Birational Geometry of Algebraic Varieties

Birkar, Caucher

doi:10.1142/9789813272880_0068

Search citation statements

Order By: Relevance

Paper Sections

Select...

Introduction3

Minimal Model Program1

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2019

2023

Publication Types

Select...

Other4

Article3

Relationship

Self Cite0

Independent7

Authors

Journals

Cited by 8 publications

(11 citation statements)

References 46 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The problem solved affirmatively in dimension ≤ 4 [69,35], for varieties of general type, for uniruled varieties [5], and in some other special cases. We refer to [3] for more comprehensive survey of the higher-dimensional MMP.…”

Section: Minimal Model Programmentioning

confidence: 99%

Effective results in the three-dimensional minimal model program

Prokhorov

2021

Preprint

View full text Add to dashboard Cite

show abstract

Section: Minimal Model Programmentioning

confidence: 99%

Effective results in the three-dimensional minimal model program

Prokhorov

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The conjecture is known up to dim X = 2 ( [Bi04]). Birkar raises the following conjectures which are related to Conjecture 1.2.…”

Section: Introductionmentioning

confidence: 99%

On singularities of threefold weighted blowups

Chen

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…Minimal Model Program (MMP), or Mori theory, is aimed at finding "economic" representatives in each birational class of projective algebraic varieties X of nonnegative Kodaira dimension. These representatives are called minimal models and they are main objectives in MMP [KM98,Mat02,Bir18a].…”

Section: Introductionmentioning

confidence: 99%

Canonical models of toric hypersurfaces

Batyrev¹

2020

Preprint

View full text Add to dashboard Cite

Let Z ⊂ T d be a non-degenerate hypersurface in d-dimensional torus T d ∼ = (C * ) d defined by a Laurent polynomial f with a given d-dimensional Newton polytope P . It follows from a theorem of Ishii that Z is birational to a smooth projective variety X of Kodaira dimension κ ≥ 0 if and only if the Fine interior F (P ) of P is nonempty. We define a unique projective model Z of Z having at worst canonical singularities which allows us to obtain minimal models Z of Z by crepant morphisms Z → Z. Moreover, we show that κ = min{d − 1, dim F (P )} and that general fibers in the Iitaka fibration of the canonical model Z are nondegenerate (d − 1 − κ)-dimensional toric hypersurfaces of Kodaira dimension 0. Using the rational polytope F (P ), we compute the stringy E-function of minimal models Z and obtain a combinatorial formula for their stringy Euler numbers.

show abstract

Birational Geometry of Algebraic Varieties

Cited by 8 publications

References 46 publications

Effective results in the three-dimensional minimal model program

Effective results in the three-dimensional minimal model program

On singularities of threefold weighted blowups

Canonical models of toric hypersurfaces

Contact Info

Product

Resources

About