Resolution of Singularities. Steven Dale Cutkosky. Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Transcription

1 Resolution of Singularities Steven Dale Cutkosky Department of Mathematics, University of Missouri, Columbia, Missouri 65211

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3 To Hema, Ashok and Maya

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5 Contents Preface vii Chapter 1. Introduction Notation 2 Chapter 2. Non-singularity and Resolution of Singularities Newton s method for determining the branches of a plane curve Smoothness and non-singularity Resolution of singularities Normalization Local uniformization and generalized resolution problems 11 Chapter 3. Curve Singularities Blowing up a point on A Completion Blowing up a point on a non-singular surface Resolution of curves embedded in a non-singular surface I Resolution of curves embedded in a non-singular surface II 29 Chapter 4. Resolution Type Theorems Blow-ups of ideals Resolution type theorems and corollaries 40 Chapter 5. Surface Singularities Resolution of surface singularities 45 v

6 vi Contents 5.2. Embedded resolution of singularities 56 Chapter 6. Resolution of Singularities in Characteristic Zero The operator and other preliminaries Hypersurfaces of maximal contact and induction in resolution Pairs and basic objects Basic objects and hypersurfaces of maximal contact General basic objects Functions on a general basic object Resolution theorems for a general basic object Resolution of singularities in characteristic zero 99 Chapter 7. Resolution of Surfaces in Positive Characteristic Resolution and some invariants τ(q) = τ(q) = Remarks and further discussion 130 Chapter 8. Local Uniformization and Resolution of Surfaces Classification of valuations in function fields of dimension Local uniformization of algebraic function fields of surfaces Resolving systems and the Zariski-Riemann manifold 148 Chapter 9. Ramification of Valuations and Simultaneous Resolution 155 Appendix. Smoothness and Non-singularity II 163 A.1. Proofs of the basic theorems 163 A.2. Non-singularity and uniformizing parameters 169 A.3. Higher derivations 171 A.4. Upper semi-continuity of ν q (I) 174 Bibliography 179 Index 185

7 Preface The notion of singularity is basic to mathematics. In elementary algebra singularity appears as a multiple root of a polynomial. In geometry a point in a space is non-singular if it has a tangent space whose dimension is the same as that of the space. Both notions of singularity can be detected through the vanishing of derivitives. Over an algebraically closed field, a variety is non-singular at a point if there exists a tangent space at the point which has the same dimension as the variety. More generally, a variety is non-singular at a point if its local ring is a regular local ring. A fundamental problem is to remove a singularity by simple algebraic mappings. That is, can a given variety be desingularized by a proper, birational morphism from a non-singular variety? This is always possible in all dimensions, over fields of characteristic zero. We give a complete proof of this in Chapter 6. We also treat positive characteristic, developing the basic tools needed for this study, and giving a proof of resolution of surface singularities in positive characteristic in Chapter 7. In Section 2.5 we discuss important open problems, such as resolution of singularities in positive characteristic and local monomialization of morphisms. Chapter 8 gives a classification of valuations in algebraic function fields of surfaces, and a modernization of Zariski s original proof of local uniformization for surfaces in characteristic zero. This book has evolved out of lectures given at the University of Missouri and at the Chennai Mathematics Institute, in Chennai, (also known as Madras), India. It can be used as part of a one year introductory sequence vii

8 viii Preface in algebraic geometry, and would provide an exciting direction after the basic notions of schemes and sheaves have been covered. A core course on resolution is covered in Chapters 2 through 6. The major ideas of resolution have been introduced by the end of Section 6.2, and after reading this far, a student will find the resolution theorems of Section 6.8 quite believable, and have a good feel for what goes into their proofs. Chapters 7 and 8 cover additional topics. These two chapters are independent, and can be chosen as possible followups to the basic material in the first 5 chapters. Chapter 7 gives a proof of resolution of singularities for surfaces in positive characteristic, and Chapter 8 gives a proof of local uniformization and resolution of singularities for algebraic surfaces. This chapter provides an introduction to valuation theory in algebraic geometry, and to the problem of local uniformization. The appendix proves foundational results on the singular locus that we need. On a first reading, I recommend that the reader simply look up the statements as needed in reading the main body of the book. Versions of almost all of these statements are much easier over algebraically closed fields of characteristic zero, and most of the results can be found in this case in standard textbooks in algebraic geometry. I assume that the reader has some familiarity with algebraic geometry and commutative algebra, such as can be obtained from an introductory course on these subjects. This material is covered in books such as Atiyah and MacDonald [13] or the basic sections of Eisenbud s book [37], and the first two chapters of Hartshorne s book on algebraic geometry [47], or Eisenbud and Harris s book on schemes [38]. I thank Professors Seshadri and Ed Dunne for their encouragement to write this book, and Laura Ghezzi, Tài Hà, Krishna Hanamanthu, Olga Kashcheyeva and Emanoil Theodorescu for their helpful comments on preliminary versions of the manuscript. For financial support during the preparation of this book I thank the National Science Foundation, the National Board of Higher Mathematics of India, the Mathematical Sciences Research Insititute and the University of Missouri. Steven Dale Cutkosky

9 Chapter 1 Introduction An algebraic variety X is defined locally by the vanishing of a system of polynomial equations f i K[x 1,..., x n ], f 1 = = f m = 0. If K is algebraically closed, points of X in this chart are α = (α 1,..., α n ) A n K which satisfy this system. The tangent space T α(x) at a point α X is the linear subspace of A n K defined by the system of linear equations where L i is defined by L i = L 1 = = L m = 0, n j=1 f i x j (α)(x j α j ). We have that dim T α (X) dim X, and X is non-singular at the point α if dim T α (X) = dim X. The locus of points in X which are singular is a proper closed subset of X. The fundamental problem of resolution of singularities is to perform simple algebraic transformations of X so that the transform Y of X is nonsingular everywhere. To be precise, we seek a resolution of singularities of X; that is, a proper birational morphism Φ : Y X such that Y is non-singular. The problem of resolution when K has characteristic zero has been studied for some time. In fact we will see (Chapters 2 and 3) that the method of Newton for determining the analytical branches of a plane curve singularity extends to give a proof of resolution for algebraic curves. The first algebraic proof of resolution of surface singularities is due to Zariski [86]. We give 1

10 2 1. Introduction a modern treatment of this proof in Chapter 8. A study of this proof is surely the best introduction to the methods and ideas underlying the recent emphasis on valuation-theoretic methods in resolution problems. The existence of a resolution of singularities has been completely solved by Hironaka [52], in all dimensions, when K has characteristic zero. We give a simplified proof of this theorem, based on the proof of canonical resolution by Encinas and Villamayor ([40], [41]), in Chapter 6. When K has positive characteristic, resolution is known for curves, surfaces and 3-folds (with char(k) > 5). The first proof in positive characteristic of resolution of surfaces and of resolution for 3-folds is due to Abhyankar [1], [4]. We give several proofs, in Chapters 2, 3 and 4, of resolution of curves in arbitrary characteristic. In Chapter 5 we give a proof of resolution of surfaces in arbitrary characteristic Notation The notation of Hartshorne [47] will be followed, with the following differences and additions. By a variety over a field K (or a K-variety), we will mean an open subset of an equidimensional reduced subscheme of the projective space P n K. Thus an integral variety is a quasi-projective variety in the classical sense. A curve is a one-dimensional variety. A surface is a two-dimensional variety, and a 3-fold is a three-dimensional variety. A subvariety Y of a variety X is a closed subscheme of X which is a variety. An affine ring is a reduced ring which is of finite type over a field K. If X is a variety, and I is an ideal sheaf on X, we denote V (I) = spec(o X /I) X. If Y is a subscheme of a variety X, we denote the ideal of Y in X by I Y. If W 1, W 2 are subschemes of a variety of X, we will denote the scheme-theoretic intersection of W 1 and W 2 by W 1 W 2. This is the subscheme W 1 W 2 = V (I W1 + I W2 ) W. A hypersurface is a codimension one subvariety of a non-singular variety.

11 Chapter 2 Non-singularity and Resolution of Singularities 2.1. Newton s method for determining the branches of a plane curve Newton s algorithm for solving f(x, y) = 0 by a fractional power series y = y(x 1 m ) can be thought of as a generalization of the implicit function theorem to general analytic functions. We begin with this algorithm because of its simplicity and elegance, and because this method contains some of the most important ideas in resolution. We will see (in Section 2.5) that the algorithm immediately gives a local solution to resolution of analytic plane curve singularities, and that it can be interpreted to give a global solution to resolution of plane curve singularities (in Section 3.5). All of the proofs of resolution in this book can be viewed as generalizations of Newton s algorithm, with the exception of the proof that curve singularities can be resolved by normalization (Theorems 2.14 and 4.3). Suppose K is an algebraically closed field of characteristic 0, K[[x, y]] is a ring of power series in two variables and f K[[x, y]] is a non-unit, such that x f. Write f = i,j a ijx i y j with a ij K. Let and mult(f) = min{i + j a ij 0}, mult(f(0, y)) = min{j a 0j 0}. 3

12 4 2. Non-singularity and Resolution of Singularities Set r 0 = mult(f(0, y)) mult(f). Set { } i δ 0 = min r 0 j : j < r 0 and a ij 0. δ 0 = if and only if f = uy r 0, where u is a unit in K[[x, y]]. Suppose that δ 0 <. Then we can write f = i+δ 0 j δ 0 r 0 a ij x i y j with a 0r0 0, and the weighted leading form L δ0 (x, y) = a ij x i y j = a 0r0 y r0 + terms of lower degree in y i+δ 0 j=δ 0 r 0 has at least two non-zero terms. We can thus choose 0 c 1 K so that L δ0 (1, c 1 ) = a ij c j 1 = 0. i+δ 0 j=δ 0 r 0 Write δ 0 = p 0 q 0, where q 0, p 0 are relatively prime positive integers. We make a transformation x = x q 0 1, y = xp 0 1 (y 1 + c 1 ). Then where (2.1) f 1 (x 1, y 1 ) = f = x r 0p 0 1 f 1 (x 1, y 1 ), i+δ 0 j=δ 0 r 0 a ij (c 1 + y 1 ) j + x 1 H(x 1, y 1 ). By our choice of c 1, f 1 (0, 0) = 0. Set r 1 = mult(f 1 (0, y 1 )). We see that r 1 r 0. We have an expansion f 1 = a ij (1)x i 1y j 1. Set { } i δ 1 = min r 1 j : j < r 1 and a ij (1) 0, and write δ 1 = p 1 q 1 with p 1, q 1 relatively prime. We can then choose c 2 K for f 1, in the same way that we chose c 1 for f, and iterate this process, obtaining a sequence of transformations (2.2) x = x q 0 1, y = xp 0 1 (y 1 + c 1 ), x 1 = x q 1 2, y 1 = x p 1 2 (y 2 + c 2 ),. Either this sequence of transformations terminates after a finite number n of steps with δ n =, or we can construct an infinite sequence of transformations with δ n < for all n. This allows us to write y as a series in ascending fractional powers of x.

13 2.1. Newton s method for determining the branches of a plane curve 5 As our first approximation, we can use our first transformation to solve for y in terms of x and y 1 : y = c 1 x δ 0 + y 1 x δ 0. Now the second transformation gives us y = c 1 x δ 0 + c 2 x δ 0+ δ 1 q0 + y 2 x δ 0+ δ 1 q0. We can iterate this procedure to get the formal fractional series (2.3) y = c 1 x δ 0 + c 2 x δ 0+ δ 1 q0 + c 3 x δ 0+ δ 1 q0 + δ 2 q 0 q 1 +. Theorem 2.1. There exists an i 0 such that δ i N for i i 0. Proof. r i = mult(f i (0, y i )) are monotonically decreasing, and positive for all i, so it suffices to show that r i = r i+1 implies δ i N. Without loss of generality, we may assume that i = 0 and r 0 = r 1. f 1 (x 1, y 1 ) is given by the expression (2.1). Set g(t) = f 1 (0, t) = a ij (c 1 + t) j. i+δ 0 j=δ 0 r 0 g(t) has degree r 0. Since r 1 = r 0, we also have mult(g(t)) = r 0. Thus g(t) = a 0r0 t r 0, and i+δ 0 j=δ 0 r 0 a ij t j = a 0r0 (t c 1 ) r0. In particular, since K has characteristic 0, the binomial theorem shows that (2.4) a i,r0 1 0, where i is a natural number with i + δ 0 (r 0 1) = δ 0 r 0. Thus δ 0 N. We can thus find a natural number m, which we can take to be the smallest possible, and a series such that (2.3) becomes p(t) = b i t i (2.5) y = p(x 1 m ). For n N, set p n (t) = Using induction, we can show that n b i t i. i=1 mult(f(t m, p n (t))

14 6 2. Non-singularity and Resolution of Singularities as n, and thus f(t m, p(t)) = 0. Thus (2.6) y = b i x i m is a branch of the curve f = 0. This expansion is called a Puiseux series (when r 0 = mult(f)), in honor of Puiseux, who introduced this theory into algebraic geometry. Remark 2.2. Our proof of Theorem 2.1 is not valid in positive characteristic, since we cannot conclude (2.4). Theorem 2.1 is in fact false over fields of positive characteristic. See Exercise 2.4 at the end of this section. Suppose that f K[[x, y]] is irreducible, and that we have found a solution y = p(x 1 m ) to f(x, y) = 0. We may suppose that m is the smallest natural number for which it is possible to write such a series. y p(x 1 m ) divides f in R 1 = K[[x 1 m, y]]. Let ω be a primitive m-th root of unity in K. Since f is invariant under the K-algebra automorphism φ of R 1 determined by x 1 m ωx 1 m and y y, it follows that y p(ω j x 1 m )) f in R 1 for all j, and thus y = p(ω j x 1 m ) is a solution to f(x, y) = 0 for all j. These solutions are distinct for 0 j m 1, by our choice of m. The series g = m 1 j=0 (y p(ω j x 1 m ) ) is invariant under φ, so g K[[x, y]] and g f in K[[x, y]], the ring of invariants of R 1 under the action of the group Z m generated by φ. Since f is irreducible, m 1 ) f = u (y p(ω j x 1 m ), where u is a unit in K[[x, y]]. j=0 Remark 2.3. Some letters of Newton developing this idea are translated (from Latin) in [18]. After we have defined non-singularity, we will return to this algorithm in (2.7) of Section 2.5, to see that we have actually constructed a resolution of singularities of a plane curve singularity. Exercise Construct a Puiseux series solution to f(x, y) = y 4 2x 3 y 2 4x 5 y + x 6 x 7 = 0 over the complex numbers.

15 2.2. Smoothness and non-singularity 7 2. Apply the algorithm of this section to the equation f(x, y) = y p + y p+1 + x = 0 over an algebraically closed field k of characteristic p. What is the resulting fractional series? 2.2. Smoothness and non-singularity Definition 2.5. Suppose that X is a scheme. X is non-singular at P X if O X,P is a regular local ring. Recall that a local ring R, with maximal ideal m, is regular if the dimension of m/m 2 as an R/m vector space is equal to the Krull dimension of R. For varieties over a field, there is a related notion of smoothness. Suppose that K[x 1,..., x n ] is a polynomial ring over a field K, We define the Jacobian matrix f 1,..., f m K[x 1,..., x n ]. J(f; x) = J(f 1,..., f m ; x 1,..., x n ) = f 1 x 1. f m x 1 f 1 x n f m x n. Let I = (f 1,..., f m ) be the ideal generated by f 1,..., f m, and define R = K[x 1,..., x n ]/I. Suppose that P spec(r) has ideal m in K[x 1,..., x n ] and ideal n in R. Let K(P ) = R n /n n. We will say that J(f; x) has rank l at P if the image of the l-th Fitting ideal I l (J(f; x)) of l l minors of J(f, x) in K(P ) is K(P ) and the image of the s-th Fitting ideal I s (J(f; x)) in K(P ) is (0) for s > l. Definition 2.6. Suppose that X is a variety of dimension s over a field K, and P X. Suppose that U = spec(r) is an affine neighborhood of P such that R = K[x 1,..., x n ]/I with I = (f 1,..., f m ). Then X is smooth over K if J(f; x) has rank n s at P. This definition depends only on P and X and not on any of the choices of U, x or f. We verify this is the appendix. Theorem 2.7. Let K be a field. The set of points in a K-variety X which are smooth over K is an open set of X. Theorem 2.8. Suppose that X is a variety over a field K and P X. 1. Suppose that X is smooth over K at P. Then P is a non-singular point of X.

16 8 2. Non-singularity and Resolution of Singularities 2. Suppose that P is a non-singular point of X and K(P ) is separably generated over K. Then X is smooth over K at P. In the case when K is algebraically closed, Theorems 2.7 and 2.8 are proven in Theorems I.5.3 and I.5.1 of [47]. Recall that an algebraically closed field is perfect. We will give the proofs of Theorems 2.7 and 2.8 for general fields in the appendix. Corollary 2.9. Suppose that X is a variety over a perfect field K and P X. Then X is non-singular at P if and only if X is smooth at P over K. Proof. This is immediate since an algebraic function field over a perfect field K is always separably generated over K (Theorem 13, Section 13, Chapter II, [92]). In the case when X is an affine variety over an algebraically closed field K, the notion of smoothness is geometrically intuitive. Suppose that X = V (I) = V (f 1,..., f m ) A n K is an s-dimensional affine variety, where I = (f 1,..., f m ) is a reduced and equidimensional ideal. We interpret the closed points of X as the set of solutions to f 1 = = f m = 0 in K n. We identify a closed point p = (a 1,..., a n ) V (I) A n K with the maximal ideal m = (x 1 a 1,..., x n a n ) of K[x 1,..., x n ]. For 1 i m, where f i f i (p) + L i mod m 2, L i = n j=1 f i x i (p)(x j a j ). But f i (p) = 0 for all i since p is a point of X. The tangent space to X in A n K at the point p is T p (X) = V (L 1,..., L m ) A n K. We see that dim T p (X) = n rank(j(f; x)(p)). Thus dim T p (X) s (by Remark A.9), and X is non-singular at p if and only if dim T p (X) = s. Theorem Suppose that X is a variety over a field K. Then the set of non-singular points of X is an open dense set of X. This theorem is proven when K is algebraically closed in Corollary I.5.3 [47]. We will prove Theorem 2.10 when K is perfect in the appendix. The general case is proven in the corollary to Theorem 11 [85]. Exercise Consider the curve y 2 x 3 = 0 in A 2 K, over an algebraically closed field K of characteristic 0 or p > 3.

17 2.3. Resolution of singularities 9 1. Show that at the point p 1 = (1, 1), the tangent space T p1 (X) is the line 3(x 1) + 2(y 1) = Show that at the point p 2 = (0, 0), the tangent space T p2 (X) is the entire plane A 2 K. 3. Show that the curve is singular only at the origin p Resolution of singularities Suppose that X is a K-variety, where K is a field. Definition A resolution of singularities of X is a proper birational morphism φ : Y X such that Y is a non-singular variety. A birational morphism is a morphism φ : Y X of varieties such that there is a dense open subset U of X such that φ 1 (U) U is an isomorphism. If X and Y are integral and K(X) and K(Y ) are the respective function fields of X and Y, φ is birational if and only if φ : K(X) K(Y ) is an isomorphism. A morphism of varieties φ : Y X is proper if for every valuation ring V with morphism α : spec(v ) X, there is a unique morphism β : spec(v ) Y such that φ β = α. If X and Y are integral, and K(X) is the function field of X, then we only need consider valuation rings V such that K V K(X) in the definition of properness. The geometric idea of properness is that every mapping of a formal curve germ into X lifts uniquely to a morphism to Y. One consequence of properness is that every proper map is surjective. The properness assumption rules out the possibility of resolving by taking the birational resolution map to be the inclusion of the open set of nonsingular points into the given variety, or the mapping of the non-singular points of a partial resolution to the variety. Birational proper morphisms of non-singular varieties (over a field of characteristic zero) can be factored by alternating sequences of blow-ups and blow-downs of non-singular subvarieties, as shown by Abramovich, Karu, Matsuki and Wlodarczyk [11]. We can extend our definition of a resolution of singularities to arbitrary schemes. A reasonable category to consider is excellent (or quasi-excellent) schemes (defined in IV.7.8 [45] and on page 260 of [66]). The definition of excellence is extremely technical, but the idea is to give minimal conditions ensuring that the the singular locus is preserved by natural base extensions such as completion. There are examples of non-excellent schemes which admit a resolution of singularities. Rotthaus [74] gives an example of a

18 10 2. Non-singularity and Resolution of Singularities regular local ring R of dimension 3 containing a field which is not excellent. In this case, spec(r) is a resolution of singularities of spec(r) Normalization Suppose that R is an affine domain with quotient field L. Let S be the integral closure of R in L. Since R is affine, S is finite over R, so that S is affine (cf. Theorem 9, Chapter V [92]). We say that spec(s) is the normalization of spec(r). Suppose that V is a valuation ring of L such that R V. V is integrally closed in L (although V need not be Noetherian). Thus S V. The morphism spec(v ) spec(r) lifts to a morphism spec(v ) spec(s), so that spec(s) spec(r) is proper. If X is an integral variety, we can cover X by open affines spec(r i ) with normalization spec(s i ). The spec(s i ) patch to a variety Y called the normalization of X. Y X is proper, by our local proof. For a general (reduced but not necessarily integral) variety X, the normalization of X is the disjoint union of the normalizations of the irreducible components of X. A ring R is normal if R p is an integrally closed domain for all p spec(r). If R is an affine ring and spec(s) is the normalization of spec(r) we constructed above, then S is a normal ring. Theorem 2.13 (Serre). A Noetherian ring A is normal if and only if A is R 1 (A p is regular if ht(p) 1) and S 2 (depth A p min(ht(p), 2) for all p spec(a)). A proof of this theorem is given in Theorem 23.8 [66]. Theorem Suppose that X is a 1-dimensional variety over a field K. Then the normalization of X is a resolution of singularities. Proof. Let X be the normalization of X. All local rings O X,p of points p X are local rings of dimension 1 which are integrally closed, so Theorem 2.13 implies they are regular. As an example, consider the curve singularity y 2 x 3 = 0 in A 2, with affine ring R = K[x, y]/(y 2 x 3 ). In the quotient field L of R we have the relation ( y x )2 x = 0. Thus y x is integral over R. Since R[ y x ] = K[ y x ] is a regular ring, it must be the integral closure of R in L. Thus spec(r[ y x ]) spec(r) is a resolution of singularities. Normalization is in general not enough to resolve singularities in dimension larger than 1. As an example, consider the surface singularity X defined

19 2.5. Local uniformization and generalized resolution problems 11 by z 2 xy = 0 in A 3 K, where K is a field. The singular locus of X is defined by the ideal J = (z 2 xy, y, x, 2z). J = (x, y, z), so the singular locus of X is the origin in A 2. Thus R = K[x, y, z]/(z 2 xy) is R 1. Since z 2 xy is a regular element in K[x, y, z], R is Cohen-Macaulay, and hence R is S 2 (Theorem 17.3 [66]). We conclude that X is normal, but singular. Kawasaki [58] has proven that under extremely mild assumptions a Noetherian scheme admits a Macaulayfication (all local rings have maximal depth). In general, Cohen-Macaulay schemes are far from being nonsingular, but they do share many good homological properties with nonsingular schemes. A scheme which does not admit a resolution of singularities is given by the example of Nagata (Example 3 in the appendix to [68]) of a onedimensional Noetherian domain R whose integral closure R is not a finite R- module. We will show that such a ring cannot have a resolution of singularities. Suppose that there exists a resolution of singularities φ : Y spec(r). That is, Y is a regular scheme and φ is proper and birational. Let L be the quotient field of R. Y is a normal scheme by Theorem Thus Y has an open cover by affine open sets U 1,..., U s such that U i = spec(ti ), with T i = R[g i1,..., g it ] where g ij L, and each T i is integrally closed. The integrally closed subring s T = Γ(Y, O Y ) = of L is a finite R-module since φ is proper (Theorem III, [44] or Corollary II.5.20 [47] if φ is assumed to be projective). Exercise Let K = Z p (t), where p is a prime and t is an indeterminate. Let R = K[x, y]/(x p + y p t), X = spec(r). Prove that X is nonsingular, but there are no points of X which are smooth over K. 2. Let K = Z p (t), where p > 2 is a prime and t is an indeterminate. Let R = K[x, y]/(x 2 + y p t), X = spec(r). Prove that X is non-singular, and X is smooth over K at every point except at the prime (y p t, x) Local uniformization and generalized resolution problems Suppose that f is irreducible in K[[x, y]]. Then the Puiseux series (2.5) of Section 2.1 determines an inclusion i=1 (2.7) R = K[[x, y]]/(f(x, y)) K[[t]] T i

20 12 2. Non-singularity and Resolution of Singularities with x = t m, y = p(t) the series of (2.5). Since the m chosen in (2.5) of Section 2.1 is the smallest possible, we can verify that t is in the quotient field L of R. Since the quotient field of K[[t]] is generated by 1, t,..., t m 1 over L, we conclude that R and K[[t]] have the same quotient field. (2.7) is an explicit realization of the normalization of the curve singularity germ f(x, y) = 0, and thus spec(k[[t]]) spec(r) is a resolution of singularities. The quotient field L of K[[t]] has a valuation ν defined for non-zero h L by ν(h(t)) = n Z if h(t) = t n u, where u is a unit in K[[t]]. This valuation can be understood in R by the Puiseux series (2.6). More generally, we can consider an algebraic function field L over a field K. Suppose that V is a valuation ring of L containing K. The problem of local uniformization is to find a regular local ring R, essentially of finite type over K and with quotient field L such that the valuation ring V dominates R (R V and the intersection of the maximal ideal of V with R is the maximal ideal of R). The exact relationship of local uniformization to resolution of singularities is explained in Section 8.3. If L has dimension 1 over K, then the valuation rings V of L which contain K are precisely the local rings of points on the unique non-singular projective curve C with function field L (cf. Section I.6 [47]). The Newton method of Section 2.1 can be viewed as a solution to the local uniformization problem for complex analytic curves. If L has dimension 2 over K, there are many valuations rings V of L which are non-noetherian (see the exercise of Section 8.1). Zariski proved local uniformization for two-dimensional function fields over an algebraically closed field of characteristic zero in [86]. He was able to patch together local solutions to prove the existence of a resolution of singularities for algebraic surfaces over an algebraically closed field of characteristic zero. He later was able to prove local uniformization for algebraic function fields of characteristic zero in [87]. This method leads to an extremely difficult patching problem in higher dimensions, which Zariski was able to solve in dimension 3 in [90]. Zariski s proof of local uniformization can be considered as an extension of the Newton method to general valuations. In Chapter 8, we present Zariski s proof of resolution of surface singularities through local uniformization. Local uniformization has been proven for two-dimensional function fields in positive characteristic by Abhyankar. Abhyankar has proven resolution of singularities in positive characteristic for surfaces and for three-dimensional varieties, [1],[3],[4].

21 2.5. Local uniformization and generalized resolution problems 13 There has recently been a resurgence of interest in local uniformization in positive characteristic, and significant progress is being made. Some of the recent papers on this area are: Hauser [50], Heinzer, Rotthaus and Wiegand [51], Kuhlmann [59], [60], Piltant [71], Spivakovsky [77], Teissier [78]. Some related resolution type problems are resolution of vector fields, resolution of differential forms and monomialization of morphisms. Suppose that X is a variety which is smooth over a perfect field K, and D Hom(Ω 1 X/K, O X) is a vector field. If p X is a closed point, and x 1,..., x n are regular parameters at p, then there is a local expression D = a 1 (x) x a n (x) x n, where a i (x) O X,p. We can associate an ideal sheaf I D to D on X by for p X. I D,p = (a 1,..., a n ) There are an effective divisor F D and an ideal sheaf J D such that V (J D ) has codimension 2 in X and I D = O X ( F D )J D. The goal of resolution of vector fields is to find a proper morphism π : Y X so that if D = π 1 (D), then the ideal I D O Y is as simple as possible. This was accomplished by Seidenberg for vector fields on non-singular surfaces over an algebraically closed field of characteristic zero in [75]. The best statement that can be attained is that the order of J D at p (Definition A.17) is ν p (J D ) 1 for all p Y. The basic invariant considered in the proof is the order ν p (J D ). While this is an upper semi-continuous function on Y, it can go up under a monoidal transform. However, we have that ν q (J D ) ν p (J D )+1 if π : Y X is the blow-up of p, and q π 1 (p). This should be compared with the classical resolution problem, where a basic result is that order cannot go up under a permissible monoidal transform (Lemma 6.4). Resolution of vector fields for smooth surfaces over a perfect field has been proven by Cano [19]. Cano has also proven a local theorem for resolution for vector fields over fields of characteristic zero [20], which implies local resolution (along a valuation) of a vector field. The statement is that we can achieve ν q (J D ) 1 (at least locally along a valuation) after a morphism π : Y X. There is an analogous problem for differential forms. A recent paper on this is [21]. We can also consider resolution problems for morphisms f : Y X of varieties. The natural question to ask is if it is possible to perform monoidal transforms (blow-ups of non-singular subvarieties) over X and Y to produce a morphism which is a monomial mapping.

22 14 2. Non-singularity and Resolution of Singularities Definition Suppose that Φ : X Y is a dominant morphism of nonsingular irreducible K-varieties (where K is a field of characteristic zero). Φ is monomial if for all p X there exists an étale neighborhood U of p, uniformizing parameters (x 1,..., x n ) on U, regular parameters (y 1,..., y m ) in O Y,Φ(p), and a matrix (a ij ) of non-negative integers (which necessarily has rank m) such that y 1 = x a 11 1 x a 1n n,. y m = x a m1 1 x amn n. Definition Suppose that Φ : X Y is a dominant morphism of integral K-varieties. A morphism Ψ : X 1 Y 1 is a monomialization of Φ [27] if there are sequences of blow-ups of non-singular subvarieties α : X 1 X and β : Y 1 Y, and a morphism Ψ : X 1 Y 1 such that the diagram X 1 Ψ Y1 X Φ Y commutes, and Ψ is a monomial morphism. If Φ : X Y is a dominant morphism from a 3-dimensional variety to a surface (over an algebraically closed field of characteristic 0), then there is a monomialization of Φ [27]. A generalized multiplicity is defined in this paper, and it can go up, causing a very high complexity in the proof. An extension of this result to strongly prepared morphisms from n-folds to surfaces is proven in [33]. It is not known if monomialization is true even for birational morphisms of varieties of dimension 3, although it is true locally along a valuation, from the following Theorem Theorem 2.18 is proven when the quotient field of S is finite over the quotient field of R in [25]. The proof for general field extensions is in [28]. Theorem 2.18 (Theorem 1.1 [26], Theorem 1.1 [28]). Suppose that R S are regular local rings, essentially of finite type over a field K of characteristic zero. Let V be a valuation ring of K which dominates S. Then there exist sequences of monoidal transforms R R and S S such that V dominates S, S dominates R and there are regular parameters (x 1,..., x m ) in R, (y 1,..., y n ) in S, units δ 1,..., δ m S and an m n matrix (a ij ) of non-negative integers such that rank(a ij ) = m is maximal and (2.8) x 1 = y a y a 1n n δ 1,. x m = y a m1 1...yn amn δ m.

23 2.5. Local uniformization and generalized resolution problems 15 Thus (since char(k) = 0) there exists an etale extension S S, where S has regular parameters y 1,..., y n such that x 1,..., x m are pure monomials in y 1,..., y n. The standard theorems on resolution of singularities allow one to easily find R and S such that (2.8) holds, but, in general, we will not have the essential condition rank(a ij ) = m. The difficulty of the problem is to achieve this condition. This result gives very simple structure theorems for the ramification of valuations in characteristic zero function fields [35]. We discuss some of these results in Chapter 9. A generalization of monomialization in characteristic p function fields of algebraic surfaces is obtained in [34] and especially in [35]. We point out that while it seems possible that Theorem 2.18 does hold in positive characteristic, there are simple examples in positive characteristic where a monomialization does not exist. The simplest example is the map of curves in characteristic p. y = x p + x p+1 A quasi-complete variety over a field K is an integral finite type K- scheme which satisfies the existence part of the valuative criterion for properness (Hironaka, Chapter 0, Section 6 of [52] and Chapter 8 of [26]). The construction of a monomialization by quasi-complete varieties follows from Theorem Theorem 2.19 is proven for generically finite morphisms in [26] and for arbitrary morphisms in Theorem 1.2 [28]. Theorem 2.19 (Theorem 1.2 [26],Theorem 1.2 [28]). Let K be a field of characteristic zero, Φ : X Y a dominant morphism of proper K-varieties. Then there exist birational morphisms of non-singular quasi-complete K- varieties α : X 1 X and β : Y 1 Y, and a monomial morphism Ψ : X 1 Y 1 such that the diagram X 1 Ψ Y 1 X Φ Y commutes and α and β are locally products of blow-ups of non-singular subvarieties. That is, for every z X 1, there exist affine neighborhoods V 1 of z and V of x = α(z) such that α : V 1 V is a finite product of monoidal transforms, and there exist affine neighborhoods W 1 of Ψ(z), W of y = β(ψ(z)) such that β : W 1 W is a finite product of monoidal transforms.

24 16 2. Non-singularity and Resolution of Singularities A monoidal transform of a non-singular K-scheme S is the map T S induced by an open subset T of proj( I n ), where I is the ideal sheaf of a non-singular subvariety of S. The proof of Theorem 2.19 follows from Theorem 2.18, by patching a finite number of local solutions. The resulting schemes may not be separated. It is an extremely interesting question to determine if a monomialization exists for all morphisms of varieties (over a field of characteristic zero). That is, the conclusions of Theorem 2.19 hold, but with the stronger conditions that α and β are products of monoidal transforms on proper varieties X 1 and Y 1.

25 Chapter 3 Curve Singularities 3.1. Blowing up a point on A 2 Suppose that K is an algebraically closed field. Let U 1 = spec(k[s, t]) = A 2 K, U 2 = spec(k[u, v]) = A 2 K. Define a K-algebra isomorphism λ : K[s, t] s = K[s, t, 1 s ] K[u, v] v = K[u, v, 1 v ] by λ(s) = 1 v, λ(t) = uv. We define a K-variety B 0 by patching U 1 to U 2 on the open sets U 2 V (v) = spec(k[u, v] v ) and U 1 V (s) = spec(k[s, t] s ) by the isomorphism λ. The K-algebra homomorphisms defined by x st, y t, and K[x, y] K[s, t] K[x, y] K[u, v] defined by x u, y uv, are compatible with the isomorphism λ, so we get a morphism π : B(p) = B 0 U 0 = spec(k[x, y]) = A 2 K, where p denotes the origin of U 0. Upon localization of the above maps, we get isomorphisms K[x, y] y = K[s, t]t and K[x, y] x = K[u, v]u. 17

26 18 3. Curve Singularities Thus π : U 1 V (t) = U 0 V (y), π : U 2 V (u) = U 0 V (x), and π is an isomorphism over U 0 V (x, y) = U 0 {p}. Moreover, π 1 (p) U 1 = V (st, t) = {V (t) spec(k[s, t])} = spec(k[s]), π 1 (p) U 2 = V (u, uv) = {V (u) spec(k[u, v])} = spec(k[ 1 s ]), by the identification s = 1 v. Thus π 1 (p) = P 1. Set E = π 1 (p). We have blown up p into a codimension 1 subvariety of B(p), isomorphic to P 1. Set R = K[x, y], m = (x, y). Then U 1 = spec(k[s, t]) = spec(k[ x y, y]) = spec(r[x y ]), We see that U 2 = spec(k[x, y x ]) = spec(r[ y x ]). B(p) = spec(r[ x y ]) spec(r[ y x ]) = proj( n 0 m n ). Suppose that q π 1 (p) is a closed point. If q U 1, then its associated ideal m q is a maximal ideal of R 1 = K[ x y, y] which contains (x, y)r 1 = yr 1. Thus m q = (y, x y α) for some α K. If we set y 1 = y, x 1 = x y α, we see that there are regular parameters (x 1, y 1 ) in O B(p),q such that x = y 1 (x 1 + α), y = y 1. By a similar calculation, if q π 1 (p) and q U 2, there are regular parameters (x 1, y 1 ) in O B(p),q such that x = x 1, y = x 1 (y 1 + β) for some β K. If the constant α or β is non-zero, then q is in both U 1 and U 2. Thus the points in π 1 (p) can be expressed (uniquely) in one of the forms x = x 1, y = x 1 (y 1 + α) with α K, or x = x 1 y 1, y = y 1. Since B(p) is projective over spec(r), it certainly is proper over spec(r). However, it is illuminating to give a direct proof. Lemma 3.1. B(p) spec(r) is proper. Proof. Suppose that V is a valuation ring containing R. Then y x or x y V. Say y x V. Then R[ y x ] V, and we have a morphism spec(v ) spec(r[ y ]) B(p) x which lifts the morphism spec(v ) spec(r).

27 3.1. Blowing up a point on A 2 19 More generally, suppose that S = spec(r) is an affine surface over a field L, and p S is a non-singular closed point. After possibly replacing S with an open subset spec(r f ), we may assume that the maximal ideal of p in R is m p = (x, y). We can then define the blow up of p in S by (3.1) π : B(p) = proj( n 0 m n p ) S. We can write B(p) as the union of two affine open subsets: B(p) = spec(r[ x y ]) spec(r[ y x ]). π is an isomorphism over S p, and π 1 (p) = P 1. Suppose that S is a surface, and p S is a non-singular point, with ideal sheaf m p O S. The blow-up of p S is π : B(p) = proj( n 0 m n p ) S. π is an isomorphism away from p, and if U = spec(r) S is an affine open neighborhood of p in S such that (x, y) = Γ(U, m p ) R is the maximal ideal of p in R, then the map π : π 1 (U) U is defined by the construction (3.1). Suppose that C A 2 K is a curve. Since K[x, y] is a unique factorization domain, there exists f K[x, y] such that V (f) = C. If q C is a closed point, we will denote the corresponding maximal ideal of K[x, y] by m q. We set ν q (C) = max{r f m r q} (more generally, see Definition A.17). ν q (C) is both the multiplicity and the order of C at q. Lemma 3.2. q A 2 K is a non-singular point of C if and only if ν q(c) = 1. Proof. Let R = K[x, y] mq, and suppose that m q = (x, y). Let T = R/fR. If ν q (C) 2, then f m 2 q, and m q T/m 2 qt = m q /m 2 q has dimension 2 > 1, so that T is not regular and q is singular on C. However, if ν q (C) = 1, then we have f αx + βy mod m 2 q, where α, β K and at least one of α and β is non-zero. Without loss of generality, we may suppose that β 0. Thus y α β x mod m2 q + (f),

28 20 3. Curve Singularities and x is a K generator of m q T/m 2 qt = m q /m 2 q + (f). We see that and q is non-singular on C. dim K m q T/m 2 qt = 1 Remark 3.3. This lemma is also true in the situation where q is a nonsingular point on a non-singular surface S (over a field K) and C is a curve contained in S. We need only modify the proof by replacing R with the regular local ring R = O S,q, which has regular parameters (x, y) which are a K(q) basis of m q /m 2 q, and since R is a unique factorization domain, there is f R such that f = 0 is a local equation of C at q. Let p be the origin in A 2 K, and suppose that ν p(c) = r > 0. Let π : B(p) A 2 K be the blow-up of p. The strict transform C of C in B(p) is the Zariski closure of π 1 (C p) = C p in B(p). Let E = π 1 (p) be the exceptional divisor. Set-theoretically, π 1 (C) = C E. For some a ij K, we have a finite sum f = a ij x i y j i+j r with a ij 0 for some i, j with i + j = r. In the open subset U 2 = spec(k[x 1, y 1 ]) of B(p), where x = x 1, y = x 1 y 1, x 1 = 0 is a local equation for E. Also, where f 1 = f = x r 1f 1, i+j r a ij x i+j r 1 y j 1. x 1 f 1 since ν p (C) = r, so that f 1 = 0 is a local equation of the strict transform C of C in U 2. In the open subset U 1 = spec(k[x 1, y 1 ]) of B(p), where y 1 = 0 is a local equation for E and where is a local equation of C in U 1. x = x 1 y 1, y = y 1, f 1 = f = y r 1f 1, i+j r a ij x i 1y i+j r 1

29 3.1. Blowing up a point on A 2 21 As a scheme, we see that π 1 (C) = C + re. The scheme-theoretic preimage of C is called the total transform of C. Define the leading form of f to be L = a ij x i y j. i+j=r Suppose that q π 1 (p). There are regular parameters (x 1, y 1 ) at q of one of the forms x = x 1, y = x 1 (y 1 + α) or x = x 1 y 1, y = y 1. In the first case f 1 = f x r = 0 is a local equation of C at q, where 1 f 1 = a ij (y 1 + α) j + x 1 Ω i+j=r for some polynomial Ω. Thus ν q ( C) r, and ν q ( C) = r implies a ij (y 1 + α) j = a 0r y1. r We then see that and i+j=r i+j=r a ij y j 1 = a 0r(y 1 α) r L = a 0r (y αx) r. In the second case f 1 = f y1 r = 0 is a local equation of C at q, where f 1 = a ij x i 1 + y 1 Ω i+j=r for some series Ω. Thus ν q ( C) r, and ν q ( C) = r implies a ij x i 1 = a r0 x r 1 and i+j=r L = a r0 x r. We then see that there exists a point q π 1 (p) such that ν q ( C) = r only when L = (ax + by) r for some constants a, b K. Since r > 0, there is at most one point q π 1 (p) where the multiplicity does not drop. Exercise 3.4. Suppose that K[x 1,..., x n ] is a polynomial ring over a perfect field K and f K[x 1,..., x n ] is such that f(0,..., 0) = 0. Let m = (x 1,..., x n ), a maximal ideal of K[x 1,..., x n ]. Define ord f(0,..., 0, x n ) = r if x r n f(0,..., 0, x n ) and x r+1 n f(0,..., 0, x n ).

30 22 3. Curve Singularities Show that R = (K[x 1,..., x n ]/(f)) m is a regular local ring if ord f(0,..., 0, x n ) = Completion Suppose that A is a local ring with maximal ideal m. A coefficient field of A is a subfield L of A which is mapped onto A/m by the quotient mapping A A/m. A basic theorem of Cohen is that an equicharacteristic complete local ring contains a coefficient field (Theorem 27 Section 12, Chapter VIII [92]). This leads to Cohen s structure theorem (Corollary, loc. cit.), which shows that an equicharacteristic complete regular local ring A is isomorphic to a formal power series ring over a field. In fact, if L is a coefficient field of A, and if (x 1,..., x n ) is a regular system of parameters of A, then A is the power series ring A = L[[x 1,..., x n ]]. We further remark that the completion of a local ring R is a regular local ring if and only if R is regular (cf. Section 11, Chapter VIII [92]). Lemma 3.5. Suppose that S is a non-singular algebraic surface defined over a field K, p S is a closed point, π : B = B(p) S is the blow-up of p, and suppose that q π 1 (p) is a closed point such that K(q) is separable over K(p). Let R 1 = ÔS,p and R 2 = ÔB,q, and suppose that K 1 is a coefficient field of R 1, (x, y) are regular parameters in R 1. Then there exist a coefficient field K 2 = K 1 (α) of R 2 and regular parameters (x 1, y 1 ) of R 2 such that ˆπ : R 1 R 2 is the map given by a ij x i y j i,j 0 where a ij K 1, or K 2 = K 1 and i,j 0 ˆπ : R 1 R 2 a ij x i+j 1 (y 1 + α) j, is the map given by a ij x i y j i,j 0 i,j 0 a ij x i 1y i+j 1. Proof. We have R 1 = K 1 [[x, y]], and a natural homomorphism O S O S,p R 1

31 3.2. Completion 23 which induces q B(p) S spec(r 1 ) = spec(r 1 [ y x ]) spec(r 1[ x y ]). Let m q be the ideal of q in B(p) S spec(r 1 ). If q spec(r 1 [ y x ]), then K(q) = R 1 [ y x ]/m q = K 1 [ y x ]/(f( y x )) for some irreducible polynomial f( y x ) in the polynomial ring K 1[ y x ]. Since K(q) is separable over K 1 = K(p), f is separable. We have mq = (x, f( y x )) R 1 [ y x ]. Let α R 1 [ y x ]/m q be the class of y x. We have the residue map φ : R 2 L where L = R 1 [ y x ]/m q. There is a natural embedding of K 1 in R 2. We have a factorization of f(t) = (t α)γ(t) in L[t], where t α and γ(t) are relatively prime. By Hensel s Lemma (Theorem 17, Section 7, Chapter VIII [92]) there is α R 2 such that φ(α) = α and f(t) = (t α)γ(t) in R 2 [t], where φ(γ(t)) = γ(t). The subfield K 2 = K 1 [α] of R 2 is thus a coefficient field of R 2, and m q R 2 = (x, y x α). Thus R 2 = K 2 [[x, y x α]]. Set x 1 = x, y 1 = y x α. The inclusion R 1 = K 1 [[x, y]] R 2 = K 2 [[x 1, y 1 ]] is natural. A series aij x i y j with coefficients a ij K 1 maps to the series aij x i+j 1 (y 1 + α) j. We now give an example to show that even if a regular local ring contains a field, there may not be a coefficient field of the completion of the ring containing that field. Thus the above lemma does not extend to non-perfect fields. Let K = Z p (t), where t is an indeterminate. Let R = K[x] (x p t). Suppose that ˆR has a coefficient field L containing K. Let φ : ˆR ˆR/m be the residue map. Since φ L is an isomorphism, there exists λ L such that λ p = t. Thus (x p t) = (x λ) p in ˆR. But this is impossible, since (x p t) is a generator of m ˆR, and is thus irreducible. Theorem 3.6. Suppose that R is a reduced affine ring over a field K, and A = R p, where p is a prime ideal of R. Then the completion  = ˆR p of A with respect to its maximal ideal is reduced.

32 24 3. Curve Singularities When K is a perfect field, this is a theorem of Chevalley (Theorem 31, Section 13, Chapter VIII [92]). The general case follows from Scholie IV (vii) [45]. However, the property of being a domain is not preserved under completion. A simple example is f = y 2 x 2 + x 3. f is irreducible in C[x, y], but is reducible in the completion C[[x, y]]: f = y 2 x 2 x 3 = (y x 1 + x)(y + x 1 + x). The first parts of the expansions of the two factors are y x 1 2 x2 + and y + x x2 +. Lemma 3.7 (Weierstrass Preparation Theorem). Let K be a field, and suppose that f K[[x 1,..., x n, y]] is such that 0 < r = ν(f(0,..., 0, y)) = max{n y n divides f(0,..., 0, y)} <. Then there exist a unit series u in K[[x 1,..., x n, y]] and non-unit series a i K[[x 1,..., x n ]] such that f = u(y r + a 1 y r a r ). A proof is given in Theorem 5, Section 1, Chapter VII [92]. A concept which will be important in this book is the Tschirnhausen transformation, which generalizes the ancient notion of completion of the square in the solution of quadratic equations. Definition 3.8. Suppose that K is a field of characteristic p 0 and f K[[x 1,..., x n, y]] has an expression f = y r + a 1 y r a r with a i K[[x 1,..., x n ]] and p = 0 or p r. The Tschirnhausen transformation of f is the change of variables replacing y with f then has an expression with b i K[[x 1,..., x n ]] for all i. y = y + a 1 r. f = (y ) r + b 2 (y ) r b r Exercise 3.9. Suppose that (R, m) is a complete local ring and L 1 R is a field such that R/m is finite and separable over L 1. Use Hensel s Lemma (Theorem 17, Section 7, Chapter VIII [92]) to prove that there exists a coefficient field L 2 of R containing L 1.

33 3.3. Blowing up a point on a non-singular surface Blowing up a point on a non-singular surface We define the strict transform, the total transform, and the multiplicity (or order) ν p (C) analogously to the definition of Section 3.1 (More generally, see Section 4.1 and Definition A.17). Lemma Suppose that X is a non-singular surface over an algebraically closed field K, and C is a curve on X. Suppose that p X and ν p (C) = r. Let π : B(p) X be the blow-up of p, C the strict transform of C, and suppose that q π 1 (p). Then ν q ( C) r, and if r > 0, there is at most one point q π 1 (p) such that ν q ( C) = r. Proof. Let f = 0 be a local equation of C at p. parameters in O X,p, we can write f = a ij x i y j If (x, y) are regular as a series of order r in ÔX,p = K[[x, y]]. Let L = a ij x i y j be the leading form of f. i+j=r If q π 1 (p), then O B(p),q has regular parameters (x 1, y 1 ) such that x = x 1, y = x 1 (y 1 + α), or x = x 1 y 1, y = y 1. This substitution induces the inclusion K[[x, y]] = ÔX,p ÔB(p),q = K[[x 1, y 1 ]]. First suppose that (x 1, y 1 ) are defined by x = x 1, y = x 1 (y 1 + α). Then a local equation for C at q is f 1 = f x r, which has the expansion 1 f 1 = a ij (y 1 + α) j + x 1 Ω for some series Ω. We finish the proof as in Section 3.1. Suppose that C is the curve y 2 x 3 = 0 in A 2 K. The only singular point of C is the origin p. Let π : B(p) A 2 K be the blow-up of p, C the strict transform of C, and E the exceptional divisor E = π 1 (p). On U 1 = spec(k[ x y, y]) B(p) we have coordinates x 1, y 1 with x = x 1 y 1, y = y 1. A local equation for E on U 1 is y 1 = 0. Also, y 2 x 3 = y 2 1(1 x 3 1y 1 ). A local equation for C on U 1 is 1 x 3 1 y 1 = 0, which is a unit on E U 1, so C E U 1 =.