Factoring birational maps in three dimensional minimal modelprogram

Jungkai Alfred ChenNational Taiwan University &

National Center for Theoretical Sciences, Taipei Office

Symposium on Projective Algebraic Varieties and ModuliFeb. 19, 2013

THE MVL Hotel, Yeosu, Korea

Minimal Model Program

Try to find good model inside a birational equivalence class.

X is minimal if and only if the canonical divisor KX is nef.

If KX is not nef, then there exists a contraction map ϕ : X → Y .

Minimal Model Program

Try to find good model inside a birational equivalence class.

X is minimal if and only if the canonical divisor KX is nef.

If KX is not nef, then there exists a contraction map ϕ : X → Y .

Minimal Model Program

Try to find good model inside a birational equivalence class.

X is minimal if and only if the canonical divisor KX is nef.

If KX is not nef, then there exists a contraction map ϕ : X → Y .

Minimal Model Program

Given a non-singular variety X , can we have a sequence ofcontraction maps

X = X0 → X1 → . . .→ Xn,

such that Xn is minimal or admits a ”Mori fiber space”?

We need some modification:We need to allow mild singularities, i.e. terminal singularities.We need to allow birational surgeries, called ”flips”.

Minimal Model Program

Given a non-singular variety X , can we have a sequence ofcontraction maps

X = X0 → X1 → . . .→ Xn,

such that Xn is minimal or admits a ”Mori fiber space”?

We need some modification:We need to allow mild singularities, i.e. terminal singularities.

We need to allow birational surgeries, called ”flips”.

Minimal Model Program

Given a non-singular variety X , can we have a sequence ofcontraction maps

X = X0 → X1 → . . .→ Xn,

such that Xn is minimal or admits a ”Mori fiber space”?

We need some modification:We need to allow mild singularities, i.e. terminal singularities.We need to allow birational surgeries, called ”flips”.

Minimal Model Program in Dimension Three

Kawamata, Mori, Reid, Kollar, Shokurov, etc.

Three dimensional terminal singularities are classified.Extremal neighborhood are classified.Existence of flips.Termination of flips.

Minimal Model Program in Dimension Three

Kawamata, Mori, Reid, Kollar, Shokurov, etc.Three dimensional terminal singularities are classified.

Extremal neighborhood are classified.Existence of flips.Termination of flips.

Minimal Model Program in Dimension Three

Kawamata, Mori, Reid, Kollar, Shokurov, etc.Three dimensional terminal singularities are classified.Extremal neighborhood are classified.

Existence of flips.Termination of flips.

Minimal Model Program in Dimension Three

Kawamata, Mori, Reid, Kollar, Shokurov, etc.Three dimensional terminal singularities are classified.Extremal neighborhood are classified.Existence of flips.

Termination of flips.

Minimal Model Program in Dimension Three

Kawamata, Mori, Reid, Kollar, Shokurov, etc.Three dimensional terminal singularities are classified.Extremal neighborhood are classified.Existence of flips.Termination of flips.

Minimal Model Program in Dimension Three

TheoremGiven a three dimensional Q-factorial terminal complex projectivevariety X , there is a sequence of divisorial contractions and flips

X = X0 99K X1 99K . . . 99K Xn,

such that either Xn is minimal or Xn admits a ”Mori fiber space”.

Explicit Minimal Model Program

QuestionCan we describe birational maps in minimal model programexplicitly, at least in dimension three?

Terminal Singularities in Dimension Three

P ∈ X is terminal, then exists a canonical cover

(P ∈ X )→ (P ∈ X )

so that(P ∈ X ) ∼= (P ∈ X )/µr

for some cyclic group µr of order r .

Terminal Singularities in Dimension Three

r = 1,P ∈ X is an isolated cDV point and Gorenstein.

1 cA: (xy + zn+1 + ug(x , y , z , u) = 0) ∈ C4.

2 cD: (x2 + y 2z + zn−1 + ug(x , y , z , u) = 0) ∈ C4.

3 cE6: (x2 + y 3 + z4 + ug(x , y , z , u) = 0) ∈ C4.

4 cE7: (x2 + y 3 + yz3 + ug(x , y , z , u) = 0) ∈ C4.

5 cE8: (x2 + y 3 + z5 + ug(x , y , z , u) = 0) ∈ C4.

Terminal Singularities in Dimension Three

r > 1,P ∈ X is a quotient of a non-singular point or an isolated cDVpoint.

1 quotient: 1r (1,−1, s), with (r , s) = 1.

2 cA/r : (xy + f (z , u) = 0) ∈ C4/1r (a, r − a, 1, r).

3 cAx/2: (x2 + y 2 + f (z , u) = 0) ∈ C4/12 (1, 0, 1, 0).

4 cAx/4: (x2 + y 2 + f (z , u) = 0) ∈ C4/14 (1, 3, 1, 2).

5 cD/2: P ∈ X is given by (ϕ = 0) ⊂ C4/12 (1, 1, 0, 1) with ϕ of

cD type.

6 cD/3: P ∈ X is given as (ϕ = 0) ⊂ C4/13 (0, 2, 1, 1) with ϕ of

cD type.

7 cE/2: (x2 + y 3 + yg(z , u) + h(z , u) = 0) ∈ C4/12 (1, 0, 1, 1).

Mori’s classification

Let f : Y → X be an extremal contraction from a smooththreefold Y .Then f is one of the following:

1 f = BlΓ, Γ ⊂ X is a smooth curve and X is smooth.

2 f = BlP , P ∈ X is a smooth point and X is smooth.

3 f = BlP , (P ∈ X ) ∼= o ∈ (x2 + y 2 + z2 + u2 = 0) ⊂ C4.

4 f = BlP , (P ∈ X ) ∼= o ∈ (x2 + y 2 + z2 + u3 = 0) ⊂ C4.

5 f = wBlv , (P ∈ X ) ∼= o ∈ C3/v with v = 12 (1, 1, 1).

Cutkosky’s classification

Let f : Y → X be an extremal contraction from a Gorensteinthreefold Y .Then f is one of the followig:

1 f = BlΓ, Γ ⊂ X is a lci curve and X is smooth.

2 f = BlP , P ∈ X is a smooth point and X is smooth.

3 f = BlP , (P ∈ X ) ∼= o ∈ (x2 + y 2 + z2 + un = 0) ⊂ C4,n ≥ 2.

4 f = wBlv , (P ∈ X ) ∼= o ∈ C3/v with v = 12 (1, 1, 1).

Divisorial Contractions to points

Let f : Y → X be a divisorial contraction to a point P ∈ X .

1 P ∈ X is nonsingular, then f = wBl(1,a,b) with(a, b) = 1.[Kawakita]

2 P ∈ X = 1r (1,−1, s), then f = wBlv with

v = 1r (s∗, r − s∗, 1).[Kawamata]

3 r(P ∈ X ) > 1 and discrepancy = 1r , then f = wBlv .[Hayakawa]

4 f is classified so that at least Sing(Y ) is known. [Kawakita]

Divisorial Contractions to points

Let f : Y → X be a divisorial contraction to a point P ∈ X .

1 P ∈ X is nonsingular, then f = wBl(1,a,b) with(a, b) = 1.[Kawakita]

2 P ∈ X = 1r (1,−1, s), then f = wBlv with

v = 1r (s∗, r − s∗, 1).[Kawamata]

3 r(P ∈ X ) > 1 and discrepancy = 1r , then f = wBlv .[Hayakawa]

4 f is classified so that at least Sing(Y ) is known. [Kawakita]

Divisorial Contractions to points

Let f : Y → X be a divisorial contraction to a point P ∈ X .

1 P ∈ X is nonsingular, then f = wBl(1,a,b) with(a, b) = 1.[Kawakita]

2 P ∈ X = 1r (1,−1, s), then f = wBlv with

v = 1r (s∗, r − s∗, 1).[Kawamata]

3 r(P ∈ X ) > 1 and discrepancy = 1r , then f = wBlv .[Hayakawa]

4 f is classified so that at least Sing(Y ) is known. [Kawakita]

Divisorial Contractions to points

Let f : Y → X be a divisorial contraction to a point P ∈ X .

1 P ∈ X is nonsingular, then f = wBl(1,a,b) with(a, b) = 1.[Kawakita]

2 P ∈ X = 1r (1,−1, s), then f = wBlv with

v = 1r (s∗, r − s∗, 1).[Kawamata]

3 r(P ∈ X ) > 1 and discrepancy = 1r , then f = wBlv .[Hayakawa]

4 f is classified so that at least Sing(Y ) is known. [Kawakita]

Divisorial Contractions to points

Let f : Y → X be a divisorial contraction to a point P ∈ X .

1 P ∈ X is nonsingular, then f = wBl(1,a,b) with(a, b) = 1.[Kawakita]

2 P ∈ X = 1r (1,−1, s), then f = wBlv with

v = 1r (s∗, r − s∗, 1).[Kawamata]

3 r(P ∈ X ) > 1 and discrepancy = 1r , then f = wBlv .[Hayakawa]

4 f is classified so that at least Sing(Y ) is known. [Kawakita]

Divisorial Contractions to Curves

Tziolas classified (assuming ”General Elephant Conjecture”):f : Y → X a divisorial contraction to a smooth curve Γ, passingthrough Sing(X ) of type cA, cD, cE .

Factorization, I

Theorem (-, Hacon)

Let g : X →W be a divisorial contraction to a curve, then g canbe factored as

X = X0f099K X199K . . . 99KXn

fn99K W ,

such that each fi is one of the following:

1 a blowdown to a LCI curve;

2 a divisorial contraction to a point;

3 the inverse of a divisorial contraction over a point of indexri > 1 with discrepancy 1

ri,

4 a flop.

Factorization, II

Theorem (-,Hacon)

Let g : X →W be a flipping contraction and φ : X99KX + be thecorresponding flip, then φ can be factored as

X = X0f099K X1 99K . . . 99K Xn

fn99K X +,

such that each fi is one of the following:

1 a blowdown to a LCI curve;

2 a divisorial contraction to a point;

3 the inverse of a divisorial contraction over a point of indexri > 1 with discrepancy 1

ri,

4 a flop.

Factorization, III

Theorem (-)

Let Y 99K X be a flip, divisorial contraction to a point or adivisorial contraction to a curve. There exists a sequence ofbirational maps

Y =: Xn 99K . . . 99K X0 =: X

such that each map Xi+1 99K Xi is one of the following:

1 a divisorial extraction over a point of index ri ≥ 1 withminimal discrepancy, or its inverse;

2 a blowup over a smooth curve, or its inverse;

3 a flop.

Remark on Minimal Discrepancy

Given a point P ∈ X of index r . A divisorial contractionf : Y → X 3 P to a point of index r with minimal discrepancy

1 2, if P ∈ X is non-singular;

2 1, if P ∈ X has index r = 1;

3 1r , if P ∈ X has index r > 1.

Hayakawa’s Result

Theorem (Hayakawa)

Let f : Y → X 3 P be a divisorial contraction over a point P ∈ Xof index r > 1 with discrepancy 1

r . Then f can be realized asweighted blowup.

TheoremGiven a terminal singularity P ∈ X of index r > 1, there is a partialresolution

Xn → ...→ X1 → X 3 P,

such that each Xi+1 → Xi is a divisorial contraction to a pointPi ∈ Xi of index ri > 1 with discrepancy 1

riand Xn has only

Gorenstein terminal singularities, i.e. terminal singularity of index1.

Hayakawa’s Result

Theorem (Hayakawa)

Let f : Y → X 3 P be a divisorial contraction over a point P ∈ Xof index r > 1 with discrepancy 1

r . Then f can be realized asweighted blowup.

TheoremGiven a terminal singularity P ∈ X of index r > 1, there is a partialresolution

Xn → ...→ X1 → X 3 P,

such that each Xi+1 → Xi is a divisorial contraction to a pointPi ∈ Xi of index ri > 1 with discrepancy 1

riand Xn has only

Gorenstein terminal singularities, i.e. terminal singularity of index1.

Depth of terminal singularities

The depth, denoted dep(X ), is defined to be the minimal length ofpartial Gorenstein resolution.

dep(P ∈ X ) = 0 iff P has index 1.Let X 99K X + be a flip, then dep(X ) > dep(X +).Let X 99K X + be a flop, then dep(X ) = dep(X +).Let Y → X be a divisorial contraction then dep(Y ) ≥ dep(X ).Equality holds only when dep(Y ) = dep(X ) = 0.Let Y → X be a divisorial contraction then dep(Y ) + 1 ≥ dep(X ).

Depth of terminal singularities

The depth, denoted dep(X ), is defined to be the minimal length ofpartial Gorenstein resolution.dep(P ∈ X ) = 0 iff P has index 1.

Let X 99K X + be a flip, then dep(X ) > dep(X +).Let X 99K X + be a flop, then dep(X ) = dep(X +).Let Y → X be a divisorial contraction then dep(Y ) ≥ dep(X ).Equality holds only when dep(Y ) = dep(X ) = 0.Let Y → X be a divisorial contraction then dep(Y ) + 1 ≥ dep(X ).

Depth of terminal singularities

The depth, denoted dep(X ), is defined to be the minimal length ofpartial Gorenstein resolution.dep(P ∈ X ) = 0 iff P has index 1.Let X 99K X + be a flip, then dep(X ) > dep(X +).

Let X 99K X + be a flop, then dep(X ) = dep(X +).Let Y → X be a divisorial contraction then dep(Y ) ≥ dep(X ).Equality holds only when dep(Y ) = dep(X ) = 0.Let Y → X be a divisorial contraction then dep(Y ) + 1 ≥ dep(X ).

Depth of terminal singularities

The depth, denoted dep(X ), is defined to be the minimal length ofpartial Gorenstein resolution.dep(P ∈ X ) = 0 iff P has index 1.Let X 99K X + be a flip, then dep(X ) > dep(X +).Let X 99K X + be a flop, then dep(X ) = dep(X +).

Let Y → X be a divisorial contraction then dep(Y ) ≥ dep(X ).Equality holds only when dep(Y ) = dep(X ) = 0.Let Y → X be a divisorial contraction then dep(Y ) + 1 ≥ dep(X ).

Depth of terminal singularities

The depth, denoted dep(X ), is defined to be the minimal length ofpartial Gorenstein resolution.dep(P ∈ X ) = 0 iff P has index 1.Let X 99K X + be a flip, then dep(X ) > dep(X +).Let X 99K X + be a flop, then dep(X ) = dep(X +).Let Y → X be a divisorial contraction then dep(Y ) ≥ dep(X ).Equality holds only when dep(Y ) = dep(X ) = 0.

Let Y → X be a divisorial contraction then dep(Y ) + 1 ≥ dep(X ).

Depth of terminal singularities

The depth, denoted dep(X ), is defined to be the minimal length ofpartial Gorenstein resolution.dep(P ∈ X ) = 0 iff P has index 1.Let X 99K X + be a flip, then dep(X ) > dep(X +).Let X 99K X + be a flop, then dep(X ) = dep(X +).Let Y → X be a divisorial contraction then dep(Y ) ≥ dep(X ).Equality holds only when dep(Y ) = dep(X ) = 0.Let Y → X be a divisorial contraction then dep(Y ) + 1 ≥ dep(X ).

Key Lemma

TheoremGiven g : X →W be a flipping contraction, a divisorialcontraction to a curve (with singularity of r > 1, or a divisorialcontraction to a point with non-minimal discrepancy. Letf : Y → X be a divisorial contraction to a point P ∈ X of highestindex n > 1 with minimal discrepancy 1

n . Then the relativeanti-canonical divisor −KY /W is nef.

2-ray Game

Let X →W be a flipping contraction. Take Y → X a divisorialcontraction to a highest index point with minimal discrepancy.There is a diagram.

Y99K−−−−→ Y ′

f

y yf ′

X X ′

g

y yg ′

W=−−−−→ W

g ′ is a divisorial contraction;

Y 99K Y1 99K . . . 99K Y ′ consists a sequence of flips and flops;

X ′ = X +;

dep(Y ) = dep(X )− 1;dep(Y ) ≥ dep(Yi ).

2-ray Game

Let X →W be a flipping contraction. Take Y → X a divisorialcontraction to a highest index point with minimal discrepancy.There is a diagram.

Y99K−−−−→ Y ′

f

y yf ′

X X ′

g

y yg ′

W=−−−−→ W

g ′ is a divisorial contraction;

Y 99K Y1 99K . . . 99K Y ′ consists a sequence of flips and flops;

X ′ = X +;

dep(Y ) = dep(X )− 1;dep(Y ) ≥ dep(Yi ).

2-ray Game

Let X →W be a flipping contraction. Take Y → X a divisorialcontraction to a highest index point with minimal discrepancy.There is a diagram.

Y99K−−−−→ Y ′

f

y yf ′

X X ′

g

y yg ′

W=−−−−→ W

g ′ is a divisorial contraction;

Y 99K Y1 99K . . . 99K Y ′ consists a sequence of flips and flops;

X ′ = X +;

dep(Y ) = dep(X )− 1;dep(Y ) ≥ dep(Yi ).

2-ray Game

Let X →W be a flipping contraction. Take Y → X a divisorialcontraction to a highest index point with minimal discrepancy.There is a diagram.

Y99K−−−−→ Y ′

f

y yf ′

X X ′

g

y yg ′

W=−−−−→ W

g ′ is a divisorial contraction;

Y 99K Y1 99K . . . 99K Y ′ consists a sequence of flips and flops;

X ′ = X +;

dep(Y ) = dep(X )− 1;dep(Y ) ≥ dep(Yi ).

2-ray Game

Let X →W be a flipping contraction. Take Y → X a divisorialcontraction to a highest index point with minimal discrepancy.There is a diagram.

Y99K−−−−→ Y ′

f

y yf ′

X X ′

g

y yg ′

W=−−−−→ W

g ′ is a divisorial contraction;

Y 99K Y1 99K . . . 99K Y ′ consists a sequence of flips and flops;

X ′ = X +;

dep(Y ) = dep(X )− 1;dep(Y ) ≥ dep(Yi ).

2-ray GameLet X →W be a divisorial contraction to a curve. Take Y → X adivisorial contraction to a highest index point with minimaldiscrepancy. There is a diagram.

Y99K−−−−→ Y ′

f

y yf ′

X X ′

g

y yg ′

W=−−−−→ W

g ′ is a divisorial contraction to a curve;

f ′ is a divisorial contraction to a point;

Y 99K Y1 99K . . . 99K Y ′ consists a sequence of flips and flops;

dep(Y ) = dep(X )− 1;dep(Yi ) ≤ dep(Y ) < dep(X );dep(X ′) < dep(X ) or dep(X ′) = 0.

2-ray GameLet X →W be a divisorial contraction to a curve. Take Y → X adivisorial contraction to a highest index point with minimaldiscrepancy. There is a diagram.

Y99K−−−−→ Y ′

f

y yf ′

X X ′

g

y yg ′

W=−−−−→ W

g ′ is a divisorial contraction to a curve;

f ′ is a divisorial contraction to a point;

Y 99K Y1 99K . . . 99K Y ′ consists a sequence of flips and flops;

dep(Y ) = dep(X )− 1;dep(Yi ) ≤ dep(Y ) < dep(X );dep(X ′) < dep(X ) or dep(X ′) = 0.

2-ray GameLet X →W be a divisorial contraction to a curve. Take Y → X adivisorial contraction to a highest index point with minimaldiscrepancy. There is a diagram.

Y99K−−−−→ Y ′

f

y yf ′

X X ′

g

y yg ′

W=−−−−→ W

g ′ is a divisorial contraction to a curve;

f ′ is a divisorial contraction to a point;

Y 99K Y1 99K . . . 99K Y ′ consists a sequence of flips and flops;

dep(Y ) = dep(X )− 1;dep(Yi ) ≤ dep(Y ) < dep(X );dep(X ′) < dep(X ) or dep(X ′) = 0.

2-ray GameLet X →W be a divisorial contraction to a curve. Take Y → X adivisorial contraction to a highest index point with minimaldiscrepancy. There is a diagram.

Y99K−−−−→ Y ′

f

y yf ′

X X ′

g

y yg ′

W=−−−−→ W

g ′ is a divisorial contraction to a curve;

f ′ is a divisorial contraction to a point;

Y 99K Y1 99K . . . 99K Y ′ consists a sequence of flips and flops;

dep(Y ) = dep(X )− 1;dep(Yi ) ≤ dep(Y ) < dep(X );dep(X ′) < dep(X ) or dep(X ′) = 0.

2-ray GameLet X →W be a divisorial contraction to a curve. Take Y → X adivisorial contraction to a highest index point with minimaldiscrepancy. There is a diagram.

Y99K−−−−→ Y ′

f

y yf ′

X X ′

g

y yg ′

W=−−−−→ W

g ′ is a divisorial contraction to a curve;

f ′ is a divisorial contraction to a point;

Y 99K Y1 99K . . . 99K Y ′ consists a sequence of flips and flops;

dep(Y ) = dep(X )− 1;dep(Yi ) ≤ dep(Y ) < dep(X );dep(X ′) < dep(X ) or dep(X ′) = 0.

2-ray GameLet X →W be a divisorial contraction to a point P ∈W withnon-minimal discrepancy a

r >1r . Take Y → X a divisorial

contraction to a highest index point with minimal discrepancy.

Y99K−−−−→ Y ′

f

y yf ′

X X ′

g

y yg ′

W=−−−−→ W

g ′ is a divisorial contraction;f ′ is a divisorial contraction to a point with discrepancya′

r <ar ;

Y 99K Y1 99K . . . 99K Y ′ consists a sequence of flips and flops;dep(Y ) = dep(X )− 1;dep(Yi ) ≤ dep(Y ) < dep(X );dep(X ′) ≤ dep(X ).

2-ray GameLet X →W be a divisorial contraction to a point P ∈W withnon-minimal discrepancy a

r >1r . Take Y → X a divisorial

contraction to a highest index point with minimal discrepancy.

Y99K−−−−→ Y ′

f

y yf ′

X X ′

g

y yg ′

W=−−−−→ W

g ′ is a divisorial contraction;

f ′ is a divisorial contraction to a point with discrepancya′

r <ar ;

Y 99K Y1 99K . . . 99K Y ′ consists a sequence of flips and flops;dep(Y ) = dep(X )− 1;dep(Yi ) ≤ dep(Y ) < dep(X );dep(X ′) ≤ dep(X ).

2-ray GameLet X →W be a divisorial contraction to a point P ∈W withnon-minimal discrepancy a

r >1r . Take Y → X a divisorial

contraction to a highest index point with minimal discrepancy.

Y99K−−−−→ Y ′

f

y yf ′

X X ′

g

y yg ′

W=−−−−→ W

g ′ is a divisorial contraction;f ′ is a divisorial contraction to a point with discrepancya′

r <ar ;

Y 99K Y1 99K . . . 99K Y ′ consists a sequence of flips and flops;dep(Y ) = dep(X )− 1;dep(Yi ) ≤ dep(Y ) < dep(X );dep(X ′) ≤ dep(X ).

2-ray GameLet X →W be a divisorial contraction to a point P ∈W withnon-minimal discrepancy a

r >1r . Take Y → X a divisorial

contraction to a highest index point with minimal discrepancy.

Y99K−−−−→ Y ′

f

y yf ′

X X ′

g

y yg ′

W=−−−−→ W

g ′ is a divisorial contraction;f ′ is a divisorial contraction to a point with discrepancya′

r <ar ;

Y 99K Y1 99K . . . 99K Y ′ consists a sequence of flips and flops;

dep(Y ) = dep(X )− 1;dep(Yi ) ≤ dep(Y ) < dep(X );dep(X ′) ≤ dep(X ).

2-ray GameLet X →W be a divisorial contraction to a point P ∈W withnon-minimal discrepancy a

r >1r . Take Y → X a divisorial

contraction to a highest index point with minimal discrepancy.

Y99K−−−−→ Y ′

f

y yf ′

X X ′

g

y yg ′

W=−−−−→ W

g ′ is a divisorial contraction;f ′ is a divisorial contraction to a point with discrepancya′

r <ar ;

Y 99K Y1 99K . . . 99K Y ′ consists a sequence of flips and flops;dep(Y ) = dep(X )− 1;dep(Yi ) ≤ dep(Y ) < dep(X );dep(X ′) ≤ dep(X ).

Sketch of the Key Lemma

A quick ”proof”:Let E := Exc(f ) and C ⊂ E be any curve. It suffices to show thatCY · KY ≤ 0 for its proper transform CY in Y .

Take SX ∈ | − KX |a general elephant.The proper transform SY ∈ | − KY |.Hence −CY · KY = CY · SY ≥ 0 if C 6⊂ SX .

Sketch of the Key Lemma

A quick ”proof”:Let E := Exc(f ) and C ⊂ E be any curve. It suffices to show thatCY · KY ≤ 0 for its proper transform CY in Y . Take SX ∈ | − KX |a general elephant.

The proper transform SY ∈ | − KY |.Hence −CY · KY = CY · SY ≥ 0 if C 6⊂ SX .

Sketch of the Key Lemma

A quick ”proof”:Let E := Exc(f ) and C ⊂ E be any curve. It suffices to show thatCY · KY ≤ 0 for its proper transform CY in Y . Take SX ∈ | − KX |a general elephant.The proper transform SY ∈ | − KY |.

Hence −CY · KY = CY · SY ≥ 0 if C 6⊂ SX .

Sketch of the Key Lemma

A quick ”proof”:Let E := Exc(f ) and C ⊂ E be any curve. It suffices to show thatCY · KY ≤ 0 for its proper transform CY in Y . Take SX ∈ | − KX |a general elephant.The proper transform SY ∈ | − KY |.Hence −CY · KY = CY · SY ≥ 0 if C 6⊂ SX .

Sketch of the Key Lemma

f : X →W is a flipping contraction or a divisorial contractionto a curve, then C 6⊂ SX by Kollar-Mori.

If C ⊂ SX , one canproceed by the classification of ”extremal neighborhood” byKollar-Mori.f : X →W is a divisorial contraction to a point withnon-minimal discrepancy and of ”ordinary type” according toKawakita’s classification, then f can be realized as a weightedblowup. Everything can be computed by using toric geometry.f : X →W is a divisorial contraction to a point withnon-minimal discrepancy and of ”exceptional type” accordingto Kawakita’s classification. Then Sing(X ) = {Q} (with oneexception). C · −KX = b

r(Q) ≥1

r(Q) .

C · −KX − CY · −KY ≤q

r 3F 3 <

1

r(Q),

where E is the f -exceptional divisor, F is the g -exceptionaldivisor and g∗E = E + q

r F .The remaining case can be treated similarly.

Sketch of the Key Lemma

f : X →W is a flipping contraction or a divisorial contractionto a curve, then C 6⊂ SX by Kollar-Mori. If C ⊂ SX , one canproceed by the classification of ”extremal neighborhood” byKollar-Mori.

f : X →W is a divisorial contraction to a point withnon-minimal discrepancy and of ”ordinary type” according toKawakita’s classification, then f can be realized as a weightedblowup. Everything can be computed by using toric geometry.f : X →W is a divisorial contraction to a point withnon-minimal discrepancy and of ”exceptional type” accordingto Kawakita’s classification. Then Sing(X ) = {Q} (with oneexception). C · −KX = b

r(Q) ≥1

r(Q) .

C · −KX − CY · −KY ≤q

r 3F 3 <

1

r(Q),

where E is the f -exceptional divisor, F is the g -exceptionaldivisor and g∗E = E + q

r F .The remaining case can be treated similarly.

Sketch of the Key Lemma

f : X →W is a flipping contraction or a divisorial contractionto a curve, then C 6⊂ SX by Kollar-Mori. If C ⊂ SX , one canproceed by the classification of ”extremal neighborhood” byKollar-Mori.f : X →W is a divisorial contraction to a point withnon-minimal discrepancy and of ”ordinary type” according toKawakita’s classification, then f can be realized as a weightedblowup. Everything can be computed by using toric geometry.

f : X →W is a divisorial contraction to a point withnon-minimal discrepancy and of ”exceptional type” accordingto Kawakita’s classification. Then Sing(X ) = {Q} (with oneexception). C · −KX = b

r(Q) ≥1

r(Q) .

C · −KX − CY · −KY ≤q

r 3F 3 <

1

r(Q),

where E is the f -exceptional divisor, F is the g -exceptionaldivisor and g∗E = E + q

r F .The remaining case can be treated similarly.

Sketch of the Key Lemma

f : X →W is a flipping contraction or a divisorial contractionto a curve, then C 6⊂ SX by Kollar-Mori. If C ⊂ SX , one canproceed by the classification of ”extremal neighborhood” byKollar-Mori.f : X →W is a divisorial contraction to a point withnon-minimal discrepancy and of ”ordinary type” according toKawakita’s classification, then f can be realized as a weightedblowup. Everything can be computed by using toric geometry.f : X →W is a divisorial contraction to a point withnon-minimal discrepancy and of ”exceptional type” accordingto Kawakita’s classification. Then Sing(X ) = {Q} (with oneexception). C · −KX = b

r(Q) ≥1

r(Q) .

C · −KX − CY · −KY ≤q

r 3F 3 <

1

r(Q),

where E is the f -exceptional divisor, F is the g -exceptionaldivisor and g∗E = E + q

r F .

The remaining case can be treated similarly.

Sketch of the Key Lemma

f : X →W is a flipping contraction or a divisorial contractionto a curve, then C 6⊂ SX by Kollar-Mori. If C ⊂ SX , one canproceed by the classification of ”extremal neighborhood” byKollar-Mori.f : X →W is a divisorial contraction to a point withnon-minimal discrepancy and of ”ordinary type” according toKawakita’s classification, then f can be realized as a weightedblowup. Everything can be computed by using toric geometry.f : X →W is a divisorial contraction to a point withnon-minimal discrepancy and of ”exceptional type” accordingto Kawakita’s classification. Then Sing(X ) = {Q} (with oneexception). C · −KX = b

r(Q) ≥1

r(Q) .

C · −KX − CY · −KY ≤q

r 3F 3 <

1

r(Q),

where E is the f -exceptional divisor, F is the g -exceptionaldivisor and g∗E = E + q

r F .The remaining case can be treated similarly.

ExampleP ∈ X is cD/2 given by

x21 + x2

2 x4 + s(x3, x4)x2x3x4 + r(x3)x2 + p(x3, x4) = 0 ⊂ C4/v ,

with v = 12 (1, 1, 1, 0).

The map f : Y → X is given by weighted blowup with vectorv1 = (2l , 2l , 1, 1). Moreover, wtv1(ϕ) = 2l and x4l

3 ∈ p(x3, x4).There is a singularity Q2 of type cA/4l . The local equation in U2

is given by

x12 + x2x4 + x3

4l + ... = 0 ⊂ C4/1

4l(0, 2l − 1, 1, 2l + 1).

There is only one weighted blowup Z → Y with minimaldiscrepancy 1

4l which is given by the weightw2 = 1

4l (4l , 2l − 1, 1, 2l + 1).

ExampleP ∈ X is cD/2 given by

x21 + x2

2 x4 + s(x3, x4)x2x3x4 + r(x3)x2 + p(x3, x4) = 0 ⊂ C4/v ,

with v = 12 (1, 1, 1, 0).

The map f : Y → X is given by weighted blowup with vectorv1 = (2l , 2l , 1, 1). Moreover, wtv1(ϕ) = 2l and x4l

3 ∈ p(x3, x4).

There is a singularity Q2 of type cA/4l . The local equation in U2

is given by

x12 + x2x4 + x3

4l + ... = 0 ⊂ C4/1

4l(0, 2l − 1, 1, 2l + 1).

There is only one weighted blowup Z → Y with minimaldiscrepancy 1

4l which is given by the weightw2 = 1

4l (4l , 2l − 1, 1, 2l + 1).

ExampleP ∈ X is cD/2 given by

x21 + x2

2 x4 + s(x3, x4)x2x3x4 + r(x3)x2 + p(x3, x4) = 0 ⊂ C4/v ,

with v = 12 (1, 1, 1, 0).

The map f : Y → X is given by weighted blowup with vectorv1 = (2l , 2l , 1, 1). Moreover, wtv1(ϕ) = 2l and x4l

3 ∈ p(x3, x4).There is a singularity Q2 of type cA/4l . The local equation in U2

is given by

x12 + x2x4 + x3

4l + ... = 0 ⊂ C4/1

4l(0, 2l − 1, 1, 2l + 1).

There is only one weighted blowup Z → Y with minimaldiscrepancy 1

4l which is given by the weightw2 = 1

4l (4l , 2l − 1, 1, 2l + 1).

ExampleP ∈ X is cD/2 given by

x21 + x2

2 x4 + s(x3, x4)x2x3x4 + r(x3)x2 + p(x3, x4) = 0 ⊂ C4/v ,

with v = 12 (1, 1, 1, 0).

The map f : Y → X is given by weighted blowup with vectorv1 = (2l , 2l , 1, 1). Moreover, wtv1(ϕ) = 2l and x4l

3 ∈ p(x3, x4).There is a singularity Q2 of type cA/4l . The local equation in U2

is given by

x12 + x2x4 + x3

4l + ... = 0 ⊂ C4/1

4l(0, 2l − 1, 1, 2l + 1).

There is only one weighted blowup Z → Y with minimaldiscrepancy 1

4l which is given by the weightw2 = 1

4l (4l , 2l − 1, 1, 2l + 1).

ExampleHence we can summarize this case into following diagram.

Z99K−−−−→ Z ′

14l

ywt=w212

ywt=w ′2

Q2 ∈ Y Y ′ 3 Q ′4

22

ywt=w112

ywt=w ′1

X=−−−−→ X

Where

w1 = v1 = (2l , 2l , 1, 1), w ′1 = v2 = 12 (2l + 1, 2l − 1, 1, 2)

w2 = 14l (4l , 2l − 1, 1, 2l + 1), w ′2 = 1

2 (2l − 1, 2l + 1, 1, 2).

In this case, both f ′ and g ′ are divisorial contractions to a cD/2point.

Corollary

Existence of flips.

Termination of flips.

Corollary

Existence of flips.Termination of flips.

Expectations and Questions

QuestionCan every divisorial contraction to a point be realized as aweighted blowup?

Expectations and Questions

QuestionCan every divisorial contraction to a point be realized as aweighted blowup?

Expectations and Questions

QuestionCan one combine this factorization with Sarkisov’s program?

QuestionDoes the given explicit resolution contain fewest exceptionaldivisors among all resolution?

Expectations and Questions

QuestionCan one combine this factorization with Sarkisov’s program?

QuestionDoes the given explicit resolution contain fewest exceptionaldivisors among all resolution?

Thank you!