Alex Küronya

Outline of talks

1. 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].
2. 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]
3. 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]
4. 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)
5. 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.
6. 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].
7. 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].
8. 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.
9. 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.
10. Finite generation of adjoint rings II.
    See above.
11. 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.
12. Proof of the main theorems of the minimal model program II.
    See above.