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1
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Geography of log models: theory and applications

100%
Shokurov V., Choi S.
Open Mathematics
|
2011
|
tom 9
|
nr 3
489-534
EN
This is an introduction to geography of log models with applications to positive cones of Fano type (FT) varieties and to geometry of minimal models and Mori fibrations.
2
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The Mukai conjecture for log Fano manifolds

81%
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).
3
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Towards the classification of weak Fano threefolds with ρ = 2

61%
Cutrone J., Marshburn N.
Open Mathematics
|
2013
|
tom 11
|
nr 9
1552-1576
EN
In this paper, examples of type II Sarkisov links between smooth complex projective Fano threefolds with Picard number one are provided. To show examples of these links, we study smooth weak Fano threefolds X with Picard number two and with a divisorial extremal ray. We assume that the pluri-anticanonical morphism of X contracts only a finite number of curves. The numerical classification of these particular smooth weak Fano threefolds is completed and the geometric existence of some numerical cases is proven.
4
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Three-dimensional terminal toric flips

52%
Fujino O., Sato H., Takano Y., Uehara H.
Open Mathematics
|
2009
|
tom 7
|
nr 1
46-53
EN
We describe three-dimensional terminal toric flips. We obtain the complete local description of three-dimensional terminal toric flips.
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