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Resolution of Singularities in Positive Characteristics

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Resolution of Singularities in Positive Characteristics RESOLUTION OF SINGULARITIES IN POSITIVE CHARACTERISTICS HEISUKE HIRONAKA TO WAKA Abstract. A proof of resolution of singularities in characteristic p > 0 and all dimensions. Contents 1. Introduction 3 1.1. Acknowledgments. 4 2. Preliminaries 4 2.1. Ideal exponents 4 2.2. Normal crossing data 5 2.3. Infinitely near singularities S(E) 5 3. Three results for basic E-operations: Diff, AR, NE 7 3.1. Differentiation Theorem 7 3.2. Ambient reductions 10 3.3. Numerical exponent theorem connecting geometric S and algebraic } 12 4. Characteristic algebra }(E) 16 4.1. Edge data of }(E) 18 4.2. Cotangent module cot(E) 22 4.3. Edge Decomposition Theorem. 24 5. The negative degree part of }(E). 25 6. Base-hikes and core-focusing of E 29 6.1. Maximum base-hike E^ 29 6.2. Core focusing Eˇ 30 6.3. Coherent LL-heads module L(Eˇ) 33 7. Permissible transformations and effects on }(E) 35 7.1. Effects on edge data of }(E) 35 7.2. The rule of transformation of }nega 36 7.3. Rule of transformation for }nega 39 7.4. Coordinate translation in p > 0 39 Date: March 23, 2017. 1 2 H. HIRONAKA 7.5. Cleaning by edge equation g = yq + 40 8. Standard unit-monomial expression 42 8.1. Finiteness of unit-monomial sum expression 42 8.2. Frobenius filtration with unit-monomials 46 8.3. Standard expressions for = g − yq 46 8.4. Cleaned standard expressions for = g − yq 46 9. Local Leverage-up Exponent-down (LLUED) 46 9.1. Master keys for the existence of LLUED 47 9.2. Introductory comments on local vs global IUED conceots 47 9.3. Frontier symbols of a standard expression 48 9.4. Frontier symbolics 49 9.5. Differentiation-product H[-operator 50 9.6. H[-operators for LLUED 50 9.7. Local H[ case by case 51 9.8. LL-heads and LL-tails of LL-chains 54 9.9. LL-chains headed by L(E;ˇ q) 54 9.10. LLED propagation from a point to a neighborhood 56 9.11. Linear combinations of LL-chains 58 10. LL-chains modifications 59 10.1. Enclosure of Sing(Eˇ) by LL-tails 60 11. Coordinate-free LL-chains and LL-tails 61 11.1. Symbolic power and degree change 61 11.2. Abstract LL-chains with }nega 62 11.3. Λ and τ operators 63 12. GLUED: the globalization of LLUED 64 12.1. Define the step down operator σ on L-levels 66 13. The global coherent GLUED-chain diagram 67 13.1. Define T(Eˇ) and Cot(Eˇ) 67 14. Local LL-chains vs global GLUED-diagram 68 14.1. Rule of transformation of T(Eˇ) 70 14.2. Cotangent AR-heading 70 14.3. How large ` e should be? 71 14.4. LLAR-type generators of T 73 15. AR-schemes and AR-reductions 74 15.1. AR-schemes tied with GLUED-tails 75 15.2. Iteration of ambient reduction of AR-scheme 78 15.3. Terminal Y and terminal plat 79 16. PROOF of RESOLUTION of SINGULARITIES, p > 0 81 16.1. Theorem ∆ and Γ-monomials 81 16.2. Theorem r I: Domino-falls by r blowups 84 16.3. Theorems r II-III: Inductive proof of resolution 87 16.4. Θ(Eˇ) headed by σL(Eˇ) 87 SINGULARITIES 3 17. Comments on the methodology of this paper 89 References 90 1. Introduction In this paper, we prove the existence of an \embedded resolution of singularities", abbreviated ERS, for an arbitrary algebraic scheme (or algebraic variety) X of any dimension, over any base field K of any characteristic p > 0. The adjective embedded means that X is given as a closed subscheme of a smooth irreducible ambient scheme Z and an ERS of X is induced by a finite permissible sequence of successive blowups with smooth centers over Z. An ERS of X ⊂ Z is a diagram: πr−1 π1 π0 Xr ⊂ Zr !··· ! X1 ⊂ Z1 ! X0 ⊂ Z0; S S D1 D0 where X0 = X ⊂ Z0 = Z. We require that for all 0 ≤ j < r, the map πj : Zj+1 ! Zj is a permissible blowup with center Dj ⊂ Xj, that is, Dj is a closed irreducible smooth subscheme contained in the singular locus of Xj; Xj+1 is the strict transform of Xj by πj; and Xr has no singular points. The main contribution of this paper is a new inductive approach to the ERS problems, local and global. The outline of our proof of ERS in characteristic p > 0 goes as follows. (1) We give a powerful application of the finitely generated graded OZ -algebra } of [23] and [27] together with its extension} ~ in- troduced in x5 of this paper. (2) We introduce a useful new technique we call local leverage up and exponent down, or LLED, in x9, and define its globalization GLUED in x12 and x13. (3) Using (1) and (2) we devise a new ambient reduction technique (which is a workable substitute in p > 0 of the \maximum contact" ambient reduction in characterisitic zero). See the formulation of induction on ambient dimension by AR-schemes of Def.(15.5) after x15.1 of x15. Our proof of ERS is given in x16 (Theorem r I) and x16.3 (Theorems r II-III). 4 H. HIRONAKA 1.1. Acknowledgments. I received invaluable mathematical influence and guidance from numerous excellent mathematical teachers and col- leagues in the course of completing this work. I have been indebted so much and for so long to many mathematical seniors, first and foremost to my teachers Oscar Zariski, Masayoshi Nagata, Alexander Grothendiek and Shreeram Abhyankar. I must also mention the valuable influence and stimulation I have received from my colleagues Michael Artin, David Mumford, Bernard Teissier, Le Dung Trang, S-Moh, Jose-Manuel Aroca, Mark Spivakowski, U.O. Villamayor, Herwig Hauser, Hiraku Kawanoue, June-E and many other algebraic geometers with strong interests in the topics surrounding singularities. Finally, I am grateful for the constant continuous encouragement, and multiple proof-reading and misprints-checking by Woo Yang Lee of Seoul National University and Tadao Oda of Tohoku University during the past decade of earnest but slow research toward the proof of resolution of singularities in pos- itive characteristics and for all dimensions. 2. Preliminaries In this section, we review some of the basic definitions used through- out this paper: the ideal exponent (x2.1), normal crossing data (x2.2), and infinitely near singularities (x2.3). Throughout this paper an ambient scheme Z always means a smooth irreducible scheme of finite type defined over a perfect base field K of characteristic p > 0. In particular we may choose K = Z=pZ. Let OZ denote the structure sheaf of Z. We may write O for OZ and Oξ for the local ring OZ,ξ at a point ξ 2 Z. Sometimes the notation Rξ or R for OZ,ξ and Mξ or M for the maximal ideal max(Oξ) of Oξ are used. We write κξ or κ for the residue field Rξ=Mξ. The symbol ρ denotes the Frobenius endomorphism, sending each element to its p-th power. It gives the filtration O ⊃ ρ(O) ⊃ ρ2(O) ⊃ · · · where each ρi(O) is a locally free ρj(O)-module for j > i. It is fre- quently used for polynomial approximations and differentiations in O. Denote by Ycl the set of closed points of a scheme Y . 2.1. Ideal exponents. For an ideal exponent E = (J; b) we write Sing(E) for the singular locus of E which is a closed reduced subscheme 0 of Z consisting of those points ξ with ordξJ ≥ b. A blowup π : Z ! Z with center D is said to be permissible for E if D is a closed smooth irreducible subscheme of Sing(E). SINGULARITIES 5 Definition 2.1. The transform E0 of E by π is E0 = (J 0; b) in Z0 with b 0 JOZ0 = I(D) J OZ0 where I(D) denotes the ideal of D in OZ . Note that I(D)OZ0 is a locally everywhere nonzero principal ideal whose b-power divides JOZ0 thanks to the permissibility of π for (J; b). Definition 2.2. The following graded algebras will be used frequently in this paper. posi L nega L (1) Bl (Z) = i>0 Bl(Z; i) and Bl (Z) = j<0 Bl(Z; j) where, Bl(Z; j) = O for all integers j; posi L (2) Bl(Z)◦ = Bl (Z) O; (3) Bl(Z) = Blposi(Z) L O L Blnega(Z); and j (4) for blξ(Z; j) = (Mξ) and ξ 2 Zcl, M blξ(Z) = blξ(Z; j) j≥0 M ⊂ Bl(Z; j) = Bl(Z): j2Z It should be noted that Bl(Z) has positive part as well as negative part in homogeneity, unlike blξ(Z). We have the natural epimorphism L j j+1 (1) blξ(Z) ! grξ(Z) = grmax(Oξ)(Oξ) = j≥0(Mξ) =(Mξ) whose kernel is Mξblξ(Z) = max(Oξ)blξ(Z). 2.2. Normal crossing data. We will be working with normal cross- ing data, NC-data for short, using the notation Γ = (Γ1; ··· ; Γs) where each Γi is a normal crossing smooth irreducible hypersurface in Z. (Some Γi may be empty and even the system may be empty to begin with.) Definition 2.3. A blowup π : Z0 ! Z is said to be permissible for Γ if its center D has normal crossing with Γ. The transform Γ0 of Γ by π 0 0 0 0 0 is defined to be Γ = (Γ1; ··· ; Γs; Γs+1) where Γi is the strict transform 0 −1 of Γi by π for every i ≤ s and Γs+1 = π (D) is the exceptional divisor of π.
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