Alterations and resolution of singularities Lecture 4: Weak resolution in characteristic 0

The main result today is:

0.1. Theorem. Let X be a variety over an algenraically closed field of characteristic 0, Z ⊂ X a proper closed subset. Then there exists a modification f : X′ → X such that X′ is regular and f^{−1}Z = Z′ is a simple normal crossings divisor. We present a proof by ℵ - de Jong.

0.2. simple reductions. As in de Jong’s theorem, we may assume:

1. X is projective

2. Z is a divisor

3. X is normal using this, we can project and reduce to the case:

4. There exists a projection X → P, equidimensional of relative dimension 1, with generically reduced connected fibers, and generic fiber smooth, such that Z → P is finite

5. We may enlarge Z as in de Jong’s theorem so it intersects each component of a fiber in at least 3 smooth points We can now repeat the stable reduction argument and get

\begin{aligned}X_{1} & → & X \\ ↓ & & ↓ \\ P_{1} & → & P \end{aligned}

such that 0.3. making P_{1} → P Galois. Note that P_{1} → P is generically étale. We can actually replace it by a Galois morphism: Let K(P) ⊂ L be the Galois closure of K(P) ⊂ K(P_{1}), with Galois group G. Let Q be the normalization of P in L. we have a natural rational map h : Q −−→ P_{1}, so for each g ∈ G we have h ∘ g : Q −−→ P_{1}. The closure P_{2} of the graph of the product map

Q → ∏_{g∈G} P_{1}

is G equivariant, and admits a morphism to P_{1}. Thus we can pull backX_{1} to P_{2}, and therefore we may as well assume that G acts on P_{1} so that P_{1}/G → P is birational.

We want G to respect X_{1} as well. We use the following:

0.4. Fact. (Separatedness of the moduli problem) Let B be normal, U ⊂ B be a dense open, and C_{i} → B two families of stable pointed curves. If (C_{1})|_{U} ≃ (C_{2})|_{U} over U then C_{1} ≃ C_{2} over B.

0.5. In our situation: let U ⊂ P be the open subset over which X is smooth and Z is étale. Let U_{1} ⊂ P_{1} the inverse image. Over U_{1} we have X_{1}|_{U_{1}} ≃ X×_{P} U_{1}, and ∪s_{i}(U_{1}) = Z×_{P} U_{1}. The group G acts on X ×_{P} U_{1} and Z ×_{P} U_{1}, thus permuting s_{i}(U_{1}). Thus we have X_{1}|_{U_{1}} ⊂ g^{∗}X_{1} for every g ∈ G. By the fact above these are all isomorphic (up to permuting the sections). Thus we have:

0.6. The action of G on P_{1} lifts to X_{1}, permuting the sections s_{i}.

We may now replace P by P_{1}/G, X by X_{1}/G and Z by Z_{1}/G. So the alterations are finite quotient maps. By induction on dimension we may resolve (P, Δ). We pull back X etc.. so we may assume P is nonsingular, and Δ, which includes the degeneracy locus of X, Z and P_{1}, is a strict normal crossings divisor. In particular (P ^{\} Δ) ⊂ P is a toroidal embedding!

0.7. Abhyankar’s lemma. If B nonsingular, D ⊂ B a d.n.c., and B_{1} → B finite, (tamely) ramified in D only, then B_{1} is toroidal. If Galois then the Galois group acts toroidally.

0.8. Proof. The formal, 1-dimensional case is obvious by the Galois theory of a power series ring. The general case follows by the Nagata-Zariski purity theorem.

0.9. Family of nodal curves. Let U ⊂ B be a toroidal embedding. Let π : C → B, s_{i} : B → C be a family of pointed nodal curves, nondegenerate over U. Then U_{C} ⊂ C is toroidal where U_{C} = π^{−1}(U)^{\} ∪ s_{i}(B).

0.10. Proof. Local equation can be written in the form uv = m, where m is a monomial. Conclusion: P_{1} is toroidal, with G acting toroidally. X_{1} is toroidal, with G acting. If G acted toroidally we would be done!

0.11. node case. Local equations for X_{1} are uv = m. now m is an eigenvector. if G does not switch the branches u = 0 and v = 0 we could choose u, v eigenvectors and be happy. To correct this, blowup Singf_{1} = zero scheme of (u, v), and normalize

0.12. Smooth (or double) case. X_{1} is locally SpecR[u], where R is the ring of P_{1}. Can choose u as an eigenvector for G_{x}. but u = 0 might not be uniquely determined, and may not be part of the toroidal divisor!!

0.13. Theorem (ℵ - de Jong). There exists a canonical G equivariant blowup X̃_{1} → X_{1} such that G_{x} acts toroidally on X̃_{1}. Thus X̃_{1}/G is toroidal, and we can resolve.

0.14. Proof. Let I_{G} be the ideal generated by u and all the monomials in R on which G acts via the same character. Then the normalized blowup { \atop B_{I_{G}}(X_{1})} does the trick!