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Resolving singularities of varieties and families Dan Abramovich Brown University Joint work with Michael Temkin and Jaros law W lodarczyk Rio de Janeiro August 7, 2018 Abramovich Resolving varieties and families August 7, 2018 1 / 29

Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

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Page 1: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Resolving singularities of varieties and families

Dan AbramovichBrown University

Joint work with Michael Temkin and Jaros law W lodarczyk

Rio de JaneiroAugust 7, 2018

Abramovich Resolving varieties and families August 7, 2018 1 / 29

Page 2: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Resolving singularities of varieties and families

Dan AbramovichBrown University

Joint work with Michael Temkin and Jaros law W lodarczyk

Rio de JaneiroAugust 7, 2018

Abramovich Resolving varieties and families August 7, 2018 1 / 29

Page 3: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Resolving singularities of varieties and families

Dan AbramovichBrown University

Joint work with Michael Temkin and Jaros law W lodarczyk

Rio de JaneiroAugust 7, 2018

Abramovich Resolving varieties and families August 7, 2018 1 / 29

Page 4: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

On singularities

Zitrus: x2 + z2 = y3(1− y)3

Kolibri: x2 = y2z2 + z3 Daisy: (x2 − y3)2 = (z2 − y2)3

Real figures by Herwig Hauser, https://imaginary.org/gallery/herwig-hauser-classic

Singularities are beautiful.Why should we “get rid of them”?Answer 1: to study singularities.Answer 2: to study the structure of varieties.

Abramovich Resolving varieties and families August 7, 2018 2 / 29

Page 5: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

On singularities

Zitrus: x2 + z2 = y3(1− y)3 Kolibri: x2 = y2z2 + z3

Daisy: (x2 − y3)2 = (z2 − y2)3

Real figures by Herwig Hauser, https://imaginary.org/gallery/herwig-hauser-classic

Singularities are beautiful.Why should we “get rid of them”?Answer 1: to study singularities.Answer 2: to study the structure of varieties.

Abramovich Resolving varieties and families August 7, 2018 2 / 29

Page 6: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

On singularities

Zitrus: x2 + z2 = y3(1− y)3 Kolibri: x2 = y2z2 + z3 Daisy: (x2 − y3)2 = (z2 − y2)3

Real figures by Herwig Hauser, https://imaginary.org/gallery/herwig-hauser-classic

Singularities are beautiful.

Why should we “get rid of them”?Answer 1: to study singularities.Answer 2: to study the structure of varieties.

Abramovich Resolving varieties and families August 7, 2018 2 / 29

Page 7: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

On singularities

Zitrus: x2 + z2 = y3(1− y)3 Kolibri: x2 = y2z2 + z3 Daisy: (x2 − y3)2 = (z2 − y2)3

Real figures by Herwig Hauser, https://imaginary.org/gallery/herwig-hauser-classic

Singularities are beautiful.Why should we “get rid of them”?

Answer 1: to study singularities.Answer 2: to study the structure of varieties.

Abramovich Resolving varieties and families August 7, 2018 2 / 29

Page 8: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

On singularities

Zitrus: x2 + z2 = y3(1− y)3 Kolibri: x2 = y2z2 + z3 Daisy: (x2 − y3)2 = (z2 − y2)3

Real figures by Herwig Hauser, https://imaginary.org/gallery/herwig-hauser-classic

Singularities are beautiful.Why should we “get rid of them”?Answer 1: to study singularities.Answer 2: to study the structure of varieties.

Abramovich Resolving varieties and families August 7, 2018 2 / 29

Page 9: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Singular and smooth points

Definition

{f (x1, . . . , xn) = 0} is singular at p if ∂f∂xi

(p) = 0 for all i .Otherwise smooth.

In other words, if smooth, {f = 0} defines a submanifold of complexcodimension 1.

In codimension c , the set {f1 = · · · = fk = 0} is smooth whend(f1, . . . , fk) has constant rank c .

Abramovich Resolving varieties and families August 7, 2018 3 / 29

Page 10: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Singular and smooth points

Definition

{f (x1, . . . , xn) = 0} is singular at p if ∂f∂xi

(p) = 0 for all i .Otherwise smooth.

In other words, if smooth, {f = 0} defines a submanifold of complexcodimension 1.

In codimension c , the set {f1 = · · · = fk = 0} is smooth whend(f1, . . . , fk) has constant rank c .

Abramovich Resolving varieties and families August 7, 2018 3 / 29

Page 11: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Singular and smooth points

Definition

{f (x1, . . . , xn) = 0} is singular at p if ∂f∂xi

(p) = 0 for all i .Otherwise smooth.

In other words, if smooth, {f = 0} defines a submanifold of complexcodimension 1.

In codimension c , the set {f1 = · · · = fk = 0} is smooth whend(f1, . . . , fk) has constant rank c .

Abramovich Resolving varieties and families August 7, 2018 3 / 29

Page 12: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

What is resolution of singularities?

Definition

A resolution of singularities X ′ → X is a modificationa with X ′ nonsingularinducing an isomorphism over the smooth locus of X .

aproper birational map. For instance, blowing up.

Theorem (Hironaka 1964)

A variety X over a field of characteristic 0 admits a resolution ofsingularities X ′ → X , so that the critical locus E ⊂ X ′ is a simple normalcrossings divisor.a

aCodim. 1, smooth components meeting transversally - as simple as possible

Always characteristic 0 . . .

Abramovich Resolving varieties and families August 7, 2018 4 / 29

Page 13: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

What is resolution of singularities?

Definition

A resolution of singularities X ′ → X is a modificationa with X ′ nonsingularinducing an isomorphism over the smooth locus of X .

aproper birational map. For instance, blowing up.

Theorem (Hironaka 1964)

A variety X over a field of characteristic 0 admits a resolution ofsingularities X ′ → X , so that the critical locus E ⊂ X ′ is a simple normalcrossings divisor.a

aCodim. 1, smooth components meeting transversally - as simple as possible

Always characteristic 0 . . .

Abramovich Resolving varieties and families August 7, 2018 4 / 29

Page 14: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Answer 1: Example of invariant - Stepanov’s theorem

If X ′ → X a resolution with critical E ⊂ X ′ a simple normal crossingsdivisor, define ∆(E ) to be the dual complex of E .

Theorem (Stepanov 2006)

The simple homotopy type of ∆(E ) is independent of the resolutionX ′ → X .

Also work by Danilov, Payne, Thuillier, Harper. . .

Abramovich Resolving varieties and families August 7, 2018 5 / 29

Page 15: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Answer 1: Example of invariant - Stepanov’s theorem

If X ′ → X a resolution with critical E ⊂ X ′ a simple normal crossingsdivisor, define ∆(E ) to be the dual complex of E .

Theorem (Stepanov 2006)

The simple homotopy type of ∆(E ) is independent of the resolutionX ′ → X .

Also work by Danilov, Payne, Thuillier, Harper. . .

Abramovich Resolving varieties and families August 7, 2018 5 / 29

Page 16: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Answer 1: Example of invariant - Stepanov’s theorem

If X ′ → X a resolution with critical E ⊂ X ′ a simple normal crossingsdivisor, define ∆(E ) to be the dual complex of E .

Theorem (Stepanov 2006)

The simple homotopy type of ∆(E ) is independent of the resolutionX ′ → X .

Also work by Danilov, Payne, Thuillier, Harper. . .

Abramovich Resolving varieties and families August 7, 2018 5 / 29

Page 17: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Answer 1: Example of invariant - Stepanov’s theorem

If X ′ → X a resolution with critical E ⊂ X ′ a simple normal crossingsdivisor, define ∆(E ) to be the dual complex of E .

Theorem (Stepanov 2006)

The simple homotopy type of ∆(E ) is independent of the resolutionX ′ → X .

Also work by Danilov, Payne, Thuillier, Harper. . .

Abramovich Resolving varieties and families August 7, 2018 5 / 29

Page 18: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Answer 2: Example of structure result: compactifications

“Working with noncompact spaces is like trying to keep change with holesin your pockets”

Angelo Vistoli

Corollary (Hironaka)

A smooth quasiprojective variety X 0 has a smooth projectivecompactification X with D = X r X 0 a simple normal crossings divisor.

Abramovich Resolving varieties and families August 7, 2018 6 / 29

Page 19: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Answer 2: Example of structure result: compactifications

“Working with noncompact spaces is like trying to keep change with holesin your pockets”

Angelo Vistoli

Corollary (Hironaka)

A smooth quasiprojective variety X 0 has a smooth projectivecompactification X with D = X r X 0 a simple normal crossings divisor.

Abramovich Resolving varieties and families August 7, 2018 6 / 29

Page 20: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Answer 2: Example of structure result: compactifications

“Working with noncompact spaces is like trying to keep change with holesin your pockets”

Angelo Vistoli

Corollary (Hironaka)

A smooth quasiprojective variety X 0 has a smooth projectivecompactification X with D = X r X 0 a simple normal crossings divisor.

Abramovich Resolving varieties and families August 7, 2018 6 / 29

Page 21: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Answer 2: Example of structure result: compactifications

“Working with noncompact spaces is like trying to keep change with holesin your pockets”

Angelo Vistoli

Corollary (Hironaka)

A smooth quasiprojective variety X 0 has a smooth projectivecompactification X with D = X r X 0 a simple normal crossings divisor.

Abramovich Resolving varieties and families August 7, 2018 6 / 29

Page 22: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Resolution of families: dimB = 1

Key Question

When are the singularities of a morphism X → B simple?

If dimB = 1 the simplest one can have by modifying X is t =∏

xaii ,

and if one also allows base change t = sk , can have s =∏

xi .[Kempf–Knudsen–Mumford–Saint-Donat 1973]

Question

What makes these special?

Abramovich Resolving varieties and families August 7, 2018 7 / 29

Page 23: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Resolution of families: dimB = 1

Key Question

When are the singularities of a morphism X → B simple?

If dimB = 1 the simplest one can have by modifying X is t =∏

xaii ,

and if one also allows base change t = sk , can have s =∏

xi .[Kempf–Knudsen–Mumford–Saint-Donat 1973]

Question

What makes these special?

Abramovich Resolving varieties and families August 7, 2018 7 / 29

Page 24: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Log smooth schemes and log smooth morphisms

A toric variety is a normal variety on which T = (C∗)n actsalgebraically with a dense free orbit.

Zariski locally defined by equations between monomials.

A variety X with divisor D is toroidal or log smooth if etale locally itlooks like a toric variety Xσ with its toric divisor Xσ r T .

Etale locally it is defined by equations between monomials.

A morphism X → Y is toroidal or log smooth if etale locally it lookslike a torus equivariant morphism of toric varieties.

The inverse image of a monomial 1 is a monomial.

1

defining equation of part of DY

Abramovich Resolving varieties and families August 7, 2018 8 / 29

Page 25: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Log smooth schemes and log smooth morphisms

A toric variety is a normal variety on which T = (C∗)n actsalgebraically with a dense free orbit.

Zariski locally defined by equations between monomials.

A variety X with divisor D is toroidal or log smooth if etale locally itlooks like a toric variety Xσ with its toric divisor Xσ r T .

Etale locally it is defined by equations between monomials.

A morphism X → Y is toroidal or log smooth if etale locally it lookslike a torus equivariant morphism of toric varieties.

The inverse image of a monomial 1 is a monomial.

1

defining equation of part of DY

Abramovich Resolving varieties and families August 7, 2018 8 / 29

Page 26: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Log smooth schemes and log smooth morphisms

A toric variety is a normal variety on which T = (C∗)n actsalgebraically with a dense free orbit.

Zariski locally defined by equations between monomials.

A variety X with divisor D is toroidal or log smooth if etale locally itlooks like a toric variety Xσ with its toric divisor Xσ r T .

Etale locally it is defined by equations between monomials.

A morphism X → Y is toroidal or log smooth if etale locally it lookslike a torus equivariant morphism of toric varieties.

The inverse image of a monomial 1 is a monomial.

1defining equation of part of DY

Abramovich Resolving varieties and families August 7, 2018 8 / 29

Page 27: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Resolution of families: higher dimensional base

Question

When are the singularities of a morphism X → B simple?

The best one can hope for, after base change, is a semistable morphism:

Definition (ℵ-Karu 2000)

A log smooth morphism, with B smooth, is semistable if locally

t1 = x1 · · · xl1...

...

tm = xlm−1+1

· · · xlm

In particular log smooth.Similar definition by Berkovich, all inspired by de Jong.

Abramovich Resolving varieties and families August 7, 2018 9 / 29

Page 28: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Ultimate goal: the semistable reduction problem

Conjecture [ℵ-Karu]

Let X → B be a dominant morphism of varieties.

(Loose) There is a base changea B1 → B and a modificationX1 → (X ×B B1)main such that X1 → B1 is semistable.

(Tight) If the geometric generic fiber Xη is smooth, such X1 → B1

can be found with Xη unchanged.

aAlteration: Proper, surjective, generically finite

One wants the tight version in order to compactify smooth families.

I’ll describe progress towards that.

Major early results by [KKMS 1973], [de Jong 1997].

Wonderful results in positive and mixed characteristics by de Jong,Gabber, Illusie and Temkin.

Abramovich Resolving varieties and families August 7, 2018 10 / 29

Page 29: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Ultimate goal: the semistable reduction problem

Conjecture [ℵ-Karu]

Let X → B be a dominant morphism of varieties.

(Loose) There is a base changea B1 → B and a modificationX1 → (X ×B B1)main such that X1 → B1 is semistable.

(Tight) If the geometric generic fiber Xη is smooth, such X1 → B1

can be found with Xη unchanged.

aAlteration: Proper, surjective, generically finite

One wants the tight version in order to compactify smooth families.

I’ll describe progress towards that.

Major early results by [KKMS 1973], [de Jong 1997].

Wonderful results in positive and mixed characteristics by de Jong,Gabber, Illusie and Temkin.

Abramovich Resolving varieties and families August 7, 2018 10 / 29

Page 30: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Ultimate goal: the semistable reduction problem

Conjecture [ℵ-Karu]

Let X → B be a dominant morphism of varieties.

(Loose) There is a base changea B1 → B and a modificationX1 → (X ×B B1)main such that X1 → B1 is semistable.

(Tight) If the geometric generic fiber Xη is smooth, such X1 → B1

can be found with Xη unchanged.

aAlteration: Proper, surjective, generically finite

One wants the tight version in order to compactify smooth families.

I’ll describe progress towards that.

Major early results by [KKMS 1973], [de Jong 1997].

Wonderful results in positive and mixed characteristics by de Jong,Gabber, Illusie and Temkin.

Abramovich Resolving varieties and families August 7, 2018 10 / 29

Page 31: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Ultimate goal: the semistable reduction problem

Conjecture [ℵ-Karu]

Let X → B be a dominant morphism of varieties.

(Loose) There is a base changea B1 → B and a modificationX1 → (X ×B B1)main such that X1 → B1 is semistable.

(Tight) If the geometric generic fiber Xη is smooth, such X1 → B1

can be found with Xη unchanged.

aAlteration: Proper, surjective, generically finite

One wants the tight version in order to compactify smooth families.

I’ll describe progress towards that.

Major early results by [KKMS 1973], [de Jong 1997].

Wonderful results in positive and mixed characteristics by de Jong,Gabber, Illusie and Temkin.

Abramovich Resolving varieties and families August 7, 2018 10 / 29

Page 32: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Ultimate goal: the semistable reduction problem

Conjecture [ℵ-Karu]

Let X → B be a dominant morphism of varieties.

(Loose) There is a base changea B1 → B and a modificationX1 → (X ×B B1)main such that X1 → B1 is semistable.

(Tight) If the geometric generic fiber Xη is smooth, such X1 → B1

can be found with Xη unchanged.

aAlteration: Proper, surjective, generically finite

One wants the tight version in order to compactify smooth families.

I’ll describe progress towards that.

Major early results by [KKMS 1973], [de Jong 1997].

Wonderful results in positive and mixed characteristics by de Jong,Gabber, Illusie and Temkin.

Abramovich Resolving varieties and families August 7, 2018 10 / 29

Page 33: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Ultimate goal: the semistable reduction problem

Conjecture [ℵ-Karu]

Let X → B be a dominant morphism of varieties.

(Loose) There is a base changea B1 → B and a modificationX1 → (X ×B B1)main such that X1 → B1 is semistable.

(Tight) If the geometric generic fiber Xη is smooth, such X1 → B1

can be found with Xη unchanged.

aAlteration: Proper, surjective, generically finite

One wants the tight version in order to compactify smooth families.

I’ll describe progress towards that.

Major early results by [KKMS 1973], [de Jong 1997].

Wonderful results in positive and mixed characteristics by de Jong,Gabber, Illusie and Temkin.

Abramovich Resolving varieties and families August 7, 2018 10 / 29

Page 34: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Toroidalization and weak semistable reduction

This is key to what’s known:

Theorem (Toroidalization, ℵ-Karu 2000, ℵ-K-Denef 2013)

There is a modification B1 → B and a modification X1 → (X ×B B1)main

such that X1 → B1 is log smooth and flat.

Theorem (Weak semistable reduction, ℵ-Karu 2000)

There is a base change B1 → B and a modification X1 → (X ×B B1)main

such that X1 → B1 is log smooth, flat, with reduced fibers.

Passing from weak semistable reduction to semistable reduction is apurely combinatorial problem [ℵ-Karu 2000],

proven by [Karu 2000] for families of surfaces and threefolds, and

whose restriction to rank-1 valuation rings is proven in a preprint by[Karim Adiprasito - Gaku Liu - Igor Pak - Michael Temkin].

Abramovich Resolving varieties and families August 7, 2018 11 / 29

Page 35: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Toroidalization and weak semistable reduction

This is key to what’s known:

Theorem (Toroidalization, ℵ-Karu 2000, ℵ-K-Denef 2013)

There is a modification B1 → B and a modification X1 → (X ×B B1)main

such that X1 → B1 is log smooth and flat.

Theorem (Weak semistable reduction, ℵ-Karu 2000)

There is a base change B1 → B and a modification X1 → (X ×B B1)main

such that X1 → B1 is log smooth, flat, with reduced fibers.

Passing from weak semistable reduction to semistable reduction is apurely combinatorial problem [ℵ-Karu 2000],

proven by [Karu 2000] for families of surfaces and threefolds, and

whose restriction to rank-1 valuation rings is proven in a preprint by[Karim Adiprasito - Gaku Liu - Igor Pak - Michael Temkin].

Abramovich Resolving varieties and families August 7, 2018 11 / 29

Page 36: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Toroidalization and weak semistable reduction

This is key to what’s known:

Theorem (Toroidalization, ℵ-Karu 2000, ℵ-K-Denef 2013)

There is a modification B1 → B and a modification X1 → (X ×B B1)main

such that X1 → B1 is log smooth and flat.

Theorem (Weak semistable reduction, ℵ-Karu 2000)

There is a base change B1 → B and a modification X1 → (X ×B B1)main

such that X1 → B1 is log smooth, flat, with reduced fibers.

Passing from weak semistable reduction to semistable reduction is apurely combinatorial problem [ℵ-Karu 2000],

proven by [Karu 2000] for families of surfaces and threefolds, and

whose restriction to rank-1 valuation rings is proven in a preprint by[Karim Adiprasito - Gaku Liu - Igor Pak - Michael Temkin].

Abramovich Resolving varieties and families August 7, 2018 11 / 29

Page 37: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Toroidalization and weak semistable reduction

This is key to what’s known:

Theorem (Toroidalization, ℵ-Karu 2000, ℵ-K-Denef 2013)

There is a modification B1 → B and a modification X1 → (X ×B B1)main

such that X1 → B1 is log smooth and flat.

Theorem (Weak semistable reduction, ℵ-Karu 2000)

There is a base change B1 → B and a modification X1 → (X ×B B1)main

such that X1 → B1 is log smooth, flat, with reduced fibers.

Passing from weak semistable reduction to semistable reduction is apurely combinatorial problem [ℵ-Karu 2000],

proven by [Karu 2000] for families of surfaces and threefolds, and

whose restriction to rank-1 valuation rings is proven in a preprint by[Karim Adiprasito - Gaku Liu - Igor Pak - Michael Temkin].

Abramovich Resolving varieties and families August 7, 2018 11 / 29

Page 38: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Applications of weak semistable reduction

This is already useful for studying families:

Theorem (Karu 2000; K-SB 97, Alexeev 94, BCHM 11)

The moduli space of stable smoothable varieties is projectivea.

ain particular bounded and proper

Theorem (Viehweg-Zuo 2004)

The moduli space of canonically polarized manifolds is Brody hyperbolic.

Theorem (Fujino 2017)

Nakayama’s numerical logarithmic Kodaira dimension is subadditive infamilies X → B with generic fiber F :

κσ(X ,DX ) ≥ κσ(F ,DF ) + κσ(B,DB).

Abramovich Resolving varieties and families August 7, 2018 12 / 29

Page 39: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Applications of weak semistable reduction

This is already useful for studying families:

Theorem (Karu 2000; K-SB 97, Alexeev 94, BCHM 11)

The moduli space of stable smoothable varieties is projectivea.

ain particular bounded and proper

Theorem (Viehweg-Zuo 2004)

The moduli space of canonically polarized manifolds is Brody hyperbolic.

Theorem (Fujino 2017)

Nakayama’s numerical logarithmic Kodaira dimension is subadditive infamilies X → B with generic fiber F :

κσ(X ,DX ) ≥ κσ(F ,DF ) + κσ(B,DB).

Abramovich Resolving varieties and families August 7, 2018 12 / 29

Page 40: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Applications of weak semistable reduction

This is already useful for studying families:

Theorem (Karu 2000; K-SB 97, Alexeev 94, BCHM 11)

The moduli space of stable smoothable varieties is projectivea.

ain particular bounded and proper

Theorem (Viehweg-Zuo 2004)

The moduli space of canonically polarized manifolds is Brody hyperbolic.

Theorem (Fujino 2017)

Nakayama’s numerical logarithmic Kodaira dimension is subadditive infamilies X → B with generic fiber F :

κσ(X ,DX ) ≥ κσ(F ,DF ) + κσ(B,DB).

Abramovich Resolving varieties and families August 7, 2018 12 / 29

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Main result

The following result is work-in-progress.

Main result (Functorial toroidalization, ℵ-Temkin-W lodarczyk)

Let X → B be a dominant morphism.

There are modifications B1 → B and X1 → (X ×B B1)main such thatX1 → B1 is log smooth and flat;

this is compatible with base change B ′ → B;

this is functorial, up to base change, with log smooth X ′′ → X .

This implies the tight version of the results of semistable reduction type.

Application:

Theorem (Deng 2018)

The moduli space of minimal complex projective manifolds of general typeis Kobayashi hyperbolic.

Abramovich Resolving varieties and families August 7, 2018 13 / 29

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Main result

The following result is work-in-progress.

Main result (Functorial toroidalization, ℵ-Temkin-W lodarczyk)

Let X → B be a dominant morphism.

There are modifications B1 → B and X1 → (X ×B B1)main such thatX1 → B1 is log smooth and flat;

this is compatible with base change B ′ → B;

this is functorial, up to base change, with log smooth X ′′ → X .

This implies the tight version of the results of semistable reduction type.Application:

Theorem (Deng 2018)

The moduli space of minimal complex projective manifolds of general typeis Kobayashi hyperbolic.

Abramovich Resolving varieties and families August 7, 2018 13 / 29

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dimB = 0: log resolution via principalizationTo resolve log singularities, one embeds X in a log smooth Y . . .

. . . which can be done locally.One reduces to principalization of IX (Hironaka, Villamayor,Bierstone–Milman).

Theorem (Principalization . . .ℵ-T-W)

Let I be an ideal on a log smooth Y . There is a functorial logarithmicmorphism Y ′ → Y , with Y ′ logarithmically smooth, and IOY ′ aninvertible monomial ideal.

Figure: The ideal (u2, x2)

and the result of blowing up the origin, I2E .

Here u is a monomial but x is not.

Abramovich Resolving varieties and families August 7, 2018 14 / 29

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dimB = 0: log resolution via principalizationTo resolve log singularities, one embeds X in a log smooth Y . . .. . . which can be done locally.

One reduces to principalization of IX (Hironaka, Villamayor,Bierstone–Milman).

Theorem (Principalization . . .ℵ-T-W)

Let I be an ideal on a log smooth Y . There is a functorial logarithmicmorphism Y ′ → Y , with Y ′ logarithmically smooth, and IOY ′ aninvertible monomial ideal.

Figure: The ideal (u2, x2)

and the result of blowing up the origin, I2E .

Here u is a monomial but x is not.

Abramovich Resolving varieties and families August 7, 2018 14 / 29

Page 45: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

dimB = 0: log resolution via principalizationTo resolve log singularities, one embeds X in a log smooth Y . . .. . . which can be done locally.One reduces to principalization of IX (Hironaka, Villamayor,Bierstone–Milman).

Theorem (Principalization . . .ℵ-T-W)

Let I be an ideal on a log smooth Y . There is a functorial logarithmicmorphism Y ′ → Y , with Y ′ logarithmically smooth, and IOY ′ aninvertible monomial ideal.

Figure: The ideal (u2, x2)

and the result of blowing up the origin, I2E .

Here u is a monomial but x is not.

Abramovich Resolving varieties and families August 7, 2018 14 / 29

Page 46: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

dimB = 0: log resolution via principalizationTo resolve log singularities, one embeds X in a log smooth Y . . .. . . which can be done locally.One reduces to principalization of IX (Hironaka, Villamayor,Bierstone–Milman).

Theorem (Principalization . . .ℵ-T-W)

Let I be an ideal on a log smooth Y . There is a functorial logarithmicmorphism Y ′ → Y , with Y ′ logarithmically smooth, and IOY ′ aninvertible monomial ideal.

Figure: The ideal (u2, x2)

and the result of blowing up the origin, I2E .

Here u is a monomial but x is not.Abramovich Resolving varieties and families August 7, 2018 14 / 29

Page 47: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

dimB = 0: log resolution via principalizationTo resolve log singularities, one embeds X in a log smooth Y . . .. . . which can be done locally.One reduces to principalization of IX (Hironaka, Villamayor,Bierstone–Milman).

Theorem (Principalization . . .ℵ-T-W)

Let I be an ideal on a log smooth Y . There is a functorial logarithmicmorphism Y ′ → Y , with Y ′ logarithmically smooth, and IOY ′ aninvertible monomial ideal.

Figure: The ideal (u2, x2) and the result of blowing up the origin, I2E .

Here u is a monomial but x is not.Abramovich Resolving varieties and families August 7, 2018 14 / 29

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Logarithmic order

Principalization is done by order reduction, using logarithmic derivatives.

for a monomial u we use u ∂∂u .

for other variables x use ∂∂x .

Definition

Write D≤a for the sheaf of logarithmic differential operators of order ≤ a.The logarithmic order of an ideal I is the minimum a such thatD≤aI = (1).

Take u, v monomials, x free variable, p the origin.logordp(u2, x) = 1 (since ∂

∂x x = 1)logordp(u2, x2) = 2 logordp(v , x2) = 2logordp(v + u) =∞ since D≤1I = D≤2I = · · · = (u, v).

Abramovich Resolving varieties and families August 7, 2018 15 / 29

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Logarithmic order

Principalization is done by order reduction, using logarithmic derivatives.

for a monomial u we use u ∂∂u .

for other variables x use ∂∂x .

Definition

Write D≤a for the sheaf of logarithmic differential operators of order ≤ a.

The logarithmic order of an ideal I is the minimum a such thatD≤aI = (1).

Take u, v monomials, x free variable, p the origin.logordp(u2, x) = 1 (since ∂

∂x x = 1)logordp(u2, x2) = 2 logordp(v , x2) = 2logordp(v + u) =∞ since D≤1I = D≤2I = · · · = (u, v).

Abramovich Resolving varieties and families August 7, 2018 15 / 29

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Logarithmic order

Principalization is done by order reduction, using logarithmic derivatives.

for a monomial u we use u ∂∂u .

for other variables x use ∂∂x .

Definition

Write D≤a for the sheaf of logarithmic differential operators of order ≤ a.The logarithmic order of an ideal I is the minimum a such thatD≤aI = (1).

Take u, v monomials, x free variable, p the origin.logordp(u2, x) = 1 (since ∂

∂x x = 1)logordp(u2, x2) = 2 logordp(v , x2) = 2logordp(v + u) =∞ since D≤1I = D≤2I = · · · = (u, v).

Abramovich Resolving varieties and families August 7, 2018 15 / 29

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Logarithmic order

Principalization is done by order reduction, using logarithmic derivatives.

for a monomial u we use u ∂∂u .

for other variables x use ∂∂x .

Definition

Write D≤a for the sheaf of logarithmic differential operators of order ≤ a.The logarithmic order of an ideal I is the minimum a such thatD≤aI = (1).

Take u, v monomials, x free variable, p the origin.logordp(u2, x) =

1 (since ∂∂x x = 1)

logordp(u2, x2) = 2 logordp(v , x2) = 2logordp(v + u) =∞ since D≤1I = D≤2I = · · · = (u, v).

Abramovich Resolving varieties and families August 7, 2018 15 / 29

Page 52: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Logarithmic order

Principalization is done by order reduction, using logarithmic derivatives.

for a monomial u we use u ∂∂u .

for other variables x use ∂∂x .

Definition

Write D≤a for the sheaf of logarithmic differential operators of order ≤ a.The logarithmic order of an ideal I is the minimum a such thatD≤aI = (1).

Take u, v monomials, x free variable, p the origin.logordp(u2, x) = 1 (since ∂

∂x x = 1)

logordp(u2, x2) = 2 logordp(v , x2) = 2logordp(v + u) =∞ since D≤1I = D≤2I = · · · = (u, v).

Abramovich Resolving varieties and families August 7, 2018 15 / 29

Page 53: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Logarithmic order

Principalization is done by order reduction, using logarithmic derivatives.

for a monomial u we use u ∂∂u .

for other variables x use ∂∂x .

Definition

Write D≤a for the sheaf of logarithmic differential operators of order ≤ a.The logarithmic order of an ideal I is the minimum a such thatD≤aI = (1).

Take u, v monomials, x free variable, p the origin.logordp(u2, x) = 1 (since ∂

∂x x = 1)logordp(u2, x2) =

2 logordp(v , x2) = 2logordp(v + u) =∞ since D≤1I = D≤2I = · · · = (u, v).

Abramovich Resolving varieties and families August 7, 2018 15 / 29

Page 54: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Logarithmic order

Principalization is done by order reduction, using logarithmic derivatives.

for a monomial u we use u ∂∂u .

for other variables x use ∂∂x .

Definition

Write D≤a for the sheaf of logarithmic differential operators of order ≤ a.The logarithmic order of an ideal I is the minimum a such thatD≤aI = (1).

Take u, v monomials, x free variable, p the origin.logordp(u2, x) = 1 (since ∂

∂x x = 1)logordp(u2, x2) = 2 logordp(v , x2) =

2logordp(v + u) =∞ since D≤1I = D≤2I = · · · = (u, v).

Abramovich Resolving varieties and families August 7, 2018 15 / 29

Page 55: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Logarithmic order

Principalization is done by order reduction, using logarithmic derivatives.

for a monomial u we use u ∂∂u .

for other variables x use ∂∂x .

Definition

Write D≤a for the sheaf of logarithmic differential operators of order ≤ a.The logarithmic order of an ideal I is the minimum a such thatD≤aI = (1).

Take u, v monomials, x free variable, p the origin.logordp(u2, x) = 1 (since ∂

∂x x = 1)logordp(u2, x2) = 2 logordp(v , x2) = 2logordp(v + u) =

∞ since D≤1I = D≤2I = · · · = (u, v).

Abramovich Resolving varieties and families August 7, 2018 15 / 29

Page 56: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Logarithmic order

Principalization is done by order reduction, using logarithmic derivatives.

for a monomial u we use u ∂∂u .

for other variables x use ∂∂x .

Definition

Write D≤a for the sheaf of logarithmic differential operators of order ≤ a.The logarithmic order of an ideal I is the minimum a such thatD≤aI = (1).

Take u, v monomials, x free variable, p the origin.logordp(u2, x) = 1 (since ∂

∂x x = 1)logordp(u2, x2) = 2 logordp(v , x2) = 2logordp(v + u) =∞ since D≤1I = D≤2I = · · · = (u, v).

Abramovich Resolving varieties and families August 7, 2018 15 / 29

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Key new ingredient: The monomial part of an ideal

Definition

M(I) is the minimal monomial ideal containing I.

Proposition (Kollar, ℵ-T-W)

(1) In characteristic 0, M(I) = D∞(I). In particularmaxp logordp(I) =∞ if and only if M(I) 6= 1.

(2) Let Y0 → Y be the normalized blowup of M(I). ThenM :=M(I)OY0 =M(IOY0), and it is an invertible monomial ideal,and so IOY0 = I0 · M with maxp logordp(I0) <∞.

(1)⇒(2)

DY0 is the pullback of DY , so (2) follows from (1) since the ideals havethe same generators.

Abramovich Resolving varieties and families August 7, 2018 16 / 29

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Key new ingredient: The monomial part of an ideal

Definition

M(I) is the minimal monomial ideal containing I.

Proposition (Kollar, ℵ-T-W)

(1) In characteristic 0, M(I) = D∞(I). In particularmaxp logordp(I) =∞ if and only if M(I) 6= 1.

(2) Let Y0 → Y be the normalized blowup of M(I). ThenM :=M(I)OY0 =M(IOY0), and it is an invertible monomial ideal,and so IOY0 = I0 · M with maxp logordp(I0) <∞.

(1)⇒(2)

DY0 is the pullback of DY , so (2) follows from (1) since the ideals havethe same generators.

Abramovich Resolving varieties and families August 7, 2018 16 / 29

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Key new ingredient: The monomial part of an ideal

Definition

M(I) is the minimal monomial ideal containing I.

Proposition (Kollar, ℵ-T-W)

(1) In characteristic 0, M(I) = D∞(I). In particularmaxp logordp(I) =∞ if and only if M(I) 6= 1.

(2) Let Y0 → Y be the normalized blowup of M(I).

ThenM :=M(I)OY0 =M(IOY0), and it is an invertible monomial ideal,and so IOY0 = I0 · M with maxp logordp(I0) <∞.

(1)⇒(2)

DY0 is the pullback of DY , so (2) follows from (1) since the ideals havethe same generators.

Abramovich Resolving varieties and families August 7, 2018 16 / 29

Page 60: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Key new ingredient: The monomial part of an ideal

Definition

M(I) is the minimal monomial ideal containing I.

Proposition (Kollar, ℵ-T-W)

(1) In characteristic 0, M(I) = D∞(I). In particularmaxp logordp(I) =∞ if and only if M(I) 6= 1.

(2) Let Y0 → Y be the normalized blowup of M(I). ThenM :=M(I)OY0 =M(IOY0), and it is an invertible monomial ideal,

and so IOY0 = I0 · M with maxp logordp(I0) <∞.

(1)⇒(2)

DY0 is the pullback of DY , so (2) follows from (1) since the ideals havethe same generators.

Abramovich Resolving varieties and families August 7, 2018 16 / 29

Page 61: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Key new ingredient: The monomial part of an ideal

Definition

M(I) is the minimal monomial ideal containing I.

Proposition (Kollar, ℵ-T-W)

(1) In characteristic 0, M(I) = D∞(I). In particularmaxp logordp(I) =∞ if and only if M(I) 6= 1.

(2) Let Y0 → Y be the normalized blowup of M(I). ThenM :=M(I)OY0 =M(IOY0), and it is an invertible monomial ideal,and so IOY0 = I0 · M with maxp logordp(I0) <∞.

(1)⇒(2)

DY0 is the pullback of DY , so (2) follows from (1) since the ideals havethe same generators.

Abramovich Resolving varieties and families August 7, 2018 16 / 29

Page 62: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Key new ingredient: The monomial part of an ideal

Definition

M(I) is the minimal monomial ideal containing I.

Proposition (Kollar, ℵ-T-W)

(1) In characteristic 0, M(I) = D∞(I). In particularmaxp logordp(I) =∞ if and only if M(I) 6= 1.

(2) Let Y0 → Y be the normalized blowup of M(I). ThenM :=M(I)OY0 =M(IOY0), and it is an invertible monomial ideal,and so IOY0 = I0 · M with maxp logordp(I0) <∞.

(1)⇒(2)

DY0 is the pullback of DY , so (2) follows from (1) since the ideals havethe same generators.

Abramovich Resolving varieties and families August 7, 2018 16 / 29

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The monomial part of an ideal - proof

Proof of (1), basic affine case.

Let OY = C[x1, . . . , xn, u1, . . . , um] and assume M = D(M).

The operators

1, u1∂

∂u1, . . . , ul

∂ul

commute and have distinct systems of eigenvalues on the eigenspacesuC[x1, . . . , xn], for distinct monomials u.

Therefore M = ⊕uMu with ideals Mu ⊂ C[x1, . . . , xn] stable underderivatives,

so each Mu is either (0) or (1).

In other words, M is monomial.

The general case requires more commutative algebra.

Abramovich Resolving varieties and families August 7, 2018 17 / 29

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The monomial part of an ideal - proof

Proof of (1), basic affine case.

Let OY = C[x1, . . . , xn, u1, . . . , um] and assume M = D(M).

The operators

1, u1∂

∂u1, . . . , ul

∂ul

commute and have distinct systems of eigenvalues on the eigenspacesuC[x1, . . . , xn], for distinct monomials u.

Therefore M = ⊕uMu with ideals Mu ⊂ C[x1, . . . , xn] stable underderivatives,

so each Mu is either (0) or (1).

In other words, M is monomial.

The general case requires more commutative algebra.

Abramovich Resolving varieties and families August 7, 2018 17 / 29

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The monomial part of an ideal - proof

Proof of (1), basic affine case.

Let OY = C[x1, . . . , xn, u1, . . . , um] and assume M = D(M).

The operators

1, u1∂

∂u1, . . . , ul

∂ul

commute and have distinct systems of eigenvalues on the eigenspacesuC[x1, . . . , xn], for distinct monomials u.

Therefore M = ⊕uMu with ideals Mu ⊂ C[x1, . . . , xn] stable underderivatives,

so each Mu is either (0) or (1).

In other words, M is monomial.

The general case requires more commutative algebra.

Abramovich Resolving varieties and families August 7, 2018 17 / 29

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The monomial part of an ideal - proof

Proof of (1), basic affine case.

Let OY = C[x1, . . . , xn, u1, . . . , um] and assume M = D(M).

The operators

1, u1∂

∂u1, . . . , ul

∂ul

commute and have distinct systems of eigenvalues on the eigenspacesuC[x1, . . . , xn], for distinct monomials u.

Therefore M = ⊕uMu with ideals Mu ⊂ C[x1, . . . , xn] stable underderivatives,

so each Mu is either (0) or (1).

In other words, M is monomial.

The general case requires more commutative algebra.

Abramovich Resolving varieties and families August 7, 2018 17 / 29

Page 67: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

The monomial part of an ideal - proof

Proof of (1), basic affine case.

Let OY = C[x1, . . . , xn, u1, . . . , um] and assume M = D(M).

The operators

1, u1∂

∂u1, . . . , ul

∂ul

commute and have distinct systems of eigenvalues on the eigenspacesuC[x1, . . . , xn], for distinct monomials u.

Therefore M = ⊕uMu with ideals Mu ⊂ C[x1, . . . , xn] stable underderivatives,

so each Mu is either (0) or (1).

In other words, M is monomial.

The general case requires more commutative algebra.

Abramovich Resolving varieties and families August 7, 2018 17 / 29

Page 68: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

The monomial part of an ideal - proof

Proof of (1), basic affine case.

Let OY = C[x1, . . . , xn, u1, . . . , um] and assume M = D(M).

The operators

1, u1∂

∂u1, . . . , ul

∂ul

commute and have distinct systems of eigenvalues on the eigenspacesuC[x1, . . . , xn], for distinct monomials u.

Therefore M = ⊕uMu with ideals Mu ⊂ C[x1, . . . , xn] stable underderivatives,

so each Mu is either (0) or (1).

In other words, M is monomial.

The general case requires more commutative algebra.

Abramovich Resolving varieties and families August 7, 2018 17 / 29

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dimB = 0: sketch of argument

In characteristic 0, if logordp(I) = a <∞, then D≤a−1I contains anelement x with derivative 1, a maximal contact element.

Carefully applying induction on dimension to an ideal on {x = 0}gives order reduction (Encinas–Villamayor, Bierstone–Milman,W lodarczyk):

Proposition (. . .ℵ-T-W)

Let I be an ideal on a logarithmically smooth Y with

maxp

logordp(I) = a.

There is a functorial logarithmic morphism Y1 → Y , with Y1

logarithmically smooth, such that IOY ′ =M · I1 with M an invertiblemonomial ideal and

maxp

logordp(I1) < a.

Abramovich Resolving varieties and families August 7, 2018 18 / 29

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dimB = 0: sketch of argument

In characteristic 0, if logordp(I) = a <∞, then D≤a−1I contains anelement x with derivative 1, a maximal contact element.

Carefully applying induction on dimension to an ideal on {x = 0}gives order reduction (Encinas–Villamayor, Bierstone–Milman,W lodarczyk):

Proposition (. . .ℵ-T-W)

Let I be an ideal on a logarithmically smooth Y with

maxp

logordp(I) = a.

There is a functorial logarithmic morphism Y1 → Y , with Y1

logarithmically smooth, such that IOY ′ =M · I1 with M an invertiblemonomial ideal and

maxp

logordp(I1) < a.

Abramovich Resolving varieties and families August 7, 2018 18 / 29

Page 71: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

dimB = 0: sketch of argument

In characteristic 0, if logordp(I) = a <∞, then D≤a−1I contains anelement x with derivative 1, a maximal contact element.

Carefully applying induction on dimension to an ideal on {x = 0}gives order reduction (Encinas–Villamayor, Bierstone–Milman,W lodarczyk):

Proposition (. . .ℵ-T-W)

Let I be an ideal on a logarithmically smooth Y with

maxp

logordp(I) = a.

There is a functorial logarithmic morphism Y1 → Y , with Y1

logarithmically smooth, such that IOY ′ =M · I1 with M an invertiblemonomial ideal and

maxp

logordp(I1) < a.

Abramovich Resolving varieties and families August 7, 2018 18 / 29

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dimB = 0: sketch of argument

In characteristic 0, if logordp(I) = a <∞, then D≤a−1I contains anelement x with derivative 1, a maximal contact element.

Carefully applying induction on dimension to an ideal on {x = 0}gives order reduction (Encinas–Villamayor, Bierstone–Milman,W lodarczyk):

Proposition (. . .ℵ-T-W)

Let I be an ideal on a logarithmically smooth Y with

maxp

logordp(I) = a.

There is a functorial logarithmic morphism Y1 → Y , with Y1

logarithmically smooth, such that IOY ′ =M · I1 with M an invertiblemonomial ideal and

maxp

logordp(I1) < a.

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Thank you for your attention!

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Adendum 1. Arbitrary B

(Work in progress)

Main result (ℵ-T-W)

Let Y → B a logarithmically smooth morphism of logarithmically smoothschemes, I ⊂ OY an ideal. There is a log morphism B ′ → B andfunctorial log morphism Y ′ → Y , with Y ′ → B ′ logarithmically smooth,and IOY ′ an invertible monomial ideal.

This is done by relative order reduction, using relative logarithmicderivatives.

Definition

Write D≤aY /B for the sheaf of relative logarithmic differential operators oforder ≤ a. The relative logarithmic order of an ideal I is the minimum asuch that D≤aY /BI = (1).

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Adendum 1. Arbitrary B

(Work in progress)

Main result (ℵ-T-W)

Let Y → B a logarithmically smooth morphism of logarithmically smoothschemes, I ⊂ OY an ideal. There is a log morphism B ′ → B andfunctorial log morphism Y ′ → Y , with Y ′ → B ′ logarithmically smooth,and IOY ′ an invertible monomial ideal.

This is done by relative order reduction, using relative logarithmicderivatives.

Definition

Write D≤aY /B for the sheaf of relative logarithmic differential operators oforder ≤ a. The relative logarithmic order of an ideal I is the minimum asuch that D≤aY /BI = (1).

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Adendum 1. The new step

relordp(I) =∞ if and only if M := D∞Y /BI is a nonunit ideal whichis monomial along the fibers.

Equivalently M = DY /BM is not the unit ideal.

Monomialization Theorem [ℵ-T-W]

Let Y → B a logarithmically smooth morphism of logarithmically smoothschemes, M⊂ OY an ideal with DY /BM =M. There is a log morphismB ′ → B with saturated pullback Y ′ → B ′, and MOY ′ a monomial ideal.

After this one can proceed as in the case “dimB = 0”.

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Adendum 1. The new step

relordp(I) =∞ if and only if M := D∞Y /BI is a nonunit ideal whichis monomial along the fibers.

Equivalently M = DY /BM is not the unit ideal.

Monomialization Theorem [ℵ-T-W]

Let Y → B a logarithmically smooth morphism of logarithmically smoothschemes, M⊂ OY an ideal with DY /BM =M. There is a log morphismB ′ → B with saturated pullback Y ′ → B ′, and MOY ′ a monomial ideal.

After this one can proceed as in the case “dimB = 0”.

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Adendum 1. Proof of Monomialization, special case

Let Y = SpecC[u, v ]→ B = SpecC[w ] with w = uv , and M = (f ).

Proof in this special case.

Every monomial is either uαwk or vαwk .

Once again the operators 1, u ∂∂u − v ∂

∂v commute and have differenteigenvalues on uα, vα.

Exanding f =∑

uαfα +∑

vβfβ, the condition M = DY /BM givesthat only one term survives,

say f = uαfα, with fα ∈ C[w ].

Blowing up (fα) on B has the effect of making it monomial, so fbecomes monomial.

The general case is surprisingly subtle.

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Adendum 1. Proof of Monomialization, special case

Let Y = SpecC[u, v ]→ B = SpecC[w ] with w = uv , and M = (f ).

Proof in this special case.

Every monomial is either uαwk or vαwk .

Once again the operators 1, u ∂∂u − v ∂

∂v commute and have differenteigenvalues on uα, vα.

Exanding f =∑

uαfα +∑

vβfβ, the condition M = DY /BM givesthat only one term survives,

say f = uαfα, with fα ∈ C[w ].

Blowing up (fα) on B has the effect of making it monomial, so fbecomes monomial.

The general case is surprisingly subtle.

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Adendum 1. Proof of Monomialization, special case

Let Y = SpecC[u, v ]→ B = SpecC[w ] with w = uv , and M = (f ).

Proof in this special case.

Every monomial is either uαwk or vαwk .

Once again the operators 1, u ∂∂u − v ∂

∂v commute and have differenteigenvalues on uα, vα.

Exanding f =∑

uαfα +∑

vβfβ, the condition M = DY /BM givesthat only one term survives,

say f = uαfα, with fα ∈ C[w ].

Blowing up (fα) on B has the effect of making it monomial, so fbecomes monomial.

The general case is surprisingly subtle.

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Adendum 1. Proof of Monomialization, special case

Let Y = SpecC[u, v ]→ B = SpecC[w ] with w = uv , and M = (f ).

Proof in this special case.

Every monomial is either uαwk or vαwk .

Once again the operators 1, u ∂∂u − v ∂

∂v commute and have differenteigenvalues on uα, vα.

Exanding f =∑

uαfα +∑

vβfβ, the condition M = DY /BM givesthat only one term survives,

say f = uαfα, with fα ∈ C[w ].

Blowing up (fα) on B has the effect of making it monomial, so fbecomes monomial.

The general case is surprisingly subtle.

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Adendum 1. Proof of Monomialization, special case

Let Y = SpecC[u, v ]→ B = SpecC[w ] with w = uv , and M = (f ).

Proof in this special case.

Every monomial is either uαwk or vαwk .

Once again the operators 1, u ∂∂u − v ∂

∂v commute and have differenteigenvalues on uα, vα.

Exanding f =∑

uαfα +∑

vβfβ, the condition M = DY /BM givesthat only one term survives,

say f = uαfα, with fα ∈ C[w ].

Blowing up (fα) on B has the effect of making it monomial, so fbecomes monomial.

The general case is surprisingly subtle.

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Adendum 1. Proof of Monomialization, special case

Let Y = SpecC[u, v ]→ B = SpecC[w ] with w = uv , and M = (f ).

Proof in this special case.

Every monomial is either uαwk or vαwk .

Once again the operators 1, u ∂∂u − v ∂

∂v commute and have differenteigenvalues on uα, vα.

Exanding f =∑

uαfα +∑

vβfβ, the condition M = DY /BM givesthat only one term survives,

say f = uαfα, with fα ∈ C[w ].

Blowing up (fα) on B has the effect of making it monomial, so fbecomes monomial.

The general case is surprisingly subtle.

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Adendum 1. Proof of Monomialization, special case

Let Y = SpecC[u, v ]→ B = SpecC[w ] with w = uv , and M = (f ).

Proof in this special case.

Every monomial is either uαwk or vαwk .

Once again the operators 1, u ∂∂u − v ∂

∂v commute and have differenteigenvalues on uα, vα.

Exanding f =∑

uαfα +∑

vβfβ, the condition M = DY /BM givesthat only one term survives,

say f = uαfα, with fα ∈ C[w ].

Blowing up (fα) on B has the effect of making it monomial, so fbecomes monomial.

The general case is surprisingly subtle.

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Adendum 1. In virtue of functoriality

Theorem (Temkin)

Resolution of singularities holds for excellent schemes, complex spaces,nonarchimedean spaces, p-adic spaces, formal spaces and for stacks.

This is a consequence of resolution for varieties and schemes,functorial for smooth morphisms (submersions). Moreover

W lodarczyk showed that if one seriously looks for a resolution functor,one is led to a resolution theorem.

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Adendum 1. In virtue of functoriality

Theorem (Temkin)

Resolution of singularities holds for excellent schemes, complex spaces,nonarchimedean spaces, p-adic spaces, formal spaces and for stacks.

This is a consequence of resolution for varieties and schemes,functorial for smooth morphisms (submersions). Moreover

W lodarczyk showed that if one seriously looks for a resolution functor,one is led to a resolution theorem.

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Adendum 1. In virtue of functoriality

Theorem (Temkin)

Resolution of singularities holds for excellent schemes, complex spaces,nonarchimedean spaces, p-adic spaces, formal spaces and for stacks.

This is a consequence of resolution for varieties and schemes,functorial for smooth morphisms (submersions). Moreover

W lodarczyk showed that if one seriously looks for a resolution functor,one is led to a resolution theorem.

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Adendum 2. Order reduction: Example 1

Consider Y1 = SpecC[u, x ] and D = {u = 0}.Let I = (u2, x2).

If one blows up (u, x) the ideal is principalized:

I on the u-chart SpecC[u, x ′] with x = x ′u we have IOY ′1

= (u2),I on the x-chart SpecC[u′, x ] with u′ = xu′ we have IOY ′ = (x2),I which is exceptional hence monomial.

This is in fact the only functorial admissible blowing up.

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Adendum 2. Order reduction: Example 1

Consider Y1 = SpecC[u, x ] and D = {u = 0}.Let I = (u2, x2).

If one blows up (u, x) the ideal is principalized:

I on the u-chart SpecC[u, x ′] with x = x ′u we have IOY ′1

= (u2),I on the x-chart SpecC[u′, x ] with u′ = xu′ we have IOY ′ = (x2),I which is exceptional hence monomial.

This is in fact the only functorial admissible blowing up.

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Adendum 2. Order reduction: Example 1

Consider Y1 = SpecC[u, x ] and D = {u = 0}.Let I = (u2, x2).

If one blows up (u, x) the ideal is principalized:

I on the u-chart SpecC[u, x ′] with x = x ′u we have IOY ′1

= (u2),I on the x-chart SpecC[u′, x ] with u′ = xu′ we have IOY ′ = (x2),I which is exceptional hence monomial.

This is in fact the only functorial admissible blowing up.

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Adendum 2. Order reduction: Example 1

Consider Y1 = SpecC[u, x ] and D = {u = 0}.Let I = (u2, x2).

If one blows up (u, x) the ideal is principalized:

I on the u-chart SpecC[u, x ′] with x = x ′u we have IOY ′1

= (u2),I on the x-chart SpecC[u′, x ] with u′ = xu′ we have IOY ′ = (x2),I which is exceptional hence monomial.

This is in fact the only functorial admissible blowing up.

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Adendum 2. Order reduction: Example 2

Consider Y2 = SpecC[v , x ] and D = {v = 0}.Let I = (v , x2).

Example 1 is the pullback of this via v = u2.

Functoriality says: we need to blow up an ideal whose pullback is(u, x).

This means we need to blow up (v1/2, x).

What is this? What is its blowup?

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Adendum 2. Order reduction: Example 2

Consider Y2 = SpecC[v , x ] and D = {v = 0}.Let I = (v , x2).

Example 1 is the pullback of this via v = u2.

Functoriality says: we need to blow up an ideal whose pullback is(u, x).

This means we need to blow up (v1/2, x).

What is this? What is its blowup?

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Adendum 2. Order reduction: Example 2

Consider Y2 = SpecC[v , x ] and D = {v = 0}.Let I = (v , x2).

Example 1 is the pullback of this via v = u2.

Functoriality says: we need to blow up an ideal whose pullback is(u, x).

This means we need to blow up (v1/2, x).

What is this? What is its blowup?

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Adendum 2. Order reduction: Example 2

Consider Y2 = SpecC[v , x ] and D = {v = 0}.Let I = (v , x2).

Example 1 is the pullback of this via v = u2.

Functoriality says: we need to blow up an ideal whose pullback is(u, x).

This means we need to blow up (v1/2, x).

What is this? What is its blowup?

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Adendum 2. Kummer ideals

Definition

A Kummer monomial is a monomial in the Kummer-etale topology ofY (like v1/2).

A Kummer monomial ideal is a monomial ideal in the Kummer-etaletopology of Y .

A Kummer center is the sum of a Kummer monomial ideal and theideal of a log smooth subscheme.

Locally (x1, . . . , xk , u1/d1 , . . . u

1/d` ).

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Adendum 2. Kummer ideals

Definition

A Kummer monomial is a monomial in the Kummer-etale topology ofY (like v1/2).

A Kummer monomial ideal is a monomial ideal in the Kummer-etaletopology of Y .

A Kummer center is the sum of a Kummer monomial ideal and theideal of a log smooth subscheme.

Locally (x1, . . . , xk , u1/d1 , . . . u

1/d` ).

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Adendum 2. Kummer ideals

Definition

A Kummer monomial is a monomial in the Kummer-etale topology ofY (like v1/2).

A Kummer monomial ideal is a monomial ideal in the Kummer-etaletopology of Y .

A Kummer center is the sum of a Kummer monomial ideal and theideal of a log smooth subscheme.

Locally (x1, . . . , xk , u1/d1 , . . . u

1/d` ).

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Adendum 2. Blowing up Kummer centers

Proposition

Let J be a Kummer center on a logarithmically smooth Y . There is auniversal proper birational Y ′ → Y such that Y ′ is logarithmically smoothand JOY ′ is an invertible ideal.

Example 0

Y = SpecC[v ], with toroidal structure associated to D = {v = 0}, andJ = (v1/2).

There is no log scheme Y ′ satisfying the proposition.

There is a stack Y ′ = Y (√D), the Cadman–Vistoli root stack,

satisfying the proposition!

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Adendum 2. Blowing up Kummer centers

Proposition

Let J be a Kummer center on a logarithmically smooth Y . There is auniversal proper birational Y ′ → Y such that Y ′ is logarithmically smoothand JOY ′ is an invertible ideal.

Example 0

Y = SpecC[v ], with toroidal structure associated to D = {v = 0}, andJ = (v1/2).

There is no log scheme Y ′ satisfying the proposition.

There is a stack Y ′ = Y (√D), the Cadman–Vistoli root stack,

satisfying the proposition!

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Adendum 2. Blowing up Kummer centers

Proposition

Let J be a Kummer center on a logarithmically smooth Y . There is auniversal proper birational Y ′ → Y such that Y ′ is logarithmically smoothand JOY ′ is an invertible ideal.

Example 0

Y = SpecC[v ], with toroidal structure associated to D = {v = 0}, andJ = (v1/2).

There is no log scheme Y ′ satisfying the proposition.

There is a stack Y ′ = Y (√D), the Cadman–Vistoli root stack,

satisfying the proposition!

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Adendum 2. Blowing up Kummer centers

Proposition

Let J be a Kummer center on a logarithmically smooth Y . There is auniversal proper birational Y ′ → Y such that Y ′ is logarithmically smoothand JOY ′ is an invertible ideal.

Example 0

Y = SpecC[v ], with toroidal structure associated to D = {v = 0}, andJ = (v1/2).

There is no log scheme Y ′ satisfying the proposition.

There is a stack Y ′ = Y (√D), the Cadman–Vistoli root stack,

satisfying the proposition!

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Adendum 2. Example 2 concluded

Consider Y2 = SpecC[v , x ] and D = {v = 0}.Let I = (v , x2) and J = (v1/2, x).

associated blowing up Y ′ → Y2 with charts:I Y ′

x := SpecC[v , x , v ′]/(v ′x2 = v), where v ′ = v/x2 (nonsingularscheme).

F Exceptional x = 0, now monomial.F I = (v , x2) transformed into (x2), invertible monomial ideal.F Kummer ideal (v 1/2, x) transformed into monomial ideal (x).

I The v1/2-chart:F stack quotient X ′

v1/2 :=[SpecC[w , y ]

/µ2

],

F where y = x/w and µ2 = {±1} acts via (w , y) 7→ (−w ,−y).F Exceptional w = 0 (monomial).F (v , x2) transformed into invertible monomial ideal (v) = (w 2).F (v 1/2, x) transformed into invertible monomial ideal (w).

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Adendum 2. Example 2 concluded

Consider Y2 = SpecC[v , x ] and D = {v = 0}.Let I = (v , x2) and J = (v1/2, x).

associated blowing up Y ′ → Y2 with charts:I Y ′

x := SpecC[v , x , v ′]/(v ′x2 = v), where v ′ = v/x2 (nonsingularscheme).

F Exceptional x = 0, now monomial.F I = (v , x2) transformed into (x2), invertible monomial ideal.F Kummer ideal (v 1/2, x) transformed into monomial ideal (x).

I The v1/2-chart:F stack quotient X ′

v1/2 :=[SpecC[w , y ]

/µ2

],

F where y = x/w and µ2 = {±1} acts via (w , y) 7→ (−w ,−y).F Exceptional w = 0 (monomial).F (v , x2) transformed into invertible monomial ideal (v) = (w 2).F (v 1/2, x) transformed into invertible monomial ideal (w).

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Adendum 2. Example 2 concluded

Consider Y2 = SpecC[v , x ] and D = {v = 0}.Let I = (v , x2) and J = (v1/2, x).

associated blowing up Y ′ → Y2 with charts:I Y ′

x := SpecC[v , x , v ′]/(v ′x2 = v), where v ′ = v/x2 (nonsingularscheme).

F Exceptional x = 0, now monomial.F I = (v , x2) transformed into (x2), invertible monomial ideal.F Kummer ideal (v 1/2, x) transformed into monomial ideal (x).

I The v1/2-chart:F stack quotient X ′

v1/2 :=[SpecC[w , y ]

/µ2

],

F where y = x/w and µ2 = {±1} acts via (w , y) 7→ (−w ,−y).F Exceptional w = 0 (monomial).F (v , x2) transformed into invertible monomial ideal (v) = (w 2).F (v 1/2, x) transformed into invertible monomial ideal (w).

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Adendum 2. Proof of proposition

Let J be a Kummer center on a logarithmically smooth Y . There is auniversal proper birational Y ′ → Y such that Y ′ is logarithmically smoothand JOY ′ is an invertible ideal.

There is a stack Y with coarse moduli space Y such that J := JOYis an ideal.

Let Y ′ → Y be the blowup of J with exceptional E .

Let Y ′ → BGm be the classifying morphism.

Y ′ is the relative coarse moduli space of Y ′ → Y × BGm.

One shows this is independent of choices. ♠

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Adendum 2. Proof of proposition

Let J be a Kummer center on a logarithmically smooth Y . There is auniversal proper birational Y ′ → Y such that Y ′ is logarithmically smoothand JOY ′ is an invertible ideal.

There is a stack Y with coarse moduli space Y such that J := JOYis an ideal.

Let Y ′ → Y be the blowup of J with exceptional E .

Let Y ′ → BGm be the classifying morphism.

Y ′ is the relative coarse moduli space of Y ′ → Y × BGm.

One shows this is independent of choices. ♠

Abramovich Resolving varieties and families August 7, 2018 29 / 29

Page 108: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Adendum 2. Proof of proposition

Let J be a Kummer center on a logarithmically smooth Y . There is auniversal proper birational Y ′ → Y such that Y ′ is logarithmically smoothand JOY ′ is an invertible ideal.

There is a stack Y with coarse moduli space Y such that J := JOYis an ideal.

Let Y ′ → Y be the blowup of J with exceptional E .

Let Y ′ → BGm be the classifying morphism.

Y ′ is the relative coarse moduli space of Y ′ → Y × BGm.

One shows this is independent of choices. ♠

Abramovich Resolving varieties and families August 7, 2018 29 / 29

Page 109: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Adendum 2. Proof of proposition

Let J be a Kummer center on a logarithmically smooth Y . There is auniversal proper birational Y ′ → Y such that Y ′ is logarithmically smoothand JOY ′ is an invertible ideal.

There is a stack Y with coarse moduli space Y such that J := JOYis an ideal.

Let Y ′ → Y be the blowup of J with exceptional E .

Let Y ′ → BGm be the classifying morphism.

Y ′ is the relative coarse moduli space of Y ′ → Y × BGm.

One shows this is independent of choices. ♠

Abramovich Resolving varieties and families August 7, 2018 29 / 29

Page 110: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Adendum 2. Proof of proposition

Let J be a Kummer center on a logarithmically smooth Y . There is auniversal proper birational Y ′ → Y such that Y ′ is logarithmically smoothand JOY ′ is an invertible ideal.

There is a stack Y with coarse moduli space Y such that J := JOYis an ideal.

Let Y ′ → Y be the blowup of J with exceptional E .

Let Y ′ → BGm be the classifying morphism.

Y ′ is the relative coarse moduli space of Y ′ → Y × BGm.

One shows this is independent of choices. ♠

Abramovich Resolving varieties and families August 7, 2018 29 / 29

Page 111: Dan Abramovich Brown Universityabrmovic/PAPERS/Resolution-ICM.pdf · Abramovich Resolving varieties and families August 7, 2018 8 / 29. Log smooth schemes and log smooth morphisms

Adendum 2. Proof of proposition

Let J be a Kummer center on a logarithmically smooth Y . There is auniversal proper birational Y ′ → Y such that Y ′ is logarithmically smoothand JOY ′ is an invertible ideal.

There is a stack Y with coarse moduli space Y such that J := JOYis an ideal.

Let Y ′ → Y be the blowup of J with exceptional E .

Let Y ′ → BGm be the classifying morphism.

Y ′ is the relative coarse moduli space of Y ′ → Y × BGm.

One shows this is independent of choices. ♠

Abramovich Resolving varieties and families August 7, 2018 29 / 29