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 / 18
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. Abramovich Resolution and logarithmic resolution New York, July 27, 2020 2 / 18
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. Abramovich Resolution and logarithmic resolution New York, July 27, 2020 2 / 18
Example: Whitney’s umbrella Consider X = V ( x 2 − y 2 z ) Abramovich Resolution and logarithmic resolution New York, July 27, 2020 3 / 18
Example: Whitney’s umbrella Consider X = V ( x 2 − y 2 z ) (image by Eleonore Faber). Abramovich Resolution and logarithmic resolution New York, July 27, 2020 3 / 18
Example: Whitney’s umbrella Consider X = V ( x 2 − y 2 z ) (image by Eleonore Faber). 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 3 / 18
Example: Whitney’s umbrella Consider X = V ( x 2 − y 2 z ) (image by Eleonore Faber). 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 3 / 18
Two theorems Nevertheless: 1 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 1 See slides “context” Abramovich Resolution and logarithmic resolution New York, July 27, 2020 4 / 18
Two theorems Nevertheless: 1 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 1 See slides “context” Abramovich Resolution and logarithmic resolution New York, July 27, 2020 4 / 18
The umbrella again For X = V ( x 2 − y 2 z ) we have inv p ( X ) = (2 , 3 , 3) Abramovich Resolution and logarithmic resolution New York, July 27, 2020 5 / 18
The umbrella again For X = V ( x 2 − y 2 z ) we have inv p ( X ) = (2 , 3 , 3) We read it from the degrees of terms. Abramovich Resolution and logarithmic resolution New York, July 27, 2020 5 / 18
The umbrella again For X = V ( x 2 − y 2 z ) we have inv p ( X ) = (2 , 3 , 3) We read it from the degrees of terms. The center is: J = ( x 2 , y 3 , z 3 ); ¯ J = ( x 1 / 3 , y 1 / 2 , z 1 / 2 ). Abramovich Resolution and logarithmic resolution New York, July 27, 2020 5 / 18
The umbrella again For X = V ( x 2 − y 2 z ) we have inv p ( X ) = (2 , 3 , 3) We read it from the degrees of terms. The center is: J = ( x 2 , y 3 , z 3 ); ¯ J = ( x 1 / 3 , y 1 / 2 , z 1 / 2 ). The blowing up Y ′ → Y makes ¯ J = ( x 1 / 3 , y 1 / 2 , z 1 / 2 ) principal. Explicitly: The z chart has x = w 3 x ′ , y = w 2 y ′ , z = w 2 with chart Y ′ = [ Spec C [ x ′ , y ′ , w ] / ( ± 1) ] , with action of ( ± 1) given by ( x ′ , y ′ , w ) �→ ( − x ′ , y ′ , − w ). Abramovich Resolution and logarithmic resolution New York, July 27, 2020 5 / 18
The umbrella again For X = V ( x 2 − y 2 z ) we have inv p ( X ) = (2 , 3 , 3) We read it from the degrees of terms. The center is: J = ( x 2 , y 3 , z 3 ); ¯ J = ( x 1 / 3 , y 1 / 2 , z 1 / 2 ). The blowing up Y ′ → Y makes ¯ J = ( x 1 / 3 , y 1 / 2 , z 1 / 2 ) principal. Explicitly: The z chart has x = w 3 x ′ , y = w 2 y ′ , z = w 2 with chart Y ′ = [ Spec C [ x ′ , y ′ , w ] / ( ± 1) ] , with action of ( ± 1) given by ( x ′ , y ′ , w ) �→ ( − x ′ , y ′ , − w ). The transformed equation is w 6 ( x ′ 2 − y ′ 2 ) , Abramovich Resolution and logarithmic resolution New York, July 27, 2020 5 / 18
The umbrella again For X = V ( x 2 − y 2 z ) we have inv p ( X ) = (2 , 3 , 3) We read it from the degrees of terms. The center is: J = ( x 2 , y 3 , z 3 ); ¯ J = ( x 1 / 3 , y 1 / 2 , z 1 / 2 ). The blowing up Y ′ → Y makes ¯ J = ( x 1 / 3 , y 1 / 2 , z 1 / 2 ) principal. Explicitly: The z chart has x = w 3 x ′ , y = w 2 y ′ , z = w 2 with chart Y ′ = [ Spec C [ x ′ , y ′ , w ] / ( ± 1) ] , with action of ( ± 1) given by ( x ′ , y ′ , w ) �→ ( − x ′ , y ′ , − w ). The transformed equation is w 6 ( x ′ 2 − y ′ 2 ) , and the invariant of the proper transform x ′ 2 − y ′ 2 is (2 , 2) < (2 , 3 , 3). Abramovich Resolution and logarithmic resolution New York, July 27, 2020 5 / 18
Order (following Koll´ ar’s book) We fix Y smooth and I ⊂ O Y . Definition For p ∈ Y let ord p ( I ) = max { a : I ⊆ m a p } . Abramovich Resolution and logarithmic resolution New York, July 27, 2020 6 / 18
Order (following Koll´ ar’s book) We fix Y smooth and I ⊂ O Y . Definition For p ∈ Y let ord p ( I ) = max { a : I ⊆ m a p } . We denote by D a the sheaf of a -th order differential operators. Abramovich Resolution and logarithmic resolution New York, July 27, 2020 6 / 18
Order (following Koll´ ar’s book) We fix Y smooth and I ⊂ O Y . Definition For p ∈ Y let ord p ( I ) = max { a : I ⊆ m a p } . We denote by D a the sheaf of a -th order differential operators. We note that ord p ( I ) = min { a : D a ( I p ) } = (1). The invariant starts with a 1 = ord p ( I ). Abramovich Resolution and logarithmic resolution New York, July 27, 2020 6 / 18
Order (following Koll´ ar’s book) We fix Y smooth and I ⊂ O Y . Definition For p ∈ Y let ord p ( I ) = max { a : I ⊆ m a p } . We denote by D a the sheaf of a -th order differential operators. We note that ord p ( I ) = min { a : D a ( I p ) } = (1). The invariant starts with a 1 = ord p ( I ). Proposition The order is upper semicontinuous. Proof. V ( D a − 1 I ) = { p : ord p ( I ) ≥ a } . ♠ Abramovich Resolution and logarithmic resolution New York, July 27, 2020 6 / 18
Maximal contact (following Koll´ ar’s book) Definition (Giraud, Hironaka) A regular parameter x 1 ∈ D a 1 − 1 I p is called a maximal contact element. The center starts with ( x a 1 1 , . . . ). Abramovich Resolution and logarithmic resolution New York, July 27, 2020 7 / 18
Maximal contact (following Koll´ ar’s book) Definition (Giraud, Hironaka) A regular parameter x 1 ∈ D a 1 − 1 I p is called a maximal contact element. The center starts with ( x a 1 1 , . . . ). Lemma (Giraud, Hironaka) In characteristic 0 a maximal contact exists on an open neighborhood of p. Since 1 ∈ D a 1 I p there is x 1 with derivative 1. This derivative is a unit in a neighborhood. Abramovich Resolution and logarithmic resolution New York, July 27, 2020 7 / 18
Maximal contact (following Koll´ ar’s book) Definition (Giraud, Hironaka) A regular parameter x 1 ∈ D a 1 − 1 I p is called a maximal contact element. The center starts with ( x a 1 1 , . . . ). Lemma (Giraud, Hironaka) In characteristic 0 a maximal contact exists on an open neighborhood of p. Since 1 ∈ D a 1 I p there is x 1 with derivative 1. This derivative is a unit in a neighborhood. Example For I = ( x 2 − y 2 z ) we have ord p I = 2 with x 1 = x (or α x +h.o.t. in D ( I )). Abramovich Resolution and logarithmic resolution New York, July 27, 2020 7 / 18
Coefficient ideals (treated following Koll´ ar) We must restrict to x 1 = 0 the data of all I , DI , . . . , D a 1 − 1 I with corresponding weights a 1 , a 1 − 1 , . . . , 1 . Abramovich Resolution and logarithmic resolution New York, July 27, 2020 8 / 18
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