarily equal wherever both denominators are nonzero, but it may be that q(P ) 6= 0 at points where g(P ) = 0 (and vice versa). Example 15.1. Consider the the zero locus X of x1 x2 − x3 x4 in A4 (which is in fact a variety) and the rational function r = x1 /x3 = x4 /x2 . At the point P = (0, 1, 0, 0) ∈ X the value x1 (P )/x3 (P ) is not defined, but x4 (P )/x2 (P ) = 0 is defined, and the reverse occurs for P = (0, 0, 1, 0) ∈ X. But we can assign a meaningful value to r(P ) at both points; the only points in X where r is not defined are those with x2 = x3 = 0. This motivates the following definition. ¯ If k[X] is a UFD then we can put r = f /g in lowest terms to get a unique representation. However, the ¯ ¯ 1 , . . . , xm ] is. A quotient of a UFD is typically coordinate ring k[X] is usually not a UFD, even though k[x not a UFD, even when it is an integral domain; consider Z[x]/(x2 + 5), for example. 1 1 ¯ ¯ Definition 15.2. A function r ∈ k(X) is said to be regular at a point P ∈ X if gr ∈ k[X] 2 ¯ ¯ for some g ∈ k[X] ⊆ k̄(X) for which g(P ) 6= 0 (we then have r = f /g for some f ∈ k[X]). ¯ The set of points at which a function r ∈ k(X) is regular form a nonempty open (hence dense) subset dom(r) of the subspace X: the complement of dom(r) in X is the closed ¯ ¯ subset of X defined by the denominator ideal {g ∈ k[X] : gr ∈ k[X]}, which we note is not 3 the zero ideal, since r = f /g for some nonzero g. We now associate to r the function from dom(r) to k¯ that maps P to r(P ) = (f /g)(P ) = f (P )/g(P ), ¯ where g is chosen so that g(P ) 6= 0 and gr = f ∈ k[X]. Now if r is actually an element of k̄[X], then r is regular at every point in X and we have dom(r) = X. The following lemma says that the converse holds. ¯ ¯ Lemma 15.3. A rational function r ∈ k(X) lies in k[X] if and only if dom(r) = X. Proof. The forward implication is clear, and if dom(r) = X then the complement of dom(r) ¯ in X is the empty set and the denominator ideal is (1), which implies r ∈ k[X]. Definition 15.4. Let X ⊆ Am and Y ⊆ An be affine varieties. We say that a tuple ¯ (φ1 , . . . , φn ) with φi ∈ k(X) is regular at P ∈ X if the φi are all regular at P . A rational map ¯ φ : X → Y is a tuple (φ1 , . . . , φn ) with φi ∈ k(X) such that φ(P ) := (φ1 (P ), . . . , φn (P )) ∈ Y for all points P ∈ X where φ is regular. If φ is regular at every point P ∈ X then we say that φ is regular. The set of points where φ is regular form an open subset dom(φ) = ∩i dom(φi ) of X. Thus φ defines a function from dom(φ) to Y , which we may also regard as a partial function from X to Y . We get a complete function from X to Y precisely when dom(φ) = X, that is, when φ is regular. This occurs precisely when φ is a morphism. Theorem 15.5. A rational map of affine varieties is a morphism if and only if it is regular. Proof. A morphism is clearly a regular rational map. For the converse, apply Lemma 15.3 to each component of φ = (φ1 , . . . , φn ). We now want to generalize the categorical equivalence between affine varieties and their function fields, analogous to what we proved in the last lecture for affine varieties and their coordinate rings (affine algebras), but with morphisms of varieties replaces by the more general notion of a rational map. But there is a problem with this. In order to even define a category of varieties with rational maps, we need to be able to compose rational maps. But this is not always possible! Example 15.6. Let X = Y = Z = A2 , and let φ : X → Y be the rational map (1/x1 , 0) and let ψ : Y → Z be the rational map (0, 1/x2 ). Then the image of φ is disjoint from dom(ψ). There is no rational function that corresponds to the composition of φ and ψ (or ψ with φ). Even formally, the fractions 1/0 that we get by naively composing φ with ψ are ¯ not elements of k(X), and the function defined by the composition of the functions defined ¯ by φ and ψ has the empty set as its domain, which is not true of any r ∈ k(X). 2 Recall that an integral domain can always be embedded in its fraction field by identifying g with the ¯ ¯ equivalence class of g/1, so we assume k[X] ⊆ k(X) henceforth. 3 ¯ When restricting our attention to a variety X in An it is simpler to work with ideals in k[X] = ¯ ¯ k[x1 , . . . , xm ]/I(X) rather than k[x1 , . . . , xm ]. The one-to-one corrsepondence between radical ideals and closed sets still holds, as does the correspondence between prime ideals and (sub-) varieties. 2 To fix this problem we want to restrict our attention to rational maps whose image is dense in its codomain. Definition 15.7. A rational map φ : X → Y is dominant if φ(dom(φ)) = Y . If φ : X → Y and ψ : Y → Z are dominant rational maps then the intersection of φ(dom(φ)) and dom(ψ) must be nonempty; the complement of dom(ψ) in Y is a proper closed subset of Y and therefore contains no sets that are dense in Y , including φ(dom(φ)). It follows that we can always compose dominant rational maps, and since the identity map is also dominant rational map, we can now speak of the category of affine varieties and rational maps. Not every morphism is a dominant rational map, but affine varieties and dominant morphisms form a subcategory of affine varieties and dominant rational maps. As you will show on the problem set, the closure of the image of a morphism of varieties is a variety, so one can always make a morphism dominant be restricting its codomain. We now prove the analog of Theorem 14.8, replacing morphisms with dominant rational maps, and coordinate rings with function fields. We now use the term function field to ¯ and we require morphisms of function fields refer to any finitely generated extension of k, ¯ ¯ to fix k (we could also call them k-algebra homomorphisms). Field homomorphisms are ¯ always injective, so a morphism of function fields is just a field embedding that fixes k. ¯ ¯ Note that the all the interesting function fields F/k are transcendental. If F/k is algebraic ¯ this corresponds to the function field of a zero-dimensional variety (a point). then F = k; ¯ let R be the k-algebra ¯ Given a function field F generated by elements α1 , . . . , αn over k, generated by α1 , . . . , αn in F ; this means that R is equal to the set of all polynomial expressions in α1 , . . . , αn , but there may be algebraic relations between the αi that make many of these expressions equivalent. In any case, R is isomorphic to the quotient of the ¯ 1 , . . . , xn ] by an ideal I corresponding to all the algebraic relations that polynomial ring k[x exist among the αi . The ring R is an integral domain (since it is contained in a field), therefore I is a prime ideal that defines a variety X whose coordinate ring is isomorphic to R and whose function field is isomorphic to F , the fraction field of R. Theorem 15.8. The following hold: (i) Every dominant rational map φ : X → Y of affine varieties, induces a morphism ¯ ) → k(X) ¯ φ∗ : k(Y of function fields such that φ∗ (r) = r ◦ φ. (ii) Every morphism θ : K → L of function fields induces a dominant rational map of ¯ ) and L ' k(X), ¯ affine varieties θ∗ : X → Y , with K ' k(Y such that the image of ∗ ¯ θ(r) in k(X) is r ◦ θ . (iii) If φ : X → Y and ψ : Y → Z are dominant rational maps of affine varieties then (ψ ◦ φ)∗ = φ∗ ◦ ψ ∗ . As in the analogous Theorem 14.8, to compute r ◦ φ one needs to lift/compose/reduce, ¯ 1 , . . . , xm ), that is, pick representatives for φ1 , . . . , φn that are rational functions in k(x ¯ 1 , . . . , yn ), then compose and reduce the numerator and pick a representative of r in k(y denominator modulo I(X). The fact that φ is dominant ensures that the denominator of ¯ the composition does not lie in I(X), so r ◦ φ is an element of k(X). Proof. The proofs of parts (i) and (iii) follow the proof of Theorem 14.8 verbatim, with coordinate rings replaced by function fields, only part (ii) merits further discussion. As ¯ ) and L ' k(X) ¯ discussed above, we can write K ' k(Y for some varieties X and Y , and 3 ¯ ) → k(Y ¯ ) that is compatible with these any morphism K → L induces a morphism k(Y ¯ ¯ isomorphisms, so let us assume θ : k(Y ) → k(X). As in the proof of Theorem 14.8 we define θ∗ : X → Y by θ∗ = (θ(y1 ), . . . , θ(yn )), where ¯ ). For any r ∈ k(Y ¯ ) we have we now regard the coordinate functions yi as elements of k(Y r ◦ θ∗ = r̂(θ(y1 ), . . . , θ(yn )) = θ(r(y1 , . . . , yn )) = θ(r). The fact that r ∈ k̄(Y ) ensures that the denominator of r̂ ∈ k̄[y1 , . . . , yn ] is not in I(Y ), so this composition is well defined. The proof that the image of θ∗ actually lies in Y is the same as in Theorem 14.8: for any f ∈ I(Y ) we have f ◦ θ∗ = θ(f ) = 0, so certainly f (θ∗ (P )) = 0 for all P ∈ dom(θ∗ ). Thus θ∗ is a rational map from X to Y . But we also need to check that θ∗ is dominant (this is the only new part of the proof). ¯ ] that vanishes on the image of θ∗ This is equivalent to showing that the only element of k[Y ¯ ) is the zero element, which is in turn equivalent to showing that the only element of k(Y ∗ that vanishes on the intersection of its domain and the image of θ is the zero element. This is in turn equivalent to showing that if r ◦ θ∗ = θ(r) vanishes at every point in its domain ¯ then r = 0. But the only element of k(X) that vanishes at every point in its domain is the zero element, and θ is injective, so we are done. Corollary 15.9. The category of affine varieties with dominant rational maps and the category of function fields are contravariantly equivalent. Proof. As in the proof of Corollary 14.9, the only thing left to show is that (φ∗ )∗ = φ and (θ∗ )∗ = θ, up to isomorphism, but both follow from Theorem 15.8 and its proof. Definition 15.10. Two affine varieties X and Y are said to be birationally equivalent if there exist dominant rational maps φ : X → Y and ψ : Y → X such that (φ ◦ ψ)(P ) = P for all P ∈ dom(φ ◦ ψ) and (ψ ◦ φ)(P ) = P for all P ∈ dom(ψ ◦ φ). Corollary 15.11. Two affine varieties are birationally equivalent if and only if their function fields are isomorphic. As with morphisms, if φ : X → Y is a rational map of varieties that are defined over k, we say that φ = (φ1 , . . . , φn ) is defined over k if the φi all lie in k(X). Corollary 15.12. Let X and Y be affine varieties defined over k. If φ : X → Y is a ¯ ) → k̄(X) restricts to a morphism dominant rational map defined over k then φ∗ : k(Y k(Y ) → k(X). 15.2 Morphisms and rational maps of projective varieties We now want to generalize everything we have done for maps between affine varieties to maps between projective varieties. This is completely straight-forward, we just need to account for the equivalence relation on Pn . ¯ Recall from Lecture 13 that although we defined the function field k(X) of a projective n variety X ⊆ P as the function field of any of its non-empty affine parts, we can always represent r by an element r̂ ∈ k̄(x0 , . . . , xn ) whose numerator and denominator are homogeneous polynomials of the same degree, and for any point P ∈ X where r̂(P ) is defined (has nonzero denominator), we can unambiguously define r(P ) = r̂(P ). 4 ¯ Definition 15.13. Let X be a projective variety. We say that r ∈ k(X) is regular at a point P ∈ X if it has a representation r̂ that is defined at P . The set of points P ∈ X at which r is regular form an open subset of X that we denote dom(r).4 For any point P ∈ dom(r) we define r(P ) = r̂(P ), where r̂ is chosen so that r̂ is defined at P . Definition 15.14. Let X ⊆ Pm and Y ⊆ Pn be projective varieties. A rational map ¯ φ : X → Y is an equivalence class of tuples φ = (φ0 : . . . : φn ) with φi ∈ k(X) not all zero such that at any point P ∈ X where all the φi are regular and at least one is nonzero, the point (φ0 (P ) : . . . : φn (P )) lies in Y . The equivalence relation is given by (φ0 : . . . : φn ) = (λφ0 : . . . : λφn ) for any λ ∈ k̄(X)× . We say that φ is regular at P if there is a tuple (λφ0 : . . . : λφn ) in its class with each component regular at P and at least one nonzero at P . The open subset of X at which φ is regular is denoted dom(φ). Remark 15.15. We can alternatively represent the rational map φ : X → Y as a tuple ¯ 0 , . . . , xm ] that all have the same degree and not all of of homogeneous polynomials in k[x which lie in I(X). To ensure that the image lies in Y one requires that for all f ∈ I(Y ) we have f (φ0 , . . . , φn ) ∈ I(X). The equivalence relation is then (φ0 : . . . : φn ) = (ψ0 : . . . : ψn ) if and only if φi ψj − φj ψi ∈ I(X) for all i.j. Remark 15.16. One can also define rational maps X → Y where one of X, Y is an affine variety and the other is a projective variety. When Y is projective the definition is exactly the same as in the case that both are projective (but we don’t use homegenized functions ¯ to represent elements of k(X) when X is affine). When X is projective and Y is affine, a rational map is no longer an equivalence class of tuples, it is a particular tuple of rational functions on X. Of course there is still a choice of representation for each rational function (and the choice may vary with P ), but note that in this case Remark 15.15 no longer applies. Now that we have defined rational maps for projective varieties we can define morphisms and dominant rational maps; the definitions are exactly the same as in the affine case, so we can now state them generically. Definition 15.17. A morphism is a regular rational map. A rational map is dominant if its image is dense in its codomain. The analogs of Theorem 15.8 and Corollary 15.9 both apply to dominant rational maps between projective varieties. The proofs are exactly the same, modulo the equivalence relations for projective points and rational maps between projective varieties. Alternatively, one can simply note that any dominant rational map X → Y of projective varieties restricts to a dominant rational map between any pair of the nonempty affine parts of X and Y (the nonempty affine parts of X (resp. Y ) are all dense in X (resp. Y ), and they are all isomorphic as affine varieties; see Corollary 14.10). Conversely, any dominant rational map of affine varieties can be extended to a dominant rational map of their projective closures (but this is not true of morphisms; see Example 15.20 below). Theorem 15.18. Theorem 15.8 and Corollary 15.9 hold for projective varieties as well as affine varieties. 4 It is clear that dom(r) is open in X; its intersection with each affine patch is an open subset of X. 5 Let us now look at a couple of examples. Example 15.19. Let X ⊆ A2 be the affine variety defined by x2 + y 2 = 1, and let P be the point (−1, 0) ∈ X. The rational map φ : X → A1 defined by     y 1−x φ(x, y) = = x+1 y sends each point Q = (x, y) ∈ X different from P to the slope of the line P Q. The map φ is not regular (hence not a morphism), because it is not regular at P , but it is dominant (even surjective). The rational map φ−1 : A1 → X defined by   1 − t2 2t −1 φ (t) = , 1 + t2 1 + t2 √ is an inverse to φ. Note that φ−1 is also not regular (it is not defined at −1), but it is dominant (but not surjective). Thus X is birationally equivalent to A1 , but not isomorphic ¯ to A1 , as expected. The function fields of X and A1 are both isomorphic to k(t). Now let us consider the corresponding map of projective varieties. The projective closure X of X in P2 is defined by x2 + y 2 = z 2 . We now define the rational map ϕ : X → P1 by       y z−x y ϕ(x : y : z) = :1 = :1 = 1: x+z y z−x Per Remark 15.15, we could also write ϕ as ϕ(x : y : z) = (y : x + z) = (z − x : y) The first RHS is defined everywhere except (1 : 0 : −1) and the second RHS is defined everywhere except (1 : 0 : 1), thus ϕ is regular everywhere, hence a morphism. We also have the rational map ϕ : P1 → X defined by  2    s − t2 2st 2st s 2 + t2 ϕ−1 (s : t) = : : 1 = 1 : : s2 + t2 s2 + t2 s2 − t2 s2 − t2 which can also be written as ϕ−1 (s : t) = (s2 − t2 : 2st : s2 + t2 ). The map ϕ−1 is regular everywhere, hence a morphism, and the compositions ϕ ◦ ϕ−1 and ϕ−1 ◦ ϕ are both the identity maps, thus X and P1 are isomorphic. Example 15.20. Recall the morphism φ : A2 → A2 defined by φ(x, y) = (x, xy) from Lecture 14, where we noted that the image of φ is not closed (but it is dense in A2 , so φ is dominant). Let us now consider the corresponding rational map ϕ : P2 → P2 defined by  x xy  ϕ(x : y : z) = : : 1 = (xz : xy : z 2 ). z z2 We might expect ϕ to be a morphism, but this is not the case! It is not regular at (0 : 1 : 0). This is not an accident. As we will see in the next lecture, morphisms of projective varieties are proper, and in particular this means that they are closed maps (so unlike the affine case, the image of a morphism of projective varieties is a variety). But there is clearly no way to extend the morphism φ : A2 → A2 to a proper morphism ϕ : P2 → P2 (the image of ϕ in the affine patch z = 6 0 must be dense but not surjective), and this means that ϕ cannot be a morphism. 6