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 X 0 → S 0 be an elliptic threefold which is not uniruled. π : ¯ X → ¯ Then there exists a birationally equivalent fibrations ¯ S, such that X has at worst terminal and ¯ ¯ S log terminal singlarities. Futhermore K ¯ X is S + ¯ Λ) , where ¯ nef and K ¯ X ≡ ¯ π ∗ ( K ¯ 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 ∗ ( K X / B ) + � � � m i − 1 m i Y i , 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: µ ¯ X X ǫ π ¯ π ( ¯ S , ¯ ( S , Λ) Λ) 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: � �� �� mm i − 1 ω ⊗ m ω B ⊗ π ∗ ( ω ⊗ m = π ∗ X / B ) ⊗ O B Y i ⊗ O X ( m ( E − G )) X m i and Kawamata showed the following isomorphism of the relative canonical bundle X / B ) ∼ � π ∗ ( ω ⊗ 12 = O B ( 12 a i D i ) ⊗ J ∞ where D i support the singular elliptic fibers, a i ∈ 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 → P 1 . 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 K X + ∆ 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 ∗ ( K X / 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 B + ¯ � X = ǫ ∗ ( K ¯ π ∗ Γ i + E − G K ˜ ∆) + c i ˜ where � c i ˜ π ∗ Γ i + 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 g ˜ X π ˜ ǫ ( ˜ B , ˜ ∆) π g h ψ ( ¯ B , ¯ ( B , ∆) ∆) David Wen (UCSB) Towards Minimal Models of Elliptic Fourfolds Jan. 11, 2018 8 / 12

Progress Towards a Generalization of Grassi’s theorem ˜ X ˜ g µ ¯ X π ˜ X ǫ ( ˜ B , ˜ ∆) π ¯ π g h ψ ( ¯ B , ¯ ( B , ∆) ∆) 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 ∗ O B ( K X / B ) such that ( B , ∆) is log terminal threefold with a log minimal model ( ¯ B , ¯ ∆) . Then there exists a birationally equivalent rational elliptic π : ¯ X ��� ¯ B, such that ¯ fibration ¯ X is a minimal model of X and B + ¯ π ∗ ( K ¯ K ¯ X ≡ ¯ ∆) . 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, K 3-fibration over 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