Smoothness, semi-stability and alterations.

*(English)*Zbl 0916.14005From the introduction: ”Let \(X\) be a variety over a field \(k\). An alteration of \(X\) is a dominant proper morphism \(X'\to X\) of varieties over \(k\), with \(\dim X=\dim X'\). We prove that any variety has an alteration which is regular. This is weaker than resolution of singularities in that we allow finite extensions of the function field \(k(X)\). In fact, we can choose \(X'\) to be a complement of a divisor with strict normal crossings in some regular projective variety \(\overline X'\) (see theorem 4.1 and remark 4.2.). If the field \(k\) is local, we can find \(X' \subset \overline X'\) such that \(\overline X'\) is actually defined over a finite extension \(k\subset k'\) and has semi-stable reduction over \({\mathcal O}_{k'}\) in the strongest possible sense (see theorem 6.5).

As an application, we note that theorem 4.1 implies that for any variety \(X\) over a perfect field \(k\), there exist \((\alpha)\) a simplicial scheme \(X_\bullet\) projective and smooth over \(k\), \((\beta)\) a strict normal crossings divisor \(D_\bullet\) in \(X_\bullet\); we put \(U_\bullet=X_\bullet\setminus D_\bullet\), and \((\gamma)\) an augmentation \(a:U_\bullet\to X\) which is a proper hypercovering of \(X\). In case \(k\) is local, we may assume that the pairs \((X_n,D_n)\) are defined over finite extensions \(k_n\) of \(k\) and extend to strict semi-stable pairs over \({\mathcal O}_{k_n}\) (see 6.3). This should be interpreted as saying that in a suitable category \({\mathcal M}{\mathcal M}_k\) of mixed motives over \(k\), any variety \(X\) may be replaced by a complex of varieties which are complements of strict normal crossing divisors in smooth projective varieties.

We give a short sketch of the argument that proves our results in case \(X\) is a proper variety. The idea is to fibre \(X\) over a variety \(Y\) such that all fibres are curves and work by induction on the dimension of \(X\). After modifying \(X\), we may assume \(X\) is projective and normal and we can choose the fibration to be a kind of Lefschetz pencil, where the morphism is smooth generically along any component of any fibre. Next one chooses a sufficiently general and sufficiently ample relative divisor \(H\) on \(X\) over \(Y\). After altering \(Y\), i.e. we take a base change with an alteration \(Y'\to Y\), we may assume that \(H\) is a union of sections \(\sigma_i:Y\to X\). The choice of \(H\) above gives that for any component of any fibre of \(X\to Y\), there are at least three sections \(\sigma_i\) intersecting it in distinct points of the smooth locus of \(X\to Y\). The generic fibre of \(X\to Y\), together with the points determined by the \(\sigma_i\) is a stable pointed curve. By the existence of proper moduli spaces of stable pointed curves, we can replace \(Y\) by an alteration such that this extends to a family \({\mathcal C}\) with sections \(\tau_i\) of stable pointed curves over \(Y\). An important step is to show that the rational morphism \({\mathcal C}\cdots\to X\) extends to a morphism, possibly after replacing \(Y\) by a modification; this follows from the condition on sections hitting components of fibres above. Thus we see that we may replace \(X\) by \({\mathcal C}\). We apply the induction hypothesis to \(Y\) and we get \(Y\) regular. However, our induction hypothesis is actually stronger and we may assume that the locus of degeneracy of \({\mathcal C}\to Y\) is a divisor with strict normal crossings. At this point it is clear that the only singularities of \({\mathcal C}\) are given by equations of the type \(xy=t_1^{n_1} \cdot\dots \cdot t_d^{n_d}\). These we resolve explicitly.

Section 2 contains definitions and results, which we assume known in the rest of the paper. In section 3 we resolve singularities for a family of semi-stable curves over a regular scheme, which is degenerate over a divisor, with normal crossings. This we use in section 4, where we prove the theorem on varieties. Section 5 deals with the problem of altering a family of curves into a family of semi-stable curves. This we use in section 6, where we do the relative case, i.e. the case of schemes over a complete discrete valuation ring.

In the final two sections we indicate how to refine the method of proof of theorem 4.1 and theorem 6.5 to get results where one has additional restraints or works over other base schemes. In section 7 we prove that our method works (over algebraically closed fields) to get resolution of singularities up to quotient singularities and purely inseparable function field extensions. In fact we deal with the situation where there is a finite group acting. In section 8 we do the arithmetic case. In particular, we show that any integral scheme \(X\), flat and projective over \(\text{Spec} \mathbb{Z}\) can be altered into a scheme \(Y\) which is semi-stable over the ring of integers in a number field (theorem 8.2).

In a follow-up of this article the author proves that one can alter any family of curves into a semi-stable family of curves [see A. J. de Jong, Ann. Inst. Fourier 47, No. 2, 599-621 (1977; Zbl 0868.14012)]. This is stronger than the result of section 5. In this cited paper the author deals with group actions as well. Thus the reader can find therein a number of results that extend the results of this article to (slightly) more general situations. For example it is shown that regular alterations exist of schemes of finite type over two-dimensional excellent base schemes.

A.Nobile:

This very important article describes a solution to a weak version of the desingularization problem for algebraic varieties over a field, as well as generalizations. The key concept to express the main results is that of alteration. An alteration of an integral scheme \(S\) is an integral scheme \(S'\) together with a morphism \(\varphi:S'\to S\) which is surjective, proper and such that, for a suitable open dense set \(U\subseteq X\), the induced morphism \(\varphi_U:\varphi^{-1}(U)\to U\) is finite.

The basic main result of this paper says: Given a variety \(X\) over an arbitrary field \(k\) and a proper closed set \(Z\subset X\), then there is an alteration \(\varphi:X_1\to X\) such that \(X_1\) is an open set of a regular scheme \(X_1'\), projective over \(k\), such that \(\varphi^{-1}(Z) \cup(X_1'-X_1)\) is a strict normal crossings divisor \(D\) of \(X_1'\) (i.e., the irreducible components of \(D\) are regular and meet transversally). There are also: (a) a \(G\)-equivariant version, where \(G\) is a finite group acting on \(X\), (b) a “relative” version, where \(X\) is an irreducible, separated scheme, flat and of finite type over \(S=\text{Spec}\,R\), with \(R\) a complete discrete valuation ring; (c) an arithmetic version, where the basic situation is as in part (b), but now \(R\) is a Dedekind domain whose field of fractions is a global field. – The result of (b) may be interpreted as “weak” semistable reduction theorem (weak because certain alterations are allowed), without restrictions on the characteristics involved. As a tool to obtain (b) or (c), some interesting results about improving a family of curves via alterations are discussed.

This is the main technique to show the relevant theorems, sketched in the case of varieties over a field.

Moreover, to simplify \(X\) is assumed projective. One tries to find an alteration \(f:X'\to X\) such that there is flat morphism \(g:X'\to T\), with \(T\) regular, \(\dim(T)= \dim (X')-1\), such that the general fiber of \(g\) is regular and any fiber is a curve with, at worst, ordinary double points as singularities, moreover the set of points of \(T\) where \(g^{-1}(t)\) is singular is contained in a strict normal crossings divisor. Then, to desingularize such a \(X'\) by means of monoidal transformations is easy. An alteration as above is obtained by using some classical projective techniques, the theory of moduli for pointed semi-stable curves and an induction hypothesis (applied to \(T\), whose dimension is one less than that of \(X)\).

These results, although in a sense weak (because they involve alterations and not birational morphisms) are strong enough to solve a number of cohomological problems that “require” resolutin) of singularities. Some are described in the introduction. See also: P. Berthelot, “Altérations de varietés algébriques, Séminaire Bourbaki, Volume 1995/96, Exposé No. 815, Astérisque 24l, 273–311 (1997; Zbl 0924.14087)].

As an application, we note that theorem 4.1 implies that for any variety \(X\) over a perfect field \(k\), there exist \((\alpha)\) a simplicial scheme \(X_\bullet\) projective and smooth over \(k\), \((\beta)\) a strict normal crossings divisor \(D_\bullet\) in \(X_\bullet\); we put \(U_\bullet=X_\bullet\setminus D_\bullet\), and \((\gamma)\) an augmentation \(a:U_\bullet\to X\) which is a proper hypercovering of \(X\). In case \(k\) is local, we may assume that the pairs \((X_n,D_n)\) are defined over finite extensions \(k_n\) of \(k\) and extend to strict semi-stable pairs over \({\mathcal O}_{k_n}\) (see 6.3). This should be interpreted as saying that in a suitable category \({\mathcal M}{\mathcal M}_k\) of mixed motives over \(k\), any variety \(X\) may be replaced by a complex of varieties which are complements of strict normal crossing divisors in smooth projective varieties.

We give a short sketch of the argument that proves our results in case \(X\) is a proper variety. The idea is to fibre \(X\) over a variety \(Y\) such that all fibres are curves and work by induction on the dimension of \(X\). After modifying \(X\), we may assume \(X\) is projective and normal and we can choose the fibration to be a kind of Lefschetz pencil, where the morphism is smooth generically along any component of any fibre. Next one chooses a sufficiently general and sufficiently ample relative divisor \(H\) on \(X\) over \(Y\). After altering \(Y\), i.e. we take a base change with an alteration \(Y'\to Y\), we may assume that \(H\) is a union of sections \(\sigma_i:Y\to X\). The choice of \(H\) above gives that for any component of any fibre of \(X\to Y\), there are at least three sections \(\sigma_i\) intersecting it in distinct points of the smooth locus of \(X\to Y\). The generic fibre of \(X\to Y\), together with the points determined by the \(\sigma_i\) is a stable pointed curve. By the existence of proper moduli spaces of stable pointed curves, we can replace \(Y\) by an alteration such that this extends to a family \({\mathcal C}\) with sections \(\tau_i\) of stable pointed curves over \(Y\). An important step is to show that the rational morphism \({\mathcal C}\cdots\to X\) extends to a morphism, possibly after replacing \(Y\) by a modification; this follows from the condition on sections hitting components of fibres above. Thus we see that we may replace \(X\) by \({\mathcal C}\). We apply the induction hypothesis to \(Y\) and we get \(Y\) regular. However, our induction hypothesis is actually stronger and we may assume that the locus of degeneracy of \({\mathcal C}\to Y\) is a divisor with strict normal crossings. At this point it is clear that the only singularities of \({\mathcal C}\) are given by equations of the type \(xy=t_1^{n_1} \cdot\dots \cdot t_d^{n_d}\). These we resolve explicitly.

Section 2 contains definitions and results, which we assume known in the rest of the paper. In section 3 we resolve singularities for a family of semi-stable curves over a regular scheme, which is degenerate over a divisor, with normal crossings. This we use in section 4, where we prove the theorem on varieties. Section 5 deals with the problem of altering a family of curves into a family of semi-stable curves. This we use in section 6, where we do the relative case, i.e. the case of schemes over a complete discrete valuation ring.

In the final two sections we indicate how to refine the method of proof of theorem 4.1 and theorem 6.5 to get results where one has additional restraints or works over other base schemes. In section 7 we prove that our method works (over algebraically closed fields) to get resolution of singularities up to quotient singularities and purely inseparable function field extensions. In fact we deal with the situation where there is a finite group acting. In section 8 we do the arithmetic case. In particular, we show that any integral scheme \(X\), flat and projective over \(\text{Spec} \mathbb{Z}\) can be altered into a scheme \(Y\) which is semi-stable over the ring of integers in a number field (theorem 8.2).

In a follow-up of this article the author proves that one can alter any family of curves into a semi-stable family of curves [see A. J. de Jong, Ann. Inst. Fourier 47, No. 2, 599-621 (1977; Zbl 0868.14012)]. This is stronger than the result of section 5. In this cited paper the author deals with group actions as well. Thus the reader can find therein a number of results that extend the results of this article to (slightly) more general situations. For example it is shown that regular alterations exist of schemes of finite type over two-dimensional excellent base schemes.

A.Nobile:

This very important article describes a solution to a weak version of the desingularization problem for algebraic varieties over a field, as well as generalizations. The key concept to express the main results is that of alteration. An alteration of an integral scheme \(S\) is an integral scheme \(S'\) together with a morphism \(\varphi:S'\to S\) which is surjective, proper and such that, for a suitable open dense set \(U\subseteq X\), the induced morphism \(\varphi_U:\varphi^{-1}(U)\to U\) is finite.

The basic main result of this paper says: Given a variety \(X\) over an arbitrary field \(k\) and a proper closed set \(Z\subset X\), then there is an alteration \(\varphi:X_1\to X\) such that \(X_1\) is an open set of a regular scheme \(X_1'\), projective over \(k\), such that \(\varphi^{-1}(Z) \cup(X_1'-X_1)\) is a strict normal crossings divisor \(D\) of \(X_1'\) (i.e., the irreducible components of \(D\) are regular and meet transversally). There are also: (a) a \(G\)-equivariant version, where \(G\) is a finite group acting on \(X\), (b) a “relative” version, where \(X\) is an irreducible, separated scheme, flat and of finite type over \(S=\text{Spec}\,R\), with \(R\) a complete discrete valuation ring; (c) an arithmetic version, where the basic situation is as in part (b), but now \(R\) is a Dedekind domain whose field of fractions is a global field. – The result of (b) may be interpreted as “weak” semistable reduction theorem (weak because certain alterations are allowed), without restrictions on the characteristics involved. As a tool to obtain (b) or (c), some interesting results about improving a family of curves via alterations are discussed.

This is the main technique to show the relevant theorems, sketched in the case of varieties over a field.

Moreover, to simplify \(X\) is assumed projective. One tries to find an alteration \(f:X'\to X\) such that there is flat morphism \(g:X'\to T\), with \(T\) regular, \(\dim(T)= \dim (X')-1\), such that the general fiber of \(g\) is regular and any fiber is a curve with, at worst, ordinary double points as singularities, moreover the set of points of \(T\) where \(g^{-1}(t)\) is singular is contained in a strict normal crossings divisor. Then, to desingularize such a \(X'\) by means of monoidal transformations is easy. An alteration as above is obtained by using some classical projective techniques, the theory of moduli for pointed semi-stable curves and an induction hypothesis (applied to \(T\), whose dimension is one less than that of \(X)\).

These results, although in a sense weak (because they involve alterations and not birational morphisms) are strong enough to solve a number of cohomological problems that “require” resolutin) of singularities. Some are described in the introduction. See also: P. Berthelot, “Altérations de varietés algébriques, Séminaire Bourbaki, Volume 1995/96, Exposé No. 815, Astérisque 24l, 273–311 (1997; Zbl 0924.14087)].

Reviewer: Augusto Nobile (Baton Rouge)

##### MSC:

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

14H10 | Families, moduli of curves (algebraic) |

14B05 | Singularities in algebraic geometry |

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\textit{A. J. de Jong}, Publ. Math., Inst. Hautes Étud. Sci. 83, 51--93 (1996; Zbl 0916.14005)

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