Alterations and resolution of singularities Lecture 1a: overview

1. Objects

1.1. Definition. A variety defined over k is a geometrically integral scheme of finite type over k. If k ⊂ k_{1} we write X_{k_{1}} for X ×_{Spec k} Spec k_{1}.

1.2. Definition. Let X be a variety. A subscheme Z ⊂ X is called a strict normal crossings divisor if for each point x ∈ Z, there is a regular system of parameters y_{1}, …, y_{k} for x in X (in particular the point x ∈ X is supposed to be a regular point on X), such that Z is given on a Zariski neighborhood of x by the equation y_{1} ⋯ y_{l} = 0.

2. Operations

2.1. Definition. A morphism of varieties Y → X is called a modification if it is proper and birational.

2.2. Definition (de Jong). A morphism of varieties Y → X is called an alteration if it is proper, surjective and generically finite. This notion of alteration will also be used for integral schemes.

2.3. Definition. Let X be a variety. A resolution of singularities in the weak sense is a modification Y → X such that Y is nonsingular.

2.4. Definition. Let X be a variety. A resolution of singularities in the strong sense is a modification Y → X, which is an isomorphism over the nonsingular locus X_{reg}, such that Y is nonsingular.

3. Results

3.1. Theorem (Hironaka). Suppose X is a variety over a fied k of characteristic 0. There is a sequence of blowings up X_{n} → ⋯ → X_{0} = X, each along a regular variety lying over SingX, such that X_{n} is regular. Thus X_{n} → X is a resolution of singularities in the strong sense.

3.2. Theorem. Let X be a variety over an algebraically closed field. There is a separable alteration Y → X such that Y is quasi projective and regular.

3.3. Corollary. Let X/k be a variety. There is a finite extension k ⊂ k_{1} and a separable alteration Y → X_{k_{1}} such that Y is quasi projective and regular.

3.4. Theorem (de Jong). Let X be a variety over an algebraically closed field, Z ⊂ X a proper closed subset. There is a separable alteration f : Y → X, and an open immersion j : Y ⊂ Ȳ, such that Ȳ is projective and regular, and the subset j(f^{−1}Z) ∪ (Ȳ ^{\} Y ) is a strict normal crossings divisor.

3.5. Theorem of semistable reduction up to alteration (de Jong). Let R be a discrete valuation ring, with fraction field K and residue field k. Let X → Spec R be an integral scheme of finite type such that X_{K} is a variety. There exists a finite extension R ⊂ R_{1}, where R_{1} is a discrete valuation ring with residue field k_{1}, and an alteration Y → X_{R_{1}}, such that Y is nonsingular, the special fiber Y_{k_{1}} is a reduced, strict normal crossings divisor.

3.6. Theorem of equivariant resolution up to alteration (de Jong). Let X be a variety over an algebraically closed field, Z ⊂ X a proper closed subset, and a finite group G ⊂ Aut(Z ⊂ X). There is an alteration f : Y → X, and a finite subgroup G_{1} ⊂ Aut Y, satisfying:

1. there is a surjection G_{1} → G such that f is G_{1} equivariant, and the field extension K(X)^{G} ⊂ K(Y)^{G_{1}} is purely inseparable;

2. Y is quasi projective and nonsingular; and 3. f^{−1}Z is a G-strict normal crossings divisor.

3.7. Corollary. Let X be a variety over an algebraically closed field. There is a purely inseparable alteration Y → X such that Y is a quotient of a nonsingular variety by the action of a finite group.

3.8. Weak Resolution Theorem. Let X be a variety over an algebraically closed field of characteristic 0. Then there is a projective resolution of singularities (weak sense) Y → X.

3.9. Question. Can we improve the methods and obtain a weak resolution of singularities in all characteristics? At least weak resolution up to purely inseparable alterations?