Towards Minimal Models of Elliptic Fourfolds David Wen University - - PowerPoint PPT Presentation

Towards Minimal Models of Elliptic Fourfolds

David Wen

University of California, Santa Barbara

Joint Math Meeting, January 11, 2018

David Wen (UCSB) Towards Minimal Models of Elliptic Fourfolds

• Jan. 11, 2018

1 / 12

Background

Grassi’s Theorem on Elliptic Threefold

Definition An elliptic fibration is a morphism f : X → B between varieties such that for a general point x ∈ B we have that f −1(x) is an elliptic curve. We say an elliptic fibration with section if in addition we have that there is a section s : B → X such that f ◦ s is the identity morphism on B. Theorem (A. Grassi) Let X0 → S0 be an elliptic threefold which is not uniruled. Then there exists a birationally equivalent fibrations ¯ π : ¯ X → ¯ S, such that ¯ X has at worst terminal and ¯ S log terminal singlarities. Futhermore K ¯

X is

nef and K ¯

X ≡ ¯

π∗(K ¯

S + ¯

Λ), where ¯ Λ is a Q-boundary divisor. Thus the canonical bundle is a pullback of a Q-bundle on ¯ S.

David Wen (UCSB) Towards Minimal Models of Elliptic Fourfolds

• Jan. 11, 2018

2 / 12

Background

Sketch of Grassi’s Argument

Let π : X → S be an elliptic threefold with reasonably nice properties. Let Λ = π∗(KX/B) +

mi−1 mi Yi

• , then (S, Λ) is a log surface. Running the

log minimal model program for surfaces gives a log minimal model ( ¯ S, ¯ Λ) with a morphism φ : S → ¯ S being the composition of the sequence of

• contractions. This gives a map ǫ : X → ¯

S, which running the relative minimal model program gives the diagram: X ¯ X (S, Λ) ( ¯ S, ¯ Λ)

π µ ǫ ¯ π φ

David Wen (UCSB) Towards Minimal Models of Elliptic Fourfolds

• Jan. 11, 2018

3 / 12

Background

Canonical Bundle Formula

For an elliptic fibration π : X → B with enough reasonable birational assumptions, Fujita established the following canonical bundle formula: ω⊗m

X

= π∗

• ωB ⊗ π∗(ω⊗m

X/B) ⊗ OB

• mmi − 1

mi Yi

• ⊗ OX(m(E − G))

and Kawamata showed the following isomorphism of the relative canonical bundle π∗(ω⊗12

X/B) ∼

= OB(

• 12aiDi) ⊗ J∞

where Di support the singular elliptic fibers, ai ∈ Q ∩ [0, 1) detemined by the Kodaira classification of singular elliptic fibers and J∞ is a divisor coming from a pullback of the J-invariant map J : B → P1.

David Wen (UCSB) Towards Minimal Models of Elliptic Fourfolds

• Jan. 11, 2018

4 / 12

Progress

Fujita-Zariski Decomposition

Definition Let D be a R-divisor on a normal variety X/Z. A Fujita-Zariski decomposition over Z for D is an expression D = P + N such that: P and N are R-Cartier P is nef over Z and N ≥ 0 If f : W → X is a projective birational morphism from a normal variety and f ∗(D) = P′ + N′ with P′ nef over Z and N′ effective, then P′ ≤ f ∗(P).

David Wen (UCSB) Towards Minimal Models of Elliptic Fourfolds

• Jan. 11, 2018

5 / 12

Progress

Birkar’s Result

Theorem (C. Birkar) Assume the log minimal model program for Q-factorial divisorial log terminal pairs in dimension n − 1. Let (X, ∆) be log canonical of dimension n, then KX + ∆ birationally has a Fujita-Zariski Decomposition [over Z] if and only if (X, ∆) has a [relative] log minimal model [over Z].

David Wen (UCSB) Towards Minimal Models of Elliptic Fourfolds

• Jan. 11, 2018

6 / 12

Progress

Key Lemma

Lemma (DW) Let π : X → B be a Weierstrass model with B a smooth threefold such that (B, ∆) has a log minimal model with ∆ ∼ = π∗(KX/B) defined in the canonical bundle formula. Then there exists a birationally equivalent fibration ǫ : ˜ X → ¯ B where ( ¯ B, ¯ ∆) is a log minimal model of the log terminal pair (B, ∆) and K ˜

X = ǫ∗(K ¯ B + ¯

∆) +

• ci ˜

π∗Γi + E − G where ci ˜ π∗Γi + E − G is effective. In fact, we will have that this is a Fujita-Zariski decomposition of K ˜

X.

David Wen (UCSB) Towards Minimal Models of Elliptic Fourfolds

• Jan. 11, 2018

7 / 12

Progress

A Commutative Diagram of Lemma

˜ X X ( ˜ B, ˜ ∆) (B, ∆) ( ¯ B, ¯ ∆)

˜ g ˜ π ǫ π g h ψ

David Wen (UCSB) Towards Minimal Models of Elliptic Fourfolds

• Jan. 11, 2018

8 / 12

Progress

Towards a Generalization of Grassi’s theorem

˜ X X ¯ X ( ˜ B, ˜ ∆) (B, ∆) ( ¯ B, ¯ ∆)

˜ g ˜ π ǫ µ π ¯ π g h ψ

David Wen (UCSB) Towards Minimal Models of Elliptic Fourfolds

• Jan. 11, 2018

9 / 12

Progress

Results

Theorem (DW) Let π : X → B be a Weierstrass model, ∆ the divisor associated π∗OB(KX/B) such that (B, ∆) is log terminal threefold with a log minimal model ( ¯ B, ¯ ∆). Then there exists a birationally equivalent rational elliptic fibration ¯ π : ¯ X ¯ B, such that ¯ X is a minimal model of X and K ¯

X ≡ ¯

π∗(K ¯

B + ¯

∆). Theorem (DW) With the assumptions from the lemma, the canonical model of ˜ X is isomorphic to the log canonical model of ( ¯ B, ¯ ∆). Equivalently, the canonical ring of ˜ X is isomorphic to the log canonical ring of ( ¯ B, ¯ ∆).

David Wen (UCSB) Towards Minimal Models of Elliptic Fourfolds

• Jan. 11, 2018

10 / 12

Future Outlook

Further Questions

Possibility to realize ¯ π as a morphism and not just a rational map? Equidimensional minimal model of Weierstrass Models? Removing the requirement of a section Generalize to higher dimensional fibers? For instance, K3-fibration

• ver surfaces?

David Wen (UCSB) Towards Minimal Models of Elliptic Fourfolds

• Jan. 11, 2018

11 / 12

Thank You!

David Wen (UCSB) Towards Minimal Models of Elliptic Fourfolds

• Jan. 11, 2018

12 / 12