[PPT] - Resolution and logarithmic resolution by weighted blowing up Dan PowerPoint Presentation

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Resolution and logarithmic resolution by weighted blowing up

Dan Abramovich, Brown University Work with Michael T¨ emkin and Jaros law W lodarczyk and work by Ming Hao Quek Also parallel work by M. McQuillan Algebraic geometry and Moduli Seminar ETH Z¨ urich, July 15, 2020

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How to resolve

To resolve a singular variety X one wants to (1) find the worst singular locus S ⊂ X, (2) Hopefully S is smooth - blow it up.

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How to resolve

To resolve a singular variety X one wants to (1) find the worst singular locus S ⊂ X, (2) Hopefully S is smooth - blow it up.

Fact

This works for curves but not in general.

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Example: Whitney’s umbrella

Consider X = V (x2 − y2z)

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Example: Whitney’s umbrella

Consider X = V (x2 − y2z) (1) The worst singularity is the origin. (2) In the z chart we get x = x′z, y = y′z, giving x′2z2 − y′2z3 = 0,

r

z2(x′2 − y′2z) = 0.

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Example: Whitney’s umbrella

Consider X = V (x2 − y2z) (1) The worst singularity is the origin. (2) In the z chart we get x = x′z, y = y′z, giving x′2z2 − y′2z3 = 0,

r

z2(x′2 − y′2z) = 0. The first term is exceptional, the second is the same as X.

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Two theorems

Nevertheless:

Theorem (ℵ-T-W, McQuillan, 2019, characteristic 0)

There is a functor F associating to a singular subvariety X ⊂ Y of a smooth variety Y , a center ¯ J with stack theoretic weighted blowing up Y ′ → Y and proper transform (X ′ ⊂ Y ′) = F(X ⊂ Y ) such that maxinv(X ′) < maxinv(X). In particular, for some n the iterate (Xn ⊂ Yn) := F ◦n(X ⊂ Y ) of F has Xn smooth.

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Two theorems

Nevertheless:

Theorem (ℵ-T-W, McQuillan, 2019, characteristic 0)

There is a functor F associating to a singular subvariety X ⊂ Y of a smooth variety Y , a center ¯ J with stack theoretic weighted blowing up Y ′ → Y and proper transform (X ′ ⊂ Y ′) = F(X ⊂ Y ) such that maxinv(X ′) < maxinv(X). In particular, for some n the iterate (Xn ⊂ Yn) := F ◦n(X ⊂ Y ) of F has Xn smooth.

Theorem (Quek, 2020, characteristic 0)

There is a functor F associating to a logarithmically singular subvariety X ⊂ Y of a logarithmically smooth variety Y , a logarithmic center ¯ J with stack theoretic logarithmic blowing up Y ′ → Y and proper transform (X ′ ⊂ Y ′) = F(X ⊂ Y ) such that maxloginv(X ′) < maxloginv(X). In particular, for some n the iterate (Xn ⊂ Yn) := F ◦n(X ⊂ Y ) of F has Xn logarithmically smooth.

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Context: families

Hironaka’s theorem resolves varieties. What can you do with families of varieties X → B?

Theorem (ℵ-Karu, 2000)

There is a modification X ′ → B′ which is logarithmically smooth.

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Context: families

Hironaka’s theorem resolves varieties. What can you do with families of varieties X → B?

Theorem (ℵ-Karu, 2000)

There is a modification X ′ → B′ which is logarithmically smooth. Logarithmically smooth = toroidal:

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Context: families

Hironaka’s theorem resolves varieties. What can you do with families of varieties X → B?

Theorem (ℵ-Karu, 2000)

There is a modification X ′ → B′ which is logarithmically smooth. Logarithmically smooth = toroidal: A toric morphism X → B of toric varieties is a torus equivariant morphism.

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Context: families

Hironaka’s theorem resolves varieties. What can you do with families of varieties X → B?

Theorem (ℵ-Karu, 2000)

There is a modification X ′ → B′ which is logarithmically smooth. Logarithmically smooth = toroidal: A toric morphism X → B of toric varieties is a torus equivariant morphism. A toroidal embedding UX ⊂ X is an open embedding ´ etale locally isomorphic to toric T ⊂ V .

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Context: families

Hironaka’s theorem resolves varieties. What can you do with families of varieties X → B?

Theorem (ℵ-Karu, 2000)

There is a modification X ′ → B′ which is logarithmically smooth. Logarithmically smooth = toroidal: A toric morphism X → B of toric varieties is a torus equivariant morphism. A toroidal embedding UX ⊂ X is an open embedding ´ etale locally isomorphic to toric T ⊂ V . A toroidal morphism X → B of toroidal embeddings is ´ etale locally isomorphic to a toric morphism.

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Examples of toroidal morphisms

A toric morphism X → B of toric varieties is a torus equivariant morphism.

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Examples of toroidal morphisms

A toric morphism X → B of toric varieties is a torus equivariant morphism.e.g. Spec C[x, y, z]/(xy − z2) → Spec C,

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Examples of toroidal morphisms

A toric morphism X → B of toric varieties is a torus equivariant morphism.e.g. Spec C[x, y, z]/(xy − z2) → Spec C, Spec C[x] → Spec C[x2],

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Examples of toroidal morphisms

A toric morphism X → B of toric varieties is a torus equivariant morphism.e.g. Spec C[x, y, z]/(xy − z2) → Spec C, Spec C[x] → Spec C[x2], toric blowups

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Context: functoriality

Hironaka’s theorem is functorial. [ℵ-Karu 2000] is not: relied on deJong’s method.

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Context: functoriality

Hironaka’s theorem is functorial. [ℵ-Karu 2000] is not: relied on deJong’s method. For K–S-B or K-moduli want functoriality.

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Context: functoriality

Hironaka’s theorem is functorial. [ℵ-Karu 2000] is not: relied on deJong’s method. For K–S-B or K-moduli want functoriality.

Theorem (ℵ-T-W 2020)

Given X → B there is a relatively functorial logarithmically smooth modification X ′ → B′.

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Context: functoriality

Hironaka’s theorem is functorial. [ℵ-Karu 2000] is not: relied on deJong’s method. For K–S-B or K-moduli want functoriality.

Theorem (ℵ-T-W 2020)

Given X → B there is a relatively functorial logarithmically smooth modification X ′ → B′. This respects AutB X. Does not modify log smooth fibers.

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Context: principalization

Following Hironaka, the above theorem is based on embedded methods:

Theorem (ℵ-T-W 2020)

Given Y → B logarithmically smooth and I ⊂ OY , there is a relatively functorial logarithmically smooth modification Y ′ → B′ such that IOY ′ is monomial.

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Context: principalization

Following Hironaka, the above theorem is based on embedded methods:

Theorem (ℵ-T-W 2020)

Given Y → B logarithmically smooth and I ⊂ OY , there is a relatively functorial logarithmically smooth modification Y ′ → B′ such that IOY ′ is monomial. This is done by a sequence of logarithmic modifications, where in each step E becomes part of the divisor DY ′.

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Example 1

Y = Spec k[x, u]; DY = V (u); B = Spec k;

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Example 1

Y = Spec k[x, u]; DY = V (u); B = Spec k; I = (x2, u2).

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Example 1

Y = Spec k[x, u]; DY = V (u); B = Spec k; I = (x2, u2). Blow up J = (x, u) IOY ′ = O(−2E)

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Example 1/2

Y = Spec k[x, u]; DY = V (u); I = (x2, u2)

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Example 1/2

Y = Spec k[x, u]; DY = V (u); I = (x2, u2) Y0 = Spec k[x, v]; DY0 = V (v); I0 = (x2, v), f : Y → Y0 v = u2 so I = f ∗I0

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Example 1/2

Y = Spec k[x, u]; DY = V (u); I = (x2, u2) Y0 = Spec k[x, v]; DY0 = V (v); I0 = (x2, v), f : Y → Y0 v = u2 so I = f ∗I0 By functoriality blow up J0 so that f ∗J0 = J = (x, u).

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Example 1/2

Y = Spec k[x, u]; DY = V (u); I = (x2, u2) Y0 = Spec k[x, v]; DY0 = V (v); I0 = (x2, v), f : Y → Y0 v = u2 so I = f ∗I0 By functoriality blow up J0 so that f ∗J0 = J = (x, u). Blow up J0 = (x, √v) Whatever J0 is, the blowup is a stack.

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Example 1/2: charts

x chart: v = v′x2: (x2, v) = (x2, v′x2) = (x2) exceptional, so monomial. √v chart: v = w2, x = x′w, with ±1 action (x′, w) → (−x′, −w): (x2, v) = (x′2w2, w2) = (w2) exceptional, so monomial. The schematic quotient of the above is not toroidal.

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Resolution again

Theorem (ℵ-T-W, McQuillan, characteristic 0)

There is a functor F associating to a singular subvariety X ⊂ Y of a smooth variety Y , a center ¯ J with stack theoretic weighted blowing up Y ′ → Y and proper transform (X ′ ⊂ Y ′) = F(X ⊂ Y ) such that maxinv(X ′) < maxinv(X). In particular, for some n the iterate (Xn ⊂ Yn) := F ◦n(X ⊂ Y ) of F has Xn smooth.

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Resolution again

Theorem (ℵ-T-W, McQuillan, characteristic 0)

There is a functor F associating to a singular subvariety X ⊂ Y of a smooth variety Y , a center ¯ J with stack theoretic weighted blowing up Y ′ → Y and proper transform (X ′ ⊂ Y ′) = F(X ⊂ Y ) such that maxinv(X ′) < maxinv(X). In particular, for some n the iterate (Xn ⊂ Yn) := F ◦n(X ⊂ Y ) of F has Xn smooth.

Example

For X = V (x2 − y2z) we have invp(X) = (2, 3, 3)

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Resolution again

Theorem (ℵ-T-W, McQuillan, characteristic 0)

There is a functor F associating to a singular subvariety X ⊂ Y of a smooth variety Y , a center ¯ J with stack theoretic weighted blowing up Y ′ → Y and proper transform (X ′ ⊂ Y ′) = F(X ⊂ Y ) such that maxinv(X ′) < maxinv(X). In particular, for some n the iterate (Xn ⊂ Yn) := F ◦n(X ⊂ Y ) of F has Xn smooth.

Example

For X = V (x2 − y2z) we have invp(X) = (2, 3, 3) We read it from the degrees of terms. The center is: J = (x2, y3, z3); ¯ J = (x1/3, y1/2, z1/2).

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Example: blowing up Whitney’s umbrella x2 = y 2z

The blowing up Y ′ → Y makes ¯ J = (x1/3, y1/2, z1/2) principal. Explicitly: The z chart has x = w3x3, y = w2y3, z = w2 with chart Y ′ = [ Spec C[x3, y3, w] / (±1) ], with action of (±1) given by (x3, y3, w) → (−x3, y3, −w).

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Example: blowing up Whitney’s umbrella x2 = y 2z

The blowing up Y ′ → Y makes ¯ J = (x1/3, y1/2, z1/2) principal. Explicitly: The z chart has x = w3x3, y = w2y3, z = w2 with chart Y ′ = [ Spec C[x3, y3, w] / (±1) ], with action of (±1) given by (x3, y3, w) → (−x3, y3, −w). The transformed equation is w6(x2

3 − y2 3 ),

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Example: blowing up Whitney’s umbrella x2 = y 2z

The blowing up Y ′ → Y makes ¯ J = (x1/3, y1/2, z1/2) principal. Explicitly: The z chart has x = w3x3, y = w2y3, z = w2 with chart Y ′ = [ Spec C[x3, y3, w] / (±1) ], with action of (±1) given by (x3, y3, w) → (−x3, y3, −w). The transformed equation is w6(x2

3 − y2 3 ),

and the invariant of the proper transform (x2

3 − y2 3 ) is

(2, 2) < (2, 3, 3).

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Order (following Koll´ ar’s book)

We fix Y smooth and I ⊂ OY .

Definition

For p ∈ Y let ordp(I) = max{a : I ⊆ ma

p}.

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Order (following Koll´ ar’s book)

We fix Y smooth and I ⊂ OY .

Definition

For p ∈ Y let ordp(I) = max{a : I ⊆ ma

p}.

We denote by Da the sheaf of a-th order differential operators.

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Order (following Koll´ ar’s book)

We fix Y smooth and I ⊂ OY .

Definition

For p ∈ Y let ordp(I) = max{a : I ⊆ ma

p}.

We denote by Da the sheaf of a-th order differential operators. We note that ordp(I) = min{a : Da(Ip)} = (1). The invariant starts with a1 = ordp(I).

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Order (following Koll´ ar’s book)

We fix Y smooth and I ⊂ OY .

Definition

For p ∈ Y let ordp(I) = max{a : I ⊆ ma

p}.

We denote by Da the sheaf of a-th order differential operators. We note that ordp(I) = min{a : Da(Ip)} = (1). The invariant starts with a1 = ordp(I).

Proposition

The order is upper semicontinuous.

Proof.

V (Da−1I) = {p : ordp(I) ≥ a}. ♠

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Maximal contact (following Koll´ ar’s book)

Definition

A regular parameter x1 ∈ Da1−1Ip is called a maximal contact element. The center starts with (xa1

1 , . . .).

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Maximal contact (following Koll´ ar’s book)

Definition

A regular parameter x1 ∈ Da1−1Ip is called a maximal contact element. The center starts with (xa1

1 , . . .).

Lemma (Hironaka,Giraud)

In characteristic 0 a maximal contact exists on an open neighborhood of p. Since 1 ∈ Da1Ip there is x1 with derivative 1. This derivative is a unit in a neighborhood.

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Maximal contact (following Koll´ ar’s book)

Definition

A regular parameter x1 ∈ Da1−1Ip is called a maximal contact element. The center starts with (xa1

1 , . . .).

Lemma (Hironaka,Giraud)

In characteristic 0 a maximal contact exists on an open neighborhood of p. Since 1 ∈ Da1Ip there is x1 with derivative 1. This derivative is a unit in a neighborhood.

Example

For I = (x2 − y2z) we have ordpI = 2 with x1 = x (or αx+h.o.t. in D(I)).

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Coefficient ideals (treated following Koll´ ar)

We must restrict to x1 = 0 the data of all I, DI, . . . , Da1−1I with corresponding weights a1, a1 − 1, . . . , 1.

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Coefficient ideals (treated following Koll´ ar)

We must restrict to x1 = 0 the data of all I, DI, . . . , Da1−1I with corresponding weights a1, a1 − 1, . . . , 1. We combine these in C(I, a1) :=

f
I, DI, . . . , Da1−1I
,

where f runs over monomials f = tb0

0 · · · t ba1−1 a1−1 with weights

bi(a1 − i) ≥ a1!.

Define I[2] = C(I, a1)|x1=0.

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Defining JI

Again a1 = ordpI and x1 maximal contact. We denoted I[2] = C(I, a1)|x1=0 (with order ≥ a1!).

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Defining JI

Again a1 = ordpI and x1 maximal contact. We denoted I[2] = C(I, a1)|x1=0 (with order ≥ a1!).

Definition

Suppose I[2] has invariant invp(I[2]) defined with parameters ¯ x2, . . . , ¯ xk, with lifts x2, . . . , xk.

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Defining JI

Again a1 = ordpI and x1 maximal contact. We denoted I[2] = C(I, a1)|x1=0 (with order ≥ a1!).

Definition

Suppose I[2] has invariant invp(I[2]) defined with parameters ¯ x2, . . . , ¯ xk, with lifts x2, . . . , xk. Set invp(I) = (a1, . . . , ak) :=

a1, invp(I[2])

(a1 − 1)!

and

JI = (xa1

1 , . . . , xak k ).

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Defining JI

Again a1 = ordpI and x1 maximal contact. We denoted I[2] = C(I, a1)|x1=0 (with order ≥ a1!).

Definition

Suppose I[2] has invariant invp(I[2]) defined with parameters ¯ x2, . . . , ¯ xk, with lifts x2, . . . , xk. Set invp(I) = (a1, . . . , ak) :=

a1, invp(I[2])

(a1 − 1)!

and

JI = (xa1

1 , . . . , xak k ).

Write (a1, . . . , ak) = ℓ(1/w1, . . . , 1/wk) with wi, ℓ ∈ N and gcd(w1, . . . , wk) = 1. We set ¯ JI = (x1/w1

1

, . . . , x1/wk

k

).

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Examples of JI

invp(I) = (a1, . . . , ak) :=

a1, invp(I[2])

(a1−1)!

,

with JI = (xa1

1 , . . . , xak k ).

Example

(0) for X = V (x2 + y2z)

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Examples of JI

invp(I) = (a1, . . . , ak) :=

a1, invp(I[2])

(a1−1)!

,

with JI = (xa1

1 , . . . , xak k ).

Example

(0) for X = V (x2 + y2z) we have I[2] = (y2z), leading to JI = (x2, y3, z3), ¯ JI = (x1/3, y1/2, z1/2) (1) for X = V (x5 + x3y3 + y8)

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Examples of JI

invp(I) = (a1, . . . , ak) :=

a1, invp(I[2])

(a1−1)!

,

with JI = (xa1

1 , . . . , xak k ).

Example

(0) for X = V (x2 + y2z) we have I[2] = (y2z), leading to JI = (x2, y3, z3), ¯ JI = (x1/3, y1/2, z1/2) (1) for X = V (x5 + x3y3 + y8) we have I[2] = (y)180, so JI = (x5, y180/24) = (x5, y15/2), ¯ JI = (x1/3, y1/2). (2) for X = V (x5 + x3y3 + y7)

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Examples of JI

invp(I) = (a1, . . . , ak) :=

a1, invp(I[2])

(a1−1)!

,

with JI = (xa1

1 , . . . , xak k ).

Example

(0) for X = V (x2 + y2z) we have I[2] = (y2z), leading to JI = (x2, y3, z3), ¯ JI = (x1/3, y1/2, z1/2) (1) for X = V (x5 + x3y3 + y8) we have I[2] = (y)180, so JI = (x5, y180/24) = (x5, y15/2), ¯ JI = (x1/3, y1/2). (2) for X = V (x5 + x3y3 + y7) we have I[2] = (y)7·24, so JI = (x5, y7), ¯ JI = (x1/7, y1/5). Implementation: Jonghyun Lee, Anne Fr¨ uhbis-Kr¨ uger.

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Properties of the invariant

Proposition

invp is well-defined. invp is lexicographically upper-semi-continuous. invp is functorial. invp takes values in a well-ordered set. We define maxinv(X) = maxp invp(X).

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Properties of the invariant

Proposition

invp is well-defined. invp is lexicographically upper-semi-continuous. invp is functorial. invp takes values in a well-ordered set. We define maxinv(X) = maxp invp(X). The invariant is well defined because of the MC-invariance property of C(I, a1). The rest is induction!

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Properties of the invariant

Proposition

invp is well-defined. invp is lexicographically upper-semi-continuous. invp is functorial. invp takes values in a well-ordered set. We define maxinv(X) = maxp invp(X). The invariant is well defined because of the MC-invariance property of C(I, a1). The rest is induction!

Theorem (MC-invariance [W lodarczyk, Koll´ ar])

Given maximal contacts x1, x′

1 there are ´

etale π, π′ : ˜ Y ⇒ Y such that π∗x1 = π′∗x′

1 . . . and π∗C(I, a1) = π′∗C(I, a1).

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Definition of Y ′ → Y

Let ¯ J = (x1/w1

1

, . . . , x1/wk

k

). Define the graded algebra A ¯

J

⊂ OY [T] as the integral closure of the image of OY [Y1, . . . , Yn]

OY [T]

Yi

✤ xiT wi.

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Definition of Y ′ → Y

Let ¯ J = (x1/w1

1

, . . . , x1/wk

k

). Define the graded algebra A ¯

J

⊂ OY [T] as the integral closure of the image of OY [Y1, . . . , Yn]

OY [T]

Yi

✤ xiT wi.

Let S0 ⊂ SpecY A ¯

J,

S0 = V ((A ¯

J)>0).

Then Bl ¯

J(Y ) := ProjY A ¯ J :=

(Spec A ¯

J S0)

Gm
.

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Description of Y ′ → Y

Charts: The x1-chart is [Spec k[u, x′

2, . . . , x′ n] / µw1],

with x1 = uw1 and xi = uwix′

i for 2 ≤ i ≤ k, and induced action:

(u, x′

2, . . . , x′ n) → (ζu , ζ−w2x′ 2 , . . . , ζ−wkx′ k, x′ k+1, . . . , x′ n).

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Description of Y ′ → Y

Charts: The x1-chart is [Spec k[u, x′

2, . . . , x′ n] / µw1],

with x1 = uw1 and xi = uwix′

i for 2 ≤ i ≤ k, and induced action:

(u, x′

2, . . . , x′ n) → (ζu , ζ−w2x′ 2 , . . . , ζ−wkx′ k, x′ k+1, . . . , x′ n).

Toric stack: Consider Spec k[x1, . . . , xn, T] with Gm action with weights (w1, . . . , wn, −1). Let U be the open set where one of the xi is a unit. Then Y ′ = [U/Gm]. It is an example of a fantastack [Geraschenko-Satriano], the stack quotient of a Cox construction.

Abramovich Resolution and logarithmic resolution ETH Z¨ urich, July 15, 2020 21 / 26

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What is J?

Definition

Consider the Zariski-Riemann space ZR(Y ) with its sheaf of ordered groups Γ, and associated sheaf of rational ordered group Γ ⊗ Q. A valuative Q-ideal is γ ∈ H0 (ZR(Y ), (Γ ⊗ Q)≥0)) .

Abramovich Resolution and logarithmic resolution ETH Z¨ urich, July 15, 2020 22 / 26

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What is J?

Definition

Consider the Zariski-Riemann space ZR(Y ) with its sheaf of ordered groups Γ, and associated sheaf of rational ordered group Γ ⊗ Q. A valuative Q-ideal is γ ∈ H0 (ZR(Y ), (Γ ⊗ Q)≥0)) . Iγ := {f ∈ OY : v(f ) ≥ γv∀v}. v(I) := (min v(f ) : f ∈ I)v.

Abramovich Resolution and logarithmic resolution ETH Z¨ urich, July 15, 2020 22 / 26

SLIDE 64

What is J?

Definition

Consider the Zariski-Riemann space ZR(Y ) with its sheaf of ordered groups Γ, and associated sheaf of rational ordered group Γ ⊗ Q. A valuative Q-ideal is γ ∈ H0 (ZR(Y ), (Γ ⊗ Q)≥0)) . Iγ := {f ∈ OY : v(f ) ≥ γv∀v}. v(I) := (min v(f ) : f ∈ I)v. A center is in particular a valuative Q-ideal. It is also an idealistic exponent or graded sequence of ideals.

Abramovich Resolution and logarithmic resolution ETH Z¨ urich, July 15, 2020 22 / 26

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Admissibility and coefficient ideals

Definition

J is I-admissible if J ≤ v(I).

Abramovich Resolution and logarithmic resolution ETH Z¨ urich, July 15, 2020 23 / 26

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Admissibility and coefficient ideals

Definition

J is I-admissible if J ≤ v(I).

Lemma

This is equivalent to IOY ′ = E ℓI′, with J = ¯ Jℓ and I′ an ideal. Indeed, on Y ′ the center J becomes E ℓ, in particular principal. This is more subtle in Quek’s theorem!

Abramovich Resolution and logarithmic resolution ETH Z¨ urich, July 15, 2020 23 / 26

SLIDE 67

Admissibility and coefficient ideals

Definition

J is I-admissible if J ≤ v(I).

Lemma

This is equivalent to IOY ′ = E ℓI′, with J = ¯ Jℓ and I′ an ideal. Indeed, on Y ′ the center J becomes E ℓ, in particular principal. This is more subtle in Quek’s theorem!

Proposition

A center J is I-admissible if and only if J(a1−1)! is C(I, a1)-admissible.

Abramovich Resolution and logarithmic resolution ETH Z¨ urich, July 15, 2020 23 / 26

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The key theorems

Theorem

invp(I) is the maximal invariant of an I-admissible center.

Theorem

JI is well-defined: it is the unique admissible center of maximal invariant.

Abramovich Resolution and logarithmic resolution ETH Z¨ urich, July 15, 2020 24 / 26

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The key theorems

Theorem

invp(I) is the maximal invariant of an I-admissible center.

Theorem

JI is well-defined: it is the unique admissible center of maximal invariant.

Theorem

C(I, a1)OY ′ = E ℓ′C ′ with invp′C ′ < invp(C(I, a1)).

Abramovich Resolution and logarithmic resolution ETH Z¨ urich, July 15, 2020 24 / 26

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The key theorems

Theorem

invp(I) is the maximal invariant of an I-admissible center.

Theorem

JI is well-defined: it is the unique admissible center of maximal invariant.

Theorem

C(I, a1)OY ′ = E ℓ′C ′ with invp′C ′ < invp(C(I, a1)).

Theorem

IOY ′ = E ℓI′ with invp′I′ < invp(I). This is a consequence of Koll´ ar’s D-balanced property of C(I, a1).

Abramovich Resolution and logarithmic resolution ETH Z¨ urich, July 15, 2020 24 / 26

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