Li Yi

Li Yi

I am a 3rd year PhD student majoring in birational geometry

Birational Geometry Reading Notes: BCHM

2 minute read

Published:

This aim of this series of notes is to study the classical paper BCHM in details. We will try to finish the proof of the main theorem and summarize some interesting applications of BCHM. We will not strictly follow the structure of the original paper, instead we will divide the paper into several topics discussion, including: Existence of flip, existence of minimal model, termination problem, non-vanishing conjecture and finite generation problem etc.

Part I. Extension Theorems

Note-I.1: Nakayama’s extension theorems [update 8.17]

Note-I.2: Hacon Mckernan extension theorem [update 2.21]

Note-I.3: deFernex-Hacon extension theorem [updage 11.12]

Note-I.4: Demailly-Hacon-Paun dlt extension

Part II. Base Point Freeeness and Variants

Note-II.1 Classical Base point free theorem for klt and dlt pairs

Note-II.2 Fujino’s some generalizations of base point free theorems

Note-II.3 Positivities in Families and Base point Freeness

Part III. Existence of flip, minimal model, good minimal model, canonical model

Note-III.1: Hacon Mckernan’s proof of existence of klt flip

Note-III.2: Hacon-Xu’s and Birkar’s proof on existence of LC flips (with some generalizations)

Note-III.3: Existence of minimal models (BCHM C and related)

Note-III.4: Dxistence of good minimal models (DHP and related)

Note-III.5: Basic properties of minimal model, good minimal models, canonical model

Note-III.6: Behavior of minimal model, good minimal model, canonical model under birational modification

Part IV. Minimal model, good minimal model, canonical model, abundance in Family

Note-IV.1: A brief survey on Minimal models, good minimal models, canonical models, abundance in Families [upd 10.13]

Note-IV.2: Existence of good minimal on the closure

Note-IV.3: Existence of good minimal model in families

Note-IV.4: Relative MMP, fiberwise MMP and absolute MMP

Note-IV.5: Restriction of MMP on the central fiber

Note-IV.6: Extension of MMP from the central fiber [update 12.3]

Part V. Finiteness of minimal model and Termination problems

Note-V.1: Polyhedron decomposition results

Note-V.2: MMP with scaling

Note-V.3: Special finiteness

Note-V.4: Global Termination Problem

Part VI. Finite Generation Problems

Finite Generation Note-VI.1: Finite generation of canonical ring, Cox ring

Finite Generation Note-VI.2: Demailly-Hacon-Paun’s analytic proof of finite generation

Finite Generation Note-VI.3: Finite generation and Abundance

Part VII. Partial Modification

Note-1: Log resolution, discrepancy

Note-2: Crepant extraction with applications

Note-3: dlt modification with applications

Note-4: Canonical modification and terminal modification, Q-factorialization

Note-5: Semi log modificaiton

Part VIII. Locus in Birational Geometry

The aim of this series of notes is to summarize the base properties about stable base locus, augmented base locus, restricted base locus which will be used over and over again in birational geometry.

Note-1: Exceptional Locus

Note-2: Zariski decomposition

Note-3: Base locus, Stable base locus, Restricted base locus and Augmented base locus

Note-4: Geometry of non-klt locus

Note-5: non-nef locus(numerical base locus), non-Kahler Locus

Note-6: Recent developments on the distribution of Hodge Locus

Part X. Non-vanishing and abundance

Note-X.1. Miyaoka’s proof of abundance for threefolds

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