Resolution of singularities of an algebraic variety over a field of characteristic zero. I.

*(English)*Zbl 0122.38603The geometric language of this vast work is largely that of A. Grothendieck [Publ. Math., Inst. Hautes Étud. Sci. 4, 1–228 (1960; Zbl 0118.36206)] and while an acquaintance with his work is desirable for an understanding of the present paper almost all of his concepts which are used here are also defined. The proof of the resolution theorem follows the general lines of the proofs given by O. Zariski [Ann. Math. (2) 45, 472–542 (1944; Zbl 0063.08361)] for the case of surfaces and threefolds, and so goes back to the early attempts of the italian geometers, especially B. Levi [Torino Atti 33, 66–86 (1897; JFM 28.0557.01); Annali di Mat. (2) 26, 218–253 (1898; JFM 28.0557.02)]. However the present proof does not depend directly on results used in these earlier proofs. The local arguments require some knowledge of the theory of local rings, in particular the notions of the multiplicity of an ideal
in a local ring and of a regular system of parameters in a regular local ring.

Chapter 0 begins with a rapid summary of some of the basic concepts in the language of schemes, and a definition of the general notion of blowing-up in terms of a universal mapping property. A proof of the existence theorem of blowing-up is outlined for certain categories. This leads to the definition and existence of monoidal transformations (and the centre of a monoidal transformation) in algebraic and analytic geometry. Let \(B\) be a commutative ring with unity. A \(\mathrm{Spec} (B)\)-scheme \(X\) is called an algebraic \(B\)-scheme if it is of finite type over \(\mathrm{Spec} (B)\). A point \(x\) of \(X\) is said to be simple (muitiple) if the local ring \(\mathcal O_{X,x}\), is (is not) regular. \(X\) is said to be noningular if each point of \(X\) is simple. The version of the resolution theorem (main theorem I) which is proved in the later chapters is formulated as follows. If \(X (= X_0)\) is an algebraic \(B\)-scheme which is reduced and irreducible, where B is a field of characteristic zero, then there exists a finite sequence of monoidal transformations \(f_i:X_{i+1}\to X_i\), \((i =0,1,\dots r-1)\) such that \(X_r\) is non-singular and (a) the centre \(D_i\) of \(f_i\) is non-singular and (b) no point of \(D_i\) is simple for \(X_i\). Let \(D\) be an algebraic subscheme of \(X\) defined by the sheaf of ideals \(\mathcal I\) on \(X\), and let \(\mathrm{gr}^p_D(X)\) be the quotient-sheaf \(\mathcal I^p/\mathcal I^{p+1}\) restricted to \(D\). \(X\) is said to be normally flat along \(D\) at a point \(x\) of \(D\) if the stalk of \(\mathrm{gr}^p_D(X)\) at \(x\) is a free \(\mathcal O_{D,p}\)-module for \(p = 0,1,2,\dots X\) is said to be normally flat along \(D\) if it is so at every point \(x\) of \(D\). It is natural to take the centre \(D_i\) of the monoidal transformation \(f_i\) to be equimultiple on \(X_i\), the present proof of the resolution theorem imposes (c) \(X_i\) is normally flat along \(D\). It is shown later that (c) is in fact a stronger condition than equi-multiplicity. The other notion that plays an important role in the proof of the resolution theorem is that of normal crossings. Let \(E\) be a reduced subscheme of a non-singular algebraic \(B\)-scheme \(X\) which is everywhere of codimension one. \(E\) is said to have only normal crossings at a point \(x\) of \(X\) if there exists a regular system of parameters \((z_1,\dots,z_n)\) of \(\mathcal O_{X,x}\), such that the ideal in \(O_{X,x}\) of each irreducible component of \(E\) which contains \(x\) is generated by one of the \(z_i\), \(E\) is said to have only normal crossings if it does so at every point of \(X\). Running alongside the inductive proof of the resolution theorem is the inductive proof of the author’s main theorem II which includes the following result. The complement of a Zariski open subset of a non-singular algebraic \(B\)-scheme (where \(B\) is a field of characteristic zero) can be transformed by a finite sequence of monoidal transformations with non-singular centres into a subscheme which has only normal crossings.

Chapter 0 also contains a discussion of the analogues of main theorems I and II in the analytic case, and indicates how his results can be used to prove the resolution theorem for an arbitrary real analytic space. For a complex analytic space the passage from the local to the global resolution of singularities apparently introduces added difficulties, and in this case the author claims a proof of the corresponding theorem for a complex analytic space of dimension \(\le 3\). The present methods make virtually no progress towards the resolution of singularities of algebraic \(B\)-schemes when \(B\) is a field of positive characteristic.

Chapter I begins by restricting the ring \(B\) to the class \(\mathcal B\) of noetherian local rings \(S\) with the properties (i) the residue field of \(S\) has characteristic zero and (ii) if \(A\) is an \(S\)-algebra of finite type and \(\hat{S}\) denotes the completion of \(S\) then, under the canonical morphism \(\mathrm{Spec} (A\otimes_S \hat{S})\to\mathrm{Spec} (A)\), the singular locus of the former is the preimage of that of the latter spectrum. An algebraic \(B\)-scheme with \(B\) in \(\mathcal B\) is called an algebraic scheme. The two main theorems of Chapter 0 are reformulated in terms of two types of resolution data giving four fundamental theorems. The fundamental theorems are of two types; two of the theorems are separation theorems for the resolution data, while the other two are resolution theorems which imply the main theorems of Chapter 0. For technical reasons the fundamental theorems are concerned only with algebraic schemes which have a given irreducible non-singular ambient scheme; the resolution theorem without such an ambient scheme is achieved by passing to the completion of a certain local ring, since every complete local ring in is a homomorphic image of a formal power series ring over a field of characteristic zero.

Chapter II is a self-contained study of normal flatness, and is of an algebraic nature. The theorems proved when interpreted geometrically imply the following results, among others.

In Ch. III the local effect on singularities of permissible monoidal transformations is studied. Two local numerical characters are introduced as a measure of the severity of a singularity. In terms of these characters it is shown that a singularity is not made worse by any permissible monoidal transformation, and that any sequence of such transformations cannot make a singularity infinitely better. The aim of this local study is to prove the existence of a special coordinate system of the non-singular ambient scheme at a point and of a special base of the ideal defining the subscheme at the point, both of which have a certain stability with respect to the sequences of permissible monoidal transformations.

Chapter IV is devoted to the inductive proofs of the four fundamental theorems. Here the arguments are of a more geometric nature and rely on suitable geometric interpretations of the algebraic results in the two preceding chapters. In conclusion it is worth noting that the inductive proofs could not be carried out should the schemes be restricted to algebraic schemes over fields of characteristic zero; i.e. an essential part of the present proof of the resolution theorem for algebraic singularities is that the corresponding theorem for algebroid singularities should be proved at the same time.

Chapter 0 begins with a rapid summary of some of the basic concepts in the language of schemes, and a definition of the general notion of blowing-up in terms of a universal mapping property. A proof of the existence theorem of blowing-up is outlined for certain categories. This leads to the definition and existence of monoidal transformations (and the centre of a monoidal transformation) in algebraic and analytic geometry. Let \(B\) be a commutative ring with unity. A \(\mathrm{Spec} (B)\)-scheme \(X\) is called an algebraic \(B\)-scheme if it is of finite type over \(\mathrm{Spec} (B)\). A point \(x\) of \(X\) is said to be simple (muitiple) if the local ring \(\mathcal O_{X,x}\), is (is not) regular. \(X\) is said to be noningular if each point of \(X\) is simple. The version of the resolution theorem (main theorem I) which is proved in the later chapters is formulated as follows. If \(X (= X_0)\) is an algebraic \(B\)-scheme which is reduced and irreducible, where B is a field of characteristic zero, then there exists a finite sequence of monoidal transformations \(f_i:X_{i+1}\to X_i\), \((i =0,1,\dots r-1)\) such that \(X_r\) is non-singular and (a) the centre \(D_i\) of \(f_i\) is non-singular and (b) no point of \(D_i\) is simple for \(X_i\). Let \(D\) be an algebraic subscheme of \(X\) defined by the sheaf of ideals \(\mathcal I\) on \(X\), and let \(\mathrm{gr}^p_D(X)\) be the quotient-sheaf \(\mathcal I^p/\mathcal I^{p+1}\) restricted to \(D\). \(X\) is said to be normally flat along \(D\) at a point \(x\) of \(D\) if the stalk of \(\mathrm{gr}^p_D(X)\) at \(x\) is a free \(\mathcal O_{D,p}\)-module for \(p = 0,1,2,\dots X\) is said to be normally flat along \(D\) if it is so at every point \(x\) of \(D\). It is natural to take the centre \(D_i\) of the monoidal transformation \(f_i\) to be equimultiple on \(X_i\), the present proof of the resolution theorem imposes (c) \(X_i\) is normally flat along \(D\). It is shown later that (c) is in fact a stronger condition than equi-multiplicity. The other notion that plays an important role in the proof of the resolution theorem is that of normal crossings. Let \(E\) be a reduced subscheme of a non-singular algebraic \(B\)-scheme \(X\) which is everywhere of codimension one. \(E\) is said to have only normal crossings at a point \(x\) of \(X\) if there exists a regular system of parameters \((z_1,\dots,z_n)\) of \(\mathcal O_{X,x}\), such that the ideal in \(O_{X,x}\) of each irreducible component of \(E\) which contains \(x\) is generated by one of the \(z_i\), \(E\) is said to have only normal crossings if it does so at every point of \(X\). Running alongside the inductive proof of the resolution theorem is the inductive proof of the author’s main theorem II which includes the following result. The complement of a Zariski open subset of a non-singular algebraic \(B\)-scheme (where \(B\) is a field of characteristic zero) can be transformed by a finite sequence of monoidal transformations with non-singular centres into a subscheme which has only normal crossings.

Chapter 0 also contains a discussion of the analogues of main theorems I and II in the analytic case, and indicates how his results can be used to prove the resolution theorem for an arbitrary real analytic space. For a complex analytic space the passage from the local to the global resolution of singularities apparently introduces added difficulties, and in this case the author claims a proof of the corresponding theorem for a complex analytic space of dimension \(\le 3\). The present methods make virtually no progress towards the resolution of singularities of algebraic \(B\)-schemes when \(B\) is a field of positive characteristic.

Chapter I begins by restricting the ring \(B\) to the class \(\mathcal B\) of noetherian local rings \(S\) with the properties (i) the residue field of \(S\) has characteristic zero and (ii) if \(A\) is an \(S\)-algebra of finite type and \(\hat{S}\) denotes the completion of \(S\) then, under the canonical morphism \(\mathrm{Spec} (A\otimes_S \hat{S})\to\mathrm{Spec} (A)\), the singular locus of the former is the preimage of that of the latter spectrum. An algebraic \(B\)-scheme with \(B\) in \(\mathcal B\) is called an algebraic scheme. The two main theorems of Chapter 0 are reformulated in terms of two types of resolution data giving four fundamental theorems. The fundamental theorems are of two types; two of the theorems are separation theorems for the resolution data, while the other two are resolution theorems which imply the main theorems of Chapter 0. For technical reasons the fundamental theorems are concerned only with algebraic schemes which have a given irreducible non-singular ambient scheme; the resolution theorem without such an ambient scheme is achieved by passing to the completion of a certain local ring, since every complete local ring in is a homomorphic image of a formal power series ring over a field of characteristic zero.

Chapter II is a self-contained study of normal flatness, and is of an algebraic nature. The theorems proved when interpreted geometrically imply the following results, among others.

- (1)
- The set of points of a reduced subscheme \(W\) of an algebraic scheme \(V\) at which \(V\) is normally flat along \(W\) form an open dense subset of \(W\).
- (2)
- When \(W\) is a non-singular irreducible subscheme of an algebraic scheme \(V\), then for \(V\) to be normally flat along \(W\) it is necessary that \(W\) should be equi-multiple on \(V\).
- (3)
- When \(W,V\) are as in (2) and \(V\) is embedded in a non-singular algebraic scheme \(X\) such that the sheaf of ideals of \(V\) on \(X\) is locally everywhere generated by a single non-zero element, then for \(V\) to be normally flat along \(W\) it is necessary and sufficient that \(W\) should be equi-muitiple on \(V\).

In Ch. III the local effect on singularities of permissible monoidal transformations is studied. Two local numerical characters are introduced as a measure of the severity of a singularity. In terms of these characters it is shown that a singularity is not made worse by any permissible monoidal transformation, and that any sequence of such transformations cannot make a singularity infinitely better. The aim of this local study is to prove the existence of a special coordinate system of the non-singular ambient scheme at a point and of a special base of the ideal defining the subscheme at the point, both of which have a certain stability with respect to the sequences of permissible monoidal transformations.

Chapter IV is devoted to the inductive proofs of the four fundamental theorems. Here the arguments are of a more geometric nature and rely on suitable geometric interpretations of the algebraic results in the two preceding chapters. In conclusion it is worth noting that the inductive proofs could not be carried out should the schemes be restricted to algebraic schemes over fields of characteristic zero; i.e. an essential part of the present proof of the resolution theorem for algebraic singularities is that the corresponding theorem for algebroid singularities should be proved at the same time.

Reviewer: David Kirby (Southampton)

##### MSC:

14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |