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The Mukai conjecture for log Fano manifolds

100%
Fujita K.
Open Mathematics
|
2014
|
tom 12
|
nr 1
14-27
EN
For a log Fano manifold (X,D) with D ≠ 0 and of the log Fano pseudoindex ≥2, we prove that the restriction homomorphism Pic(X) → Pic(D 1) of Picard groups is injective for any irreducible component D 1 ⊂ D. The strategy of our proof is to run a certain minimal model program and is similar to Casagrande’s argument. As a corollary, we prove that the Mukai conjecture (resp. the generalized Mukai conjecture) implies the log Mukai conjecture (resp. the log generalized Mukai conjecture).
2
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Threefolds with big and nef anticanonical bundles II

76%
Jahnke P., Peternell T., Radloff I.
Open Mathematics
|
2011
|
tom 9
|
nr 3
449-488
EN
In a follow-up to our paper [Threefolds with big and nef anticanonical bundles I, Math. Ann., 2005, 333(3), 569–631], we classify smooth complex projective threefolds Xwith −K X big and nef but not ample, Picard number γ(X) = 2, and whose anticanonical map is small. We assume also that the Mori contraction of X and of its flop X + are not both birational.
3
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Halphen pencils on weighted Fano threefold hypersurfaces

64%
Cheltsov I., Park J.
Open Mathematics
|
2009
|
tom 7
|
nr 1
1-45
EN
On a general quasismooth well-formed weighted hypersurface of degree Σi=14 a i in ℙ(1, a 1, a 2, a 3, a 4), we classify all pencils whose general members are surfaces of Kodaira dimension zero.
4
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Quartic del Pezzo surfaces over function fields of curves

52%
Hassett B., Tschinkel Y.
Open Mathematics
|
2014
|
tom 12
|
nr 3
395-420
EN
We classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of sections to the fibration, and mappings to the intermediate Jacobian of the total space. We exhibit examples where these are birational, which has applications to arithmetic questions, especially over finite fields.
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