This is an attempt to understand the work of Vlad Lazic about the finite generation of the canonical ring in detail. Although the statement has been first proved by Bircar, Cascini, Hacon, and McKernan, the proof given by Lazic appears to be more approachable. Since most of us are non-specialists, we will need a lengthy warm-up in birational geometry, which will cover the first 8 weeks. A rough introduction can be found here.
There is a recent survey article of Cascini and Lazic, which can be very helpful, since it is not overly technical. You can have a look at it
here.
This series of talks also doubles as preparation for the summer school on birational automorphisms of varieties of general type.
For every talk, there will be suggestions of (mostly standard technical) material that needs to be remembered beforehand, and there will be consequences, examples that could enhance the experience when read afterwards.
Note: the handouts are of informal nature and not fully vetted. Use at your own risk.
Note: the handouts are of informal nature and not fully vetted. Use at your own risk.
Schedule of talks
- Introduction (May 4th, Alex Küronya)
- Linear series on surfaces and Zariski decomposition (May 11th, Oliver Straser, handout )
- Finite generation and the MMP on surfaces I. (May 18th, Sebastian Goette)
- Finite generation and the MMP on surfaces II. (May 25th, Sarah Kitchen)
- Singularities of pairs via lots of examples (June 1st, Stefan Kebekus, handout)
- Asymptotic invariants of linear series (June 8th, Clemens Jörder, handout)
- Divisorial rings and Cox rings (June 22th, Fritz Hörmann, handout)
- Vanishing theorems and lifting of sections (June 29th, Andreas Höring) *
- Finite generation of adjoint rings I. (July 6th, Patrick Graf and Daniel Lohmann, handout) *
- Finite generation of adjoint rings II. (July 13th, Patrick Graf and Daniel Lohmann) *
- Proof of the main theorems of the minimal model program I. (July 20th, Daniel Greb, handout) *
- Proof of the main theorems of the minimal model program II. (July 27th) *
Literature
- [And] Marco Andreatta: An introduction to Mori theory: the case of surfaces,notes for a PhD school
- [AnM] Marco Andreatta, Massimiliano Mella: Morphisms of projective varieties from the viewpoint of minimal model theory, Dissertationes Mathematicae, vol. 413 (2003).
- [AHDL] Ivan Arzhantsev, Ulrich Derenthal, Jürgen Hausen, Antonio Laface: Cox rings
- [Bad] Lucian Badescu: Algebraic surfaces, Universitext, Springer Verlag, Berlin, 2001.
- [Bau] Thomas Bauer: A simple proof for the existence of Zariski decompositions on surfaces.
- [BaF] Thomas Bauer, Michael Funke: Weyl and Zariski chambers on K3 surfaces
- [BCK] Thomas Bauer, Mirel Caibar, Gary Kennedy: Zariski decomposition: a new (old) chapter of linear algebra
- [BKS] Thomas Bauer, Alex Küronya, Tomasz Szemberg: Zariski chambers, volumes, and stable base loci
- [CaL] Paolo Cascini, Vladimir Lazic: New outlook on Mori theory, I.
- [CaL2] Paolo Cascini, Vladimir Lazic: The minimal model program revisited
- [CoL] Alessio Corti, Vladimir Lazic: New outlook on Mori theory, II.
- [Cor] Alessio Corti: Finite generation of adjoint rings after Lazic: an introduction.
- [CKL] Alessio Corti, Anne-Sophie Kaloghiros, Vladimir Lazic: Introduction to the Minimal Model Program and the existence of flips
- [CHKLM] Alessio Corti, Paul Hacking, János Kollár, Robert Lazarsfeld, Mircea Mustaţă: Lectures on flips and minimal models
- [DebN] Olivier Debarre: Introduction to Mori theory
- [Deb] Olivier Debarre: Higher-dimensional algebraic geometry, Universitext 233, Springer Verlag, Berlin, 2001.
- [ELMNP] Lawrence Ein, Robert Lazarsfeld, Mircea Mustata, Michael Nakamaye, Mihnea Popa: Asymptotic invariants of base loci
- [Gat] Andreas Gathmann: Algebraic Geometry
- [HK] Christopher Hacon, Sándor Kovács: Higher-dimensional algebraic geometry, Oberwolfach Seminars 43, Birkhäuser, 2010.
- [Ha] Robin Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics 77, Springer Verlag, Berlin, 1977.
- [KoM] János Kollár, Shigefumi Mori: Birational geometry of algebraic varieties. With the collaboration of C. H. Clemens and A. Corti. Cambridge Tracts in Mathematics, 134. Cambridge University Press, Cambridge, 1998.
- [PAG] Robert Lazarsfeld: Positivity in algebraic geometry I.-II.
- [Miy] Masayoshi Miyanishi: Open algebraic surfaces, CRM Monograph Series 12, AMS, 1996.
- [Mat] Kenji Matsuki: Introduction to the Mori program, Universitext, Springer Verlag, Berlin, 2002.
- [Nak] Noboru Nakayama: Zariski-decomposition and abundance. MSJ Memoirs, 14. Mathematical Society of Japan, Tokyo, 2004.
- [RCh] Miles Reid: Chapters on algebraic surfaces.
Outline of talks
- Introduction
The idea of the birational classification of algebraic varieties, significance of the sign of the canonical divisor, case of curves. Finite generation of the canonical ring of a curve. Linear series and the associated rational maps, base loci, stable base loci, decomposition into moving part and free part, resolution of indeterminacies, the difference between dimensions two and more. Cartier divisors with rational and real coefficients, numerical equivalence of divisors, important notions of positivity (ample/nef/big), Zariski decomposition. Relation between birational models and linear series, finite generation of section rings and the Proj construction. The algorithmic approach: the idea of the Mori program. Correspondence between extremal rays of the Mori cone, faces of the nef cone, and the associated morphisms. Main theorems of the minimal model program.
Literature: [KoM] Chapters 1,2; [PAG] Chapter 1, [Deb] Chapters I. and VII.
Before: [Ha] Section II.7. and [PAG] Section 1.1
After: Examples of Mori cones and ample cones in [KoM], [PAG], and Section 6.2 of [Deb]. - Linear series on surfaces and Zariski decomposition
Definition of Zariski decomposition on surfaces, with the projective plane blown up in two points as an illustration (including the various notable cones of divisors in the Néron-Severi space, and the morphisms corresponding to chambers). Presentation of Bauer's proof in detail, sketch of Fujita's extension to pseudo-effective divisors. Important consequences (positive part carries all the sections, etc). Birational Zariski decomposition, Cutkosky's counterexample to the existence of such (without proof). Kawamata's observation: existence of ZD for the canonical divisor implies the finite generation of the canonical ring (without proof). Zariski's example of a non-finitely generated section ring. The locally finite polyhedral property of the nef cone. Variation of Zariski decomposition in the big cone (mostly without proof), continuity of Zariski decomposition (mostly without proof).
Literature: [Bau], [BCK], [Bad], [PAG] Section 2.3, [Miy], [BKS]
Before: [Ha] Section V.1., beginning of Section V.3.
After: [BCK], [BaF], the case of del Pezzo surfaces in [BKS] - Finite generation and the MMP on surfaces I.
The Mori cone on surfaces, examples. The statement of Mori's cone theorem and the connection to rational maps. Castelnuovo's contraction theorem (rough sketch of proof) and the possibility of contracting curves. The vanishing theorems of Kodaira and Kawamata-Viehweg (without proof). The statements of the main theorems of MMP on surfaces, and the relationships among them. Proofs of the base-point free and rationality theorems (sketches with some important parts worked out in full detail).
Literature: [And], [RCh], Chapter VI of [Deb], [KoM], Chapter I. of [Mat]
Before: [Ha], Sections V.3. and V.5.
After: Read through the appropriate parts of [RCh] - Finite generation and the MMP on surfaces II.
Proof of the Cone and Contraction Theorems. The minimal model program on surfaces. Finite generation of the canonical ring of a surface. Zariski's result on semiampleness and finite generation (sketch).
Literature: [CaL2], [And], [RCh], Chapter VI of [Deb], [KoM], Chapter I. of [Mat]
Before: [Ha], Sections V.3. and V.5.
After: Read Section 2.1 of [KoM] focusing on explanations (and not necessarily on technical details) - Singularities of pairs via lots of examples
Valuations of the functions field, divisorial valuations, discrepancies. Log pairs, log resolutions, good resolutions. Important classes of singularities of pairs: log canonical, klt, dlt, plt, canonical, terminal. The significance of each of these classes from the point of view of the Mori program. Examples. Behaviour under restriction to hypersurfaces, adjunction, inversion of adjunction (mostly without proofs).
Literature: Section 2.3 of [KoM], [Mat], Chapter 3 of [HK].
Before: Recall how to blow up a variety along a subvariety from [Gat] Chapter 4, [Ha] Sections I.4 and II.7.
After: Memorize the contents of this talk, since they are fundamental for most of what is to come. - Asymptotic invariants of linear series
Asymptotic order of vanishing with respect to a divisorial valuation, basic properties. Asymptotic invariants as functions on the big cone. Stable base locus for divisors with real coefficients. Nakayama-Zariski decomposition. Examples.
Literature: Sections 2 and 3 of [ELMNP], Chapter III. of [Nak], Section 9.A of [HK], Subsection 2.3 of [CaL].
Before: Read through [PAG] 1.3 and 2.2.A.
After: Look at the examples in [ELMNP]. - Divisorial rings and Cox rings
Rings graded by monoids and abelian groups, Definition of Cox rings, divisorial rings, and adjoint rings. Finite generation of a Veronese subring is a necessary and sufficient condition for the finite generation of the original ring. Example: section rings of ample or semi-ample line bundles are finitely generated (with proof). Manipulating divisorial rings.
Literature: [ADHL], Section 2 of [Cor], Chapter 7 of [Deb], Subsection 2.2 and 2.4 of [CaL].
Before: Skim through Sections I.1 and I.4. of [AHDL].
After: Read Section 4 of [ELMNP]. - Vanishing theorems and lifting of sections
Notions of positivity for line bundles: ample versus big and nef. The latter is the birationally stable version of the former. The vanishing theorems of Serre and Kodaira, applications: lifting of sections in simple cases (give a concrete example), and surjectivity of multiplications of sections (explain the connection to finite generations). Kawamata-Viehweg vanishing theorem first for integral, then for Q-divisors. The lifting statement in Siu's proof of invariance of plurigenera (with suitable (over)simplifications). Hacon-McKernan's lifting theorem.
Literature: [PAG] Sections 1.2, 1.4, 2.2B, 4.3, 11.5, Chapter 7. of [Deb], Chapter 3 of [KoM], [CKL].
Before: Read [Ha] Section II. 6 for Cartier divisors/line bundles/invertible sheaves, [Ha] Section II.5 or [Gat] for the short exact sequence associated to an ideal sheaf, and [Ha] Section III.1 or any book on homological algebra for the associated long exact sequence.
After: Make sure you understand how to use vanishing theorems to lift sections. - Finite generation of adjoint rings I.
Explain all the main ideas and some typical details of [CaL] and [Cor]. Main focus: understand why induction on dimension works.
Literature: primarily [CaL], [CaL2]. and [Cor], perhaps [HK] as well.
Before: Recall what we have learnt about linear series and adjoint/divisorial rings.
After: Understand the main claims and the structure of the induction in the proof. - Finite generation of adjoint rings II.
See above. - Proof of the main theorems of the minimal model program I.
Prove the main theorems of the minimal model program (non-vanishing, base-point free, rationality, cone, contraction) using the finite generation of certain adjoint rings.
Literature: mainly [CoL], maybe [CKL], [Deb], [KoM] as well.
Before: Recall the main theorems of the MMP.
After: Take one of the five theorems and try to prove it in detail. - Proof of the main theorems of the minimal model program II.
See above.