Perverse obstructions to flat regular compactifications

Brosnan, Patrick

doi:10.1007/s00209-017-2010-0

Cited by 6 publications

(4 citation statements)

References 13 publications

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“…We will say that X is general in the sense of LSV if the construction of [47] works for J U .X /, and we refer to J D J.X / as in Theorem 1.1 as the LSV fibration. A necessary condition for this to happen is that the hyperplane sections of X are palindromic; see [17]. For example, a cubic fourfold containing a plane is not general in the sense of LSV.…”

Section: A Hyper-kähler Compactification Of the Intermediate Jacobian...mentioning

confidence: 99%

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Birational geometry of the intermediate Jacobian fibration of a cubic fourfold

Saccà

2023

Geom. Topol.

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Birational geometry of the intermediate Jacobian fibration of a cubic fourfold GIULIA SACCÀ APPENDIX BY CLAIRE VOISINWe show that the intermediate Jacobian fibration associated to any smooth cubic fourfold X admits a hyper-Kähler compactification J.X / with a regular Lagrangian fibration W J ! P 5 . This builds upon work of Laza, Saccà and Voisin (2017), where the result is proved for general X , as well as on the degeneration techniques introduced in the work of Kollár, Laza, Saccà and Voisin, and the minimal model program. We then study some aspects of the birational geometry of J.X /: for very general X we compute the movable and nef cones of J.X /, showing that J.X / is not birational to the twisted version of the intermediate Jacobian fibration, nor to an OG10-type moduli space of objects in the Kuznetsov component of X ; for any smooth X we show, using normal functions, that the Mordell-Weil group MW. / of the fibration is isomorphic to the integral degree-4 primitive algebraic cohomology of X , ie MW. / Š H 2;2 .X; Z/ 0 . 14D06, 14J42Proof Let x J ! P 5 be any normal projective compactification of J U 1 with a regular morphism x W x J ! P 5 . By Lemma 1.4, there is a holomorphic two-form x on the smooth locus of x J extending J U 1 , the canonical class K x J 0 is effective, and K x J D 0 if and only if x J is a symplectic variety. Since K x J is supported on the complement of J U 1 , codim x .Supp.K x J // 2. By definition [40, Definition 7], this means that K x J is x -exceptional, if it is nontrivial. If this is the case, then by [61, III 5.1] (see also [45, Lemma 2.10]), K x J is not x -nef. More precisely, there is a component of K x J that is covered by curves that are contracted by x and that intersect K x J negatively.Let z J ! P 5 be a smooth projective compactification of J U 1 admitting a regular morphism z W z J ! P 5 , and let K z J be its canonical class. If the effective divisor K z

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Section: A Hyper-kähler Compactification Of the Intermediate Jacobian...mentioning

confidence: 99%

“…Remark 3.12 As already mentioned just below Theorem 1.1, a necessary condition for the irreducibility of the fibers of J ! P 5 is given in [17]. 4 Birational geometry of J.X / for general X…”

Section: Induced Automorphismsmentioning

confidence: 99%

Birational geometry of the intermediate Jacobian fibration of a cubic fourfold

Saccà

2023

Geom. Topol.

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“…We will say that X is general in the sense of LSV if the construction of [LSV17] works for J U (X), and we refer to J = J(X) as in Theorem 1.1 as the LSV fibration. A necessary condition for this to happen is that the hyperplane sections of X are palindromic (see [Bro18]). For example, a cubic fourfold containing a plane is not general in the sense of LSV.…”

Section: A Hyper-kähler Compactification Of the Intermediate Jacobian...mentioning

confidence: 99%

Birational geometry of the intermediate Jacobian fibration of a cubic fourfold

Saccà¹,

Voisin²

2020

Preprint

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We show that the intermediate Jacobian fibration associated to any smooth cubic fourfold X admits a hyper-Kähler compactification J(X) with a regular Largangian fibration J → P 5 . This builds upon [LSV17], where the result is proved for general X, as well as on the degeneration techniques on [KLSV17] and techniques from the minimal model program. We then study some aspects of the birational geometry of J(X): for very general X we compute the movable and nef cones of J(X), showing that J(X) is not birational to the twisted version of the intermediate Jacobian fibration [Voi18a], nor to an OG10-type moduli space of objects in the Kuznetsov component of X; for any smooth X we show, using normal functions, that the Mordell-Weil group M W (π) of the abelian fibration π : J → P 5 is isomorphic to the integral degree 4 primitive algebraic cohomology of X, i.e., M W (π) ∼ = H 2,2 (X, Z)0.Contents 12 3. The relative theta divisor on the intermediate Jacobian fibration. 15 4. Birational geometry of J(X), for general X 20 5. The Mordell-Weil group of J(X) 25 Appendix A. On the Beauville conjecture for LSV varieties, by Claire Voisin 32 References 34

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“…The method used for both of these studies is reduction to curves via Mumford's Prym construction for the intermediate Jacobian. It would be interesting to study the degenerations of intermediate Jacobians directly in terms of cubics (see [Bro18] for a step in this direction).…”

Section: Introductionmentioning

confidence: 99%