Notation and Conventions

In this paper, we always work over the complex number field C.

Notation:

ζ = ζ_r: a primitive r-th root of unity in C µr: the group of all r-th roots of unity in C e₁, e₂, . . . , e_n: the standard basis of Rⁿ

All schemes and varieties are assumed to be at least quasi-projective. For Q-Cartier divisors, the intersection numbers are defined by extending the intersection number of Cartier divisors byQ-linearity.

2 Singularities

2.1 Basic properties

We study singularities formally isomorphic to cyclic quotients of Aⁿ. First, we recall some facts about quotient varieties.

Given an affine variety X = SpecA with a finite group G action, we construct the affine quotient Y = X^G = SpecA^G and the natural quotient map f : X → Y induced by the inclusion A^G ↩→ A. When X is merely quasi-projective, one covers X with G-invariant affine open sets, and finds the affine quotients can be glued together. It is easy to see that

Fact 2.1. i) X is normal =⇒ Y is normal, ii) X is Q-factorial =⇒ Y is Q-factorial.

Conversely, given a normal variety Y , we may pick the normalization X of Y in a Galois extension over the function field of Y . Then Y = X^G, where G is the Galois group.

Definition 2.2. (Quotient singularities)

(i) Given a₁, . . . , a_n∈ N, let µr acts onAⁿ by

ζ· (x1, x₂, . . . , x_n)7→ (ζ^a1x₁, ζ^a2x₂, . . . , ζ^anx_n)

. Denote the quotient X₀ byAⁿ/µ_r orAⁿ/¹_r(a₁, . . . , a_n) to be the standard quotient singularity of type ¹_r(a1, . . . , an). Usually we assume gcd(a1, . . . , an) = 1.

(ii) Let p∈ X be a point. If locally near p, there is a map to a standard quotient singularity: φ : (p ∈ X) → (0 ∈ X0) inducing formal isomorphism ˆO0,X0 → ˆOp,X, we call (p∈ X) a quotient singularity of type ¹_r(a₁, . . . , a_n), or a ¹_r(a₁, . . . , a_n) point.

The morphism required in the definition is actually étale, that make it possible to pullback resolutions of standard quotient singularities to general ones.

Proposition 2.3. Keep the notation as in Definition 2.2. φ is étale near p.

Proof. Since étaleness is an open condition, it suffices to prove this for local ho-momorphism O0,X0 → Op,X. By faithful flatness of completion, we only need that Oˆ0,X0 → ˆOp,X is étale, which is an isomorphism by assumption, and hence the proof.

The following proposition says that quotients of smooth varieties by µ_r actions indeed give rise to quotient singularities.

Proposition 2.4. Let µr act on a smooth variety X, and p∈ X be an isolated fixed point. Let the quotient be (¯p∈ ¯X). Then it is a quotient singularity.

Proof. We may assume X = SpecA, an invariant open set, is affine. Since µ_r acts on the maximal ideal p, we get the eigenspace decomposition:

p =

⊕r−1 i=0

I_i

where the action is given by ζ· s = ζⁱs,∀s ∈ Ii.

We may choose s1, . . . sn ∈ A satisfying ζ · si = ζ^aisi so that they form a regular system of parameters of A_p, and obtain a morphism: s : A → Aⁿ defined by x1 = s1, . . . , xn= sn. s induces an isomorphism of completed local rings.

Let µ_r act on Aⁿ by ζ · (x1, x₂, . . . , x_n) 7→ (ζ^a1x₁, ζ^a2x₂, . . . , ζ^anx_n). Then s is equivariant, inducing the natural map ¯s : ¯X → X0 between quotient varieties. We need to check that ¯s also induces isomorphisms of completed local rings. This is done by Theorem2.5 below.

Theorem 2.5. Suppose A₁ → A2 is a G-equivariant local morphism such that the maximal ideals m1, m2 are G-invariant. Set Bi = A^G_i , for i = 1, 2. Then Bi is local.

If the induced map ˆA₁ → ˆA₂ is an isomorphism, then so is ˆB₁ → ˆB₂. We prove Theorem 2.5 with a series of lemmata.

Lemma 2.6. Suppose B ⊆ A be a finite extension of noetherian rings. Let I be an ideal of B and J_n = IⁿA∩ B. Then the I-adic and Jn-adic topologies are the same.

Proof. Firstly, Iⁿ⊆ Jn is trivial.

Conversely, T = A⊕ (⊕_∞

n=1(IA)ⁿ) is finite over B⊕ (⊕_∞

n=1Iⁿ), and hence so is the subalgebra S = B⊕ (⊕_∞

n=1J_n) of T .

This implies IJ_i = J_i+1 for i ≥ N for some N. We have then Jn = I^n−NJ_N ⊆ I^n−N.

Lemma 2.6 holds in particular for B = A^G, where G is a finite group acting on A. For p∈ SpecB, Bp = (Ap)^G. Now we assume (B, n) local. A is then semilocal.

Denote by m its Jacobson radical.

Corollary 2.7. i) In A, nA-adic and m-adic topologies are the same.

ii) In B, n-adic and mⁿ∩ B-adic topologies are the same.

Proof. i) Note that A/nA is finite over B/n, and hence an artin ring. √

nA = m.

ii) By Lemma2.6, n-adic, (nA)ⁿ∩ B-adic topologies, and mⁿ∩ B-adic topology are the same.

Lemma 2.8. Let ˆB be the n-adic completion of B, ˆA be the m-adic completion of A. Then ˆA = A⊗BB, and ˆˆ B ⊆ ˆA as topological subspace under ˆn-adic, and ˆm-adic topologies respectively.

Proof. Since A⊗BB is the nA-adic completion of A, which is the same as the m-ˆ adic completion of A (c.f. Corollary 2.7). Since ˆB is flat over B, we obtain the inclusion.

Now assume that in Theorem 2.5, |G| ∈ A^× is a unit. This holds in particular, when A contains a field k, and char k̸ ||G|.

Lemma 2.9. ˆB = ˆA^G.

Proof. Consider ι : B ↩→ A, and ϵ : A → B , which is defined by ϵ(x) = _|G|¹ ∑

g∈Gg·x.

Then ϵ◦ ι =id.

Tensoring with ˆB over B, we obtain that the composition ˆB ↩→ ˆA → ˆB is the identity map, and x = ϵ(x)∈ ˆB, for any x∈ ˆA^G.

We are now ready to prove Theorem 2.5.

Proof of Theorem 2.5. The theorem follows from taking G-invariants of the induced map ˆA₁ → ˆA₂.

Remark 2.10. Singularities of normal surfaces are always isolated. We shall call a surface quotient singularity a ¹_r(a, b) point, if the µ_r action is given by ζ · (x, y) 7→

(ζ^ax, ζ^by).

Quotient singularities only occur on the locus where the group action is not free.

First we state an application of Hurwitz formula.

Definition 2.11. Suppose that X = SpecA, Y = SpecB are normal varieties and f : X → Y is a finite map.

For any height-1 prime p of A, and q = p∩ B of B, the map f induces a local morphism of discrete valuation rings B_q → Ap. Denote the local parameters of A_p and B_q by s and t respectively. We define the ramification index e = e(p, q) to be the integer such that t = us^e for some u∈ A^×p/

Theorem 2.12. Suppose X = SpecA, Y = SpecB are normal varieties and f : X → Y is a finite map. Then KX = f^∗K_Y +∑

ht(p)=1(e(p, p∩ B) − 1)p, and the sum is a finite sum.

Proof. Denote the residue fields of A_p, B_q by k(p), k(q) respectively. Pick a tran-scendental basis y1, . . . , yn−1 ∈ B of k(q) over C, which is also a transcendental basis of k(p) over C. Consider the rational n-forms ω1 = ds∧ dy1∧ dy2. . .∧ dyn−1 and ω2 = dt∧ dy1∧ dy2. . .∧ dyn−1.

First we note that ω₁ is the local generator of ω_X at p. Indeed, by the second exact sequence of differential [10, II. 8.4], we have the following exact sequence

pA_p/(pA_p)² → ΩAp/C → Ωκ(p)/C → 0,

and thus Ω_X,p is generated by ds, dy₁, . . . , dy_n−1. Similarly, ω₂ is the local generator of ω_Y.

Fix a rational n-form ω on Y , we may also regard it as an n-form on X by pulling back. We find the coefficient c of q in K_Y satisfies ω = u₁t^cω₁, where u₁ ∈ A^×. The coefficient of p in f^∗K_Y is ec, and the coefficient c^′ of p in K_X satisfies ω = u₂s^c′ω₂, where u₂ ∈ A^×.

Now we have

ω2 = dt∧ dy1∧ dy2. . .∧ dyn−1

= d(us^e)∧ dy1∧ dy2. . .∧ dyn−1

= s^e−1(ue + sb)ω1

= s^e−1u^′ω₁. for some b∈ B, u^′ ∈ Bq^×.

We obtain c^′ = ec + (e− 1), which is the desired equality.

Remark 2.13. In terms of divisors, write KX + D = f^∗(KY + E), D = ∑ diDi,

E = ∑

b_jE_j. Suppose f (D_i) = E_j for some i, j, then d_i = b_je− (e − 1), i.e., (1− di) = e(1− bj).

Theorem 2.14. Suppose G is a finite group acting on the normal variety X, and the action is free generically. Let f : X → Y be the quotient variety. Then

K_X = π^∗K_Y +∑

(|GD| − 1)D, where GD is the subgroup fixing D.

Proof. We may assume that X = SpecA, Y = SpecB are affine, with fraction fields K, L respectively. Then L = K^G.

We only need to check that for any prime divisor D, corresponding to height-1 prime p of A, we have e(p, q) = |GD|, where q = p ∩ B. Consider the Dedekind domains B_q ⊆ Aq = A⊗BB_q.

Denote the subgroups I_p ={g | g(a) − a ∈ pAq,∀a ∈ Aq} and Dp ={g | g(a) ∈ pAq,∀a ∈ pAq} of G. By Hilbert’s ramification theory (cf. [15, I Prop.9.4]) and since in characteristic 0, the residue field extension k(p)/k(q) is always separable,

we know that

the subgroup G_D fixing D, and hence the proof.

Definition 2.15. Let G be a finite group acting on a variety X. Say the action is free if Φ : G× X → X × X defined by (g, x) 7→ (g · x, x) is a closed immersion.

The definition may seem strange at the first glance. However, the following lemma tells us that, it is the same as the intuitive definition. The advantage is that Definition 2.15 can be generalized straightforwardly to algebraic group actions on any schemes.

Lemma 2.16. Let G be a finite group acting on a variety X. The action is free if and only if for any closed point m∈ X, there is no non-identity element g ∈ G, such that g· m = m.

Conversely, we are given that{g(a) − a | a ∈ A} generates A for all non-identity elements g. For g ̸= h , we also have {g(a)−h(a) | a ∈ A} generates A. That is, there are a_i, b_i such that ∑

ig(a_i)b_i −∑

ih(a_i)b_i = 1. So the components corresponding to g and h of ∑

ia_i⊗ bi−∑

i1⊗ h(ai)b_i are 1 and 0 respectively. From this, we see that ϕ is a surjective ring homomorphism.

Proposition 2.17. Suppose G is a finite group acting on an integral scheme X = SpecA freely. Then A is locally free over B = A^G of rank |G|. Moreover, A is étale over B.

Proof. For p∈ Spec B, let k be the algebraic closure of the residue field k(p). We observe that the G also acts freely on the geometric fiber X_p = SpecA^′, where A^′ = A⊗B k . Indeed, the surjective homomorphism ϕ : A⊗CA → A^|G| remains surjective after tensoring with k. Moreover, since we have the splitting ϵ : A→ B ϵ(a) = _|G|¹ ∑

g∈Gg(a) in characteristics zero, tensoring with k gives we A^′G = k.

Now, by the structure theorem of artin rings, A^′ =A₁ × A2 × . . . × Am, and there is a canonically defined subring A0 = k × . . . × k (|G| copies) such that the composition A₀ ↩→ A^′ → A^′/√

0 ∼= A0 is the identity map. G also acts on A₀ by permuting components. Since we must have A^G₀ = k, G acts transitively on components. From this we see all A_i are isomorphic, and G acts on A^′ by permuting components as well. But then A^′G = {(x, x, . . . , x)|x ∈ A1} = k. We must have A₁ = k, and A^′ = A₀

Counting dimensions over k gives dimkA^′ =|G|. We find dimk(p)A⊗ k(p) = |G|

for all p ∈ SpecB, and hence A is locally free over B of rank |G|. A being flat, we only need to check A over B is unramified, which can be checked on geometric fibers. But A^′ = k× . . . × k, it is then obvious that ΩA^′/k = 0.

Remark 2.18. When no divisor of X is fixed by a non-identity element in G, we see the quotient map π : X → Y is étale in codimension 1, and the Hurwitz formula reads K_X = π^∗K_Y.