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

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 Simons Conference on Rationality New York, July 27, 2020 Abramovich Resolution and logarithmic resolution New York, July 27, 2020 1 / 24

Example: Whitney’s umbrella “You can’t just blow up the worst singular locus” Consider X = V ( x 2 − y 2 z ) Abramovich Resolution and logarithmic resolution New York, July 27, 2020 2 / 24

Example: Whitney’s umbrella “You can’t just blow up the worst singular locus” Consider X = V ( x 2 − y 2 z ) The worst singularity is the origin. In the z chart we get x = x ′ z , y = y ′ z , giving z 2 ( x ′ 2 − y ′ 2 z ) = 0 . x ′ 2 z 2 − y ′ 2 z 3 = 0, or Abramovich Resolution and logarithmic resolution New York, July 27, 2020 2 / 24

Example: Whitney’s umbrella “You can’t just blow up the worst singular locus” Consider X = V ( x 2 − y 2 z ) The worst singularity is the origin. In the z chart we get x = x ′ z , y = y ′ z , giving z 2 ( x ′ 2 − y ′ 2 z ) = 0 . x ′ 2 z 2 − y ′ 2 z 3 = 0, or The first term is exceptional, the second is the same as X . Abramovich Resolution and logarithmic resolution New York, July 27, 2020 2 / 24

Two theorems Nevertheless: Theorem ( ℵ -T-W, McQuillan, 2019, characteristic 0) There is a functor F associating to a singular subvariety a X ⊂ Y of a smooth J with stack theoretic weighted blowing up Y ′ → Y and variety Y , a center ¯ proper transform ( X ′ ⊂ Y ′ ) = F ( X ⊂ Y ) such that maxinv( X ′ ) < maxinv( X ) . In particular, for some n the iterate ( X n ⊂ Y n ) := F ◦ n ( X ⊂ Y ) of F has X n smooth. a or substack Abramovich Resolution and logarithmic resolution New York, July 27, 2020 3 / 24

Two theorems Nevertheless: Theorem ( ℵ -T-W, McQuillan, 2019, characteristic 0) There is a functor F associating to a singular subvariety a X ⊂ Y of a smooth J with stack theoretic weighted blowing up Y ′ → Y and variety Y , a center ¯ proper transform ( X ′ ⊂ Y ′ ) = F ( X ⊂ Y ) such that maxinv( X ′ ) < maxinv( X ) . In particular, for some n the iterate ( X n ⊂ Y n ) := F ◦ n ( X ⊂ Y ) of F has X n smooth. a or substack Theorem (Quek, 2020, characteristic 0) There is a functor F associating to a logarithmically singular subvariety a 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 ( X n ⊂ Y n ) := F ◦ n ( X ⊂ Y ) of F has X n logarithmically smooth. a or subtack Abramovich Resolution and logarithmic resolution New York, July 27, 2020 3 / 24

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. Abramovich Resolution and logarithmic resolution New York, July 27, 2020 4 / 24

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: Abramovich Resolution and logarithmic resolution New York, July 27, 2020 4 / 24

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 toroidal morphism X → B of toroidal embeddings is ´ etale locally isomorphic to a torus equivariant dominant morphism. Abramovich Resolution and logarithmic resolution New York, July 27, 2020 4 / 24

Examples of toroidal morphisms e.g. Spec C [ x , y , z ] / ( xy − z 2 ) → Spec C , Spec C [ x 2 ] , Spec C [ x ] → toric blowups Abramovich Resolution and logarithmic resolution New York, July 27, 2020 5 / 24

Context: functoriality Hironaka’s theorem is functorial. [Bierstone–Milman, Villamayor,...] [ ℵ -Karu 2000] is not: relied on deJong’s method. Abramovich Resolution and logarithmic resolution New York, July 27, 2020 6 / 24

Context: functoriality Hironaka’s theorem is functorial. [Bierstone–Milman, Villamayor,...] [ ℵ -Karu 2000] is not: relied on deJong’s method. For higher dimensional moduli one wants functoriality. Abramovich Resolution and logarithmic resolution New York, July 27, 2020 6 / 24

Context: functoriality Hironaka’s theorem is functorial. [Bierstone–Milman, Villamayor,...] [ ℵ -Karu 2000] is not: relied on deJong’s method. For higher dimensional moduli one wants functoriality. Theorem ( ℵ -T-W 2020) Given X → B there is a relatively functorial logarithmically smooth modification X ′ → B ′ . Abramovich Resolution and logarithmic resolution New York, July 27, 2020 6 / 24

Context: functoriality Hironaka’s theorem is functorial. [Bierstone–Milman, Villamayor,...] [ ℵ -Karu 2000] is not: relied on deJong’s method. For higher dimensional moduli one wants functoriality. Theorem ( ℵ -T-W 2020) Given X → B there is a relatively functorial logarithmically smooth modification X ′ → B ′ . This respects Aut B X . Does not modify log smooth fibers. Abramovich Resolution and logarithmic resolution New York, July 27, 2020 6 / 24

Context: principalization Following Hironaka, the above theorem is based on embedded methods: Theorem ( ℵ -T-W 2020) Given Y → B logarithmically smooth and I ⊂ O Y , there is a relatively functorial logarithmically smooth modification Y ′ → B ′ such that IO Y ′ is monomial. Abramovich Resolution and logarithmic resolution New York, July 27, 2020 7 / 24

Context: principalization Following Hironaka, the above theorem is based on embedded methods: Theorem ( ℵ -T-W 2020) Given Y → B logarithmically smooth and I ⊂ O Y , there is a relatively functorial logarithmically smooth modification Y ′ → B ′ such that IO Y ′ is monomial. This is done by a sequence of logarithmic modifications, where in each step E becomes part of the divisor D Y ′ . Abramovich Resolution and logarithmic resolution New York, July 27, 2020 7 / 24

Example 1 Y = Spec k [ x , u ]; D Y = V ( u ); B = Spec k ; Abramovich Resolution and logarithmic resolution New York, July 27, 2020 8 / 24

Example 1 I = ( x 2 , u 2 ). Y = Spec k [ x , u ]; D Y = V ( u ); B = Spec k ; Abramovich Resolution and logarithmic resolution New York, July 27, 2020 8 / 24

Example 1 I = ( x 2 , u 2 ). Y = Spec k [ x , u ]; D Y = V ( u ); B = Spec k ; Blow up J = ( x , u ) IO Y ′ = O ( − 2 E ) Abramovich Resolution and logarithmic resolution New York, July 27, 2020 8 / 24

Example 1/2 I = ( x 2 , u 2 ) Y = Spec k [ x , u ]; D Y = V ( u ); Abramovich Resolution and logarithmic resolution New York, July 27, 2020 9 / 24

Example 1/2 I = ( x 2 , u 2 ) Y = Spec k [ x , u ]; D Y = V ( u ); I 0 = ( x 2 , v ), Y 0 = Spec k [ x , v ]; D Y 0 = V ( v ); f ∗ v = u 2 I = f ∗ I 0 f : Y → Y 0 so Abramovich Resolution and logarithmic resolution New York, July 27, 2020 9 / 24

Example 1/2 I = ( x 2 , u 2 ) Y = Spec k [ x , u ]; D Y = V ( u ); I 0 = ( x 2 , v ), Y 0 = Spec k [ x , v ]; D Y 0 = V ( v ); f ∗ v = u 2 I = f ∗ I 0 f : Y → Y 0 so By functoriality blow up J 0 so that f ∗ J 0 = J = ( x , u ). Abramovich Resolution and logarithmic resolution New York, July 27, 2020 9 / 24

Example 1/2 I = ( x 2 , u 2 ) Y = Spec k [ x , u ]; D Y = V ( u ); I 0 = ( x 2 , v ), Y 0 = Spec k [ x , v ]; D Y 0 = V ( v ); f ∗ v = u 2 I = f ∗ I 0 f : Y → Y 0 so By functoriality blow up J 0 so that f ∗ J 0 = J = ( x , u ). Blow up J 0 = ( x , √ v ) Whatever J 0 is, the blowup is a stack. Abramovich Resolution and logarithmic resolution New York, July 27, 2020 9 / 24

Example 1/2: charts x chart: v = v ′ x 2 : ( x 2 , v ) = ( x 2 , v ′ x 2 ) = ( x 2 ) exceptional, so monomial. Abramovich Resolution and logarithmic resolution New York, July 27, 2020 10 / 24

Example 1/2: charts x chart: v = v ′ x 2 : ( x 2 , v ) = ( x 2 , v ′ x 2 ) = ( x 2 ) exceptional, so monomial. √ v chart: v = w 2 , x = x ′ w , with ± 1 action ( x ′ , w ) �→ ( − x ′ , − w ): ( x 2 , v ) = ( x ′ 2 w 2 , w 2 ) = ( w 2 ) exceptional, so monomial. Abramovich Resolution and logarithmic resolution New York, July 27, 2020 10 / 24

Example 1/2: charts x chart: v = v ′ x 2 : ( x 2 , v ) = ( x 2 , v ′ x 2 ) = ( x 2 ) exceptional, so monomial. √ v chart: v = w 2 , x = x ′ w , with ± 1 action ( x ′ , w ) �→ ( − x ′ , − w ): ( x 2 , v ) = ( x ′ 2 w 2 , w 2 ) = ( w 2 ) exceptional, so monomial. The schematic quotient of the above is not toroidal. Abramovich Resolution and logarithmic resolution New York, July 27, 2020 10 / 24