72 Andreas Gathmann 9. Birational Maps and Blowing Up In the course of this class we have already seen many examples of varieties that are “almost the same” in the sense that they contain isomorphic dense open subsets (although the varieties are not isomorphic themselves). Let us quickly recall some of them. Example 9.1 (Irreducible varieties with isomorphic non-empty open subsets). n n n (a) The affine space A and the projective space P have the common open subset A by Propo- m n m+n m n m+n sition 7.2. Consequently, P ×P and P have the common open subset A ×A = A — but they are not isomorphic by Exercise 7.4. k(n−k) (b) Similarly, the affine space A and the Grassmannian G(k;n) have the common open k(n−k) subset A by Construction 8.15. 1 2 3 2 (c) The affine line A and the curve X = V(x1 − x2) ⊂ A of Example 4.9 have the isomorphic open subsets A1nf0g resp. Xnf0g — in fact, the morphism f given there is an isomorphism after removing the origin from both the source and the target curve. We now want to study this situation in more detail and present a very general construction — the so-called blow-ups — that gives rise to many examples of this type. But first of all we have to set up some notation to deal with morphisms that are defined on dense open subsets. For simplicity, we will do this only for the case of irreducible varieties, in which every non-empty open subset is automatically dense by Remark 2.18. Definition 9.2 (Rational maps). Let X and Y be irreducible varieties. A rational map f from X to Y, written f : X 99K Y, is a morphism f : U ! Y (denoted by the same letter) from a non-empty open subset U ⊂ X to Y. We say that two such rational maps f1 : U1 ! Y and f2 : U2 ! Y with U1;U2 ⊂ X are the same if f1 = f2 on a non-empty open subset of U1 \U2. Remark 9.3. Strictly speaking, Definition 9.2 means that a rational map f : X 99K Y is an equiva- lence class of morphisms from non-empty open subsets of X to Y. Note that the given relation is in fact an equivalence relation: reflexivity and symmetry are obvious, and if f1 : U1 ! Y agrees with f2 : U2 ! Y on a non-empty open subset U1;2 and f2 with f3 : U3 ! Y on a non-empty open subset U2;3 then f1 and f3 agree on U1;2 \U1;3, which is again non-empty by Remark 2.18 (a) since X is irreducible. For the sake of readability it is customary however not to indicate these equivalence classes in the notation and to denote the rational map f : X 99K Y and the morphism f : U ! Y by the same letter. If we now want to consider “rational maps with an inverse”, i. e. rational maps f : X 99K Y such that there is another rational map g : Y 99K X with g ◦ f = idX and f ◦ g = idY , we run into problems: if e. g. f is a constant map and g is not defined at the point f (X) then there is no meaningful way to compose it with f . So we need to impose a technical condition first to ensure that compositions are well-defined: Definition 9.4 (Birational maps). Again let X and Y be irreducible varieties. (a) A rational map f : X 99K Y is called dominant if its image contains a non-empty open subset U of Y. In this case, if g : Y 99K Z is another rational map, defined on a non-empty open subset V of Y, we can construct the composition g ◦ f : X 99K Z as a rational map since we have such a composition of ordinary morphisms on the non-empty open subset f −1(U \V). (b) A rational map f : X 99K Y is called birational if it is dominant, and if there is another dominant rational map g : Y 99K X with g ◦ f = idX and f ◦ g = idY . (c) We say that X and Y are birational if there is a birational map f : X 99K Y between them. 9. Birational Maps and Blowing Up 73 Remark 9.5. By definition, two irreducible varieties are birational if and only if they contain iso- morphic non-empty open subsets. In particular, Exercise 5.25 then implies that birational irreducible varieties have the same dimension. 14 An important case of rational maps is when the target space is just the ground field, i. e. if we consider regular functions on open subsets. Construction 9.6 (Rational functions and function fields). Let X be an irreducible variety. 1 A rational map j : X 99K A = K is called a rational function on X. In other words, a rational function on X is given by a regular function j 2 OX (U) on some non-empty open subset U ⊂ X, with two such regular functions defining the same rational function if and only if they agree on a non-empty open subset. The set of all rational functions on X will be denoted K(X). Note that K(X) is a field: for j1 2 OX (U1) and j2 2 OX (U2) we can define j1 + j2 and j1 j2 −1 on U1 \U2 6= /0,the additive inverse −j1 on U1, and for j1 6= 0 the multiplicative inverse j1 on U1nV(j1). We call K(X) the function field of X. Remark 9.7. If U ⊂ X is a non-empty open subset of an irreducible variety X then K(U) =∼ K(X): an isomorphism is given by K(U) ! K(X) K(X) ! K(U) with inverse j 2 OU (V) 7! j 2 OX (V) j 2 OX (V) 7! jjV\U 2 OU (V \U): In particular, birational irreducible varieties have isomorphic function fields. Exercise 9.8. Let X be an irreducible affine variety. Show: (a) The function field K(X) is isomorphic to the so-called quotient field of the coordinate ring A(X), i. e. to the localization of the integral domain A(X) at the multiplicatively closed subset A(X)nf0g. (b) Every local ring OX;a for a 2 X is naturally a subring of K(X). n Exercise 9.9. Let X ⊂ P be a quadric, i. e. an irreducible variety which is the zero locus of an irreducible homogeneous polynomial of degree 2. Show that X is birational, but in general not n− isomorphic, to the projective space P 1. The main goal of this chapter is now to describe and study a general procedure to modify an irre- ducible variety to a birational one. In its original form, this construction depends on given polyno- mial functions f1;:::; fr on an affine variety X — but we will see in Construction 9.17 that it can also be performed with a given ideal in A(X) or subvariety of X instead, and that it can be glued in order to work on arbitrary varieties. n Construction 9.10 (Blowing up). Let X ⊂ A be an affine variety. For some r 2 N>0 let f1;:::; fr 2 A(X) be polynomial functions on X, and set U = XnV( f1;:::; fr). As f1;:::; fr then do not vanish simultaneously at any point of U, we obtain a well-defined morphism r−1 f : U ! P ; x 7! ( f1(x): ::: : fr(x)): We consider its graph r−1 G f = f(x; f (x)) : x 2 Ug ⊂ U × P which is isomorphic to U (with inverse morphism the projection to the first factor). Note that G f is r−1 r−1 closed in U × P by Proposition 5.21 (a), but in general not closed in X × P . The closure of G f r−1 in X × P then contains G f as a dense open subset. It is called the blow-up of X at f1;:::; fr; we will usually denote it by X˜ . Note that there is a natural projection morphism p : X˜ ! X to the first factor. Sometimes we will also say that this morphism p is the blow-up of X at f1;:::; fr. Before we give examples of blow-ups let us introduce some more notation and easy general results that will help us to deal with them. 74 Andreas Gathmann Remark 9.11 (Exceptional sets). In construction 9.10, the graph G f is isomorphic to U, with pjG f : G f ! U being an isomorphism. By abuse of notation, one often uses this isomorphism to identify −1 G f with U, so that U becomes an open subset of X˜ . Its complement X˜ nU = p (V( f1;:::; fr)), on which p is usually not an isomorphism, is called the exceptional set of the blow-up. If X is irreducible and f1;:::; fr do not vanish simultaneously on all of X, then U = XnV( f1;:::; fr) is a non-empty and hence dense open subset of X. So its closure in the blow-up, which is all of X˜ by definition, is also irreducible. We therefore conclude that X and X˜ are birational in this case, with common dense open subset U. Remark 9.12 (Strict transforms and blow-ups of subvarieties). In the notation of Construction 9.10, let Y be a closed subvariety of X.