Resolution by weighted blowing up Dan Abramovich, Brown University - PowerPoint PPT Presentation

Resolution by weighted blowing up Dan Abramovich, Brown University Joint work with Michael T¨ emkin and Jaros� law W� lodarczyk Also parallel work by M. McQuillan with G. Marzo Rational points on irrational varieties Columbia, September 13, 2019 Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 1 / 18

How to resolve a curve To resolve a singular curve C (1) find a singular point x ∈ C , (2) blow it up. Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 2 / 18

How to resolve a curve To resolve a singular curve C (1) find a singular point x ∈ C , (2) blow it up. Fact p a gets smaller. Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 2 / 18

How to resolve a surface To resolve a singular surface S one wants to (1) find the worst singular locus C ∈ S , (2) C is smooth - blow it up. Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 3 / 18

How to resolve a surface To resolve a singular surface S one wants to (1) find the worst singular locus C ∈ S , (2) C is smooth - blow it up. Fact This in general does not get better. Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 3 / 18

Example: Whitney’s umbrella Consider S = V ( x 2 − y 2 z ) Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 4 / 18

Example: Whitney’s umbrella Consider S = V ( x 2 − y 2 z ) (image by Eleonore Faber). Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 4 / 18

Example: Whitney’s umbrella Consider S = V ( x 2 − y 2 z ) (image by Eleonore Faber). (1) The worst singularity is the origin. (2) In the z chart we get x = x 3 z , y = y 3 z , giving 3 z 2 − y 2 3 z 3 = 0, x 2 z 2 ( x 2 3 − y 2 or 3 z ) = 0 . Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 4 / 18

Example: Whitney’s umbrella Consider S = V ( x 2 − y 2 z ) (image by Eleonore Faber). (1) The worst singularity is the origin. (2) In the z chart we get x = x 3 z , y = y 3 z , giving 3 z 2 − y 2 3 z 3 = 0, x 2 z 2 ( x 2 3 − y 2 or 3 z ) = 0 . The first term is exceptional, the second is the same as S . Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 4 / 18

Example: Whitney’s umbrella Consider S = V ( x 2 − y 2 z ) (image by Eleonore Faber). (1) The worst singularity is the origin. (2) In the z chart we get x = x 3 z , y = y 3 z , giving 3 z 2 − y 2 3 z 3 = 0, x 2 z 2 ( x 2 3 − y 2 or 3 z ) = 0 . The first term is exceptional, the second is the same as S . Classical solution: (a) Remember exceptional divisors (this is OK) (b) Remember their history (this is a pain) Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 4 / 18

Main result Nevertheless: Theorem ( ℵ -T-W, MM, “weighted Hironaka”, characteristic 0) There is a procedure F associating to a singular subvariety X ⊂ Y embedded with pure codimension c in a smooth variety Y , a center ¯ J with 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 ( X n ⊂ Y n ) := F ◦ n ( X ⊂ Y ) of F has X n smooth. Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 5 / 18

Main result Nevertheless: Theorem ( ℵ -T-W, MM, “weighted Hironaka”, characteristic 0) There is a procedure F associating to a singular subvariety X ⊂ Y embedded with pure codimension c in a smooth variety Y , a center ¯ J with 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 ( X n ⊂ Y n ) := F ◦ n ( X ⊂ Y ) of F has X n smooth. Here procedure means a functor for smooth surjective morphisms: if f : Y 1 ։ Y smooth then J 1 = f − 1 J and Y ′ 1 = Y 1 × Y Y ′ . Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 5 / 18

Preview on invariants For p ∈ X we define Q ≤ n inv p ( X ) ∈ Γ ⊂ ≥ 0 , with Γ well-ordered, and show Proposition it is lexicographically upper-semi-continuous, and p ∈ X is smooth ⇔ inv p ( X ) = min Γ . We define maxinv( X ) = max p inv p ( X ). Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 6 / 18

Preview on invariants For p ∈ X we define Q ≤ n inv p ( X ) ∈ Γ ⊂ ≥ 0 , with Γ well-ordered, and show Proposition it is lexicographically upper-semi-continuous, and p ∈ X is smooth ⇔ inv p ( X ) = min Γ . We define maxinv( X ) = max p inv p ( X ). Example inv p ( V ( x 2 − y 2 z )) = (2 , 3 , 3) Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 6 / 18

Preview on invariants For p ∈ X we define Q ≤ n inv p ( X ) ∈ Γ ⊂ ≥ 0 , with Γ well-ordered, and show Proposition it is lexicographically upper-semi-continuous, and p ∈ X is smooth ⇔ inv p ( X ) = min Γ . We define maxinv( X ) = max p inv p ( X ). Example inv p ( V ( x 2 − y 2 z )) = (2 , 3 , 3) Remark These invariants have been in our arsenal for ages. Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 6 / 18

Preview of centers If inv p ( X ) = maxinv( X ) = ( a 1 , . . . , a k ) then, locally at p 1 , . . . , x a k J = ( x a 1 k ) . Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 7 / 18

Preview of centers If inv p ( X ) = maxinv( X ) = ( a 1 , . . . , a k ) then, locally at p 1 , . . . , x a k J = ( x a 1 k ) . Write ( a 1 , . . . , a k ) = ℓ (1 / w 1 , . . . , 1 / w k ) with w i , ℓ ∈ N and gcd( w 1 , . . . , w k ) = 1 . We set J = ( x 1 / w 1 , . . . , x 1 / w k ¯ ) . 1 k Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 7 / 18

Preview of centers If inv p ( X ) = maxinv( X ) = ( a 1 , . . . , a k ) then, locally at p 1 , . . . , x a k J = ( x a 1 k ) . Write ( a 1 , . . . , a k ) = ℓ (1 / w 1 , . . . , 1 / w k ) with w i , ℓ ∈ N and gcd( w 1 , . . . , w k ) = 1 . We set J = ( x 1 / w 1 , . . . , x 1 / w k ¯ ) . 1 k Example For X = V ( x 2 − y 2 z ) we have J = ( x 2 , y 3 , z 3 ); ¯ J = ( x 1 / 3 , y 1 / 2 , z 1 / 2 ). Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 7 / 18

Preview of centers If inv p ( X ) = maxinv( X ) = ( a 1 , . . . , a k ) then, locally at p 1 , . . . , x a k J = ( x a 1 k ) . Write ( a 1 , . . . , a k ) = ℓ (1 / w 1 , . . . , 1 / w k ) with w i , ℓ ∈ N and gcd( w 1 , . . . , w k ) = 1 . We set J = ( x 1 / w 1 , . . . , x 1 / w k ¯ ) . 1 k Example For X = V ( x 2 − y 2 z ) we have J = ( x 2 , y 3 , z 3 ); ¯ J = ( x 1 / 3 , y 1 / 2 , z 1 / 2 ). Remark J has been staring in our face for a while. Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 7 / 18

Example: blowing up Whitney’s umbrella x 2 = y 2 z 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 3 , y = w 2 y 3 , z = w 2 with chart Y ′ = [ Spec C [ x 3 , y 3 , w ] / ( ± 1) ] , with action of ( ± 1) given by ( x 3 , y 3 , w ) �→ ( − x 3 , y 3 , − w ). Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 8 / 18

Example: blowing up Whitney’s umbrella x 2 = y 2 z 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 3 , y = w 2 y 3 , z = w 2 with chart Y ′ = [ Spec C [ x 3 , y 3 , w ] / ( ± 1) ] , with action of ( ± 1) given by ( x 3 , y 3 , w ) �→ ( − x 3 , y 3 , − w ). The transformed equation is w 6 ( x 2 3 − y 2 3 ) , Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 8 / 18

Example: blowing up Whitney’s umbrella x 2 = y 2 z 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 3 , y = w 2 y 3 , z = w 2 with chart Y ′ = [ Spec C [ x 3 , y 3 , w ] / ( ± 1) ] , with action of ( ± 1) given by ( x 3 , y 3 , w ) �→ ( − x 3 , y 3 , − w ). The transformed equation is w 6 ( x 2 3 − y 2 3 ) , and the invariant of the proper transform ( x 2 3 − y 2 3 ) is (2 , 2) < (2 , 3 , 3). Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 8 / 18

Example: blowing up Whitney’s umbrella x 2 = y 2 z 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 3 , y = w 2 y 3 , z = w 2 with chart Y ′ = [ Spec C [ x 3 , y 3 , w ] / ( ± 1) ] , with action of ( ± 1) given by ( x 3 , y 3 , w ) �→ ( − x 3 , y 3 , − w ). The transformed equation is w 6 ( x 2 3 − y 2 3 ) , and the invariant of the proper transform ( x 2 3 − y 2 3 ) is (2 , 2) < (2 , 3 , 3). In fact, X has begged to be blown up like this all along. Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 8 / 18

✤ Definition of Y ′ → Y J = ( x 1 / w 1 , . . . , x 1 / w k Let ¯ ) . Define the graded algebra 1 k A ¯ ⊂ O Y [ T ] J as the integral closure of the image of � O Y [ T ] O Y [ Y 1 , . . . , Y n ] � x i T w i . Y i Abramovich Resolution by weighted blowing up Columbia, September 13, 2019 9 / 18