Groupe de travail sur la géométrie birationnelle 

Program 

1. Birational geometry and introduction to MMP techniques 

• Talk 1 (MMP for smooth surfaces (3h) J.-B.). In this talk we should explain some fundamental results of birational geometry of surfaces (curves also). The main result should be the Castelnuovo contraction theorem, see [Har13, Theorem 5.7] and [Bea78]. If time permits discuss minimal models of smooth projective surfaces: except for rational and ruled surfaces all surfaces admit a unique minimal model, see [KM98, Theorem 1.29], [Har13, Remark 5.8.4] and references therein.
• Talk 2 (Finding rational curves (3h) Vanja). Explain bend and break! Introduce some cones of curves and divisors as in Def. 1.15, 1.16, 1.17, Section 1.4 [KM98].
  1. Sections 1.1 and 1.2 [KM98]
  2. Main theorem: Theorem 1.10 [KM98]
  3. You will find in [Deb01] a very good exposition and in [Kol96].
• Talk 3 (Singularities of the minimal model program I (2h) Thomas). Give the standard definitions of singularities as treated [KM98, Section 2.3]: the definition to reach is [KM98, Definition 2.34]. Compare them to [KM98, Definitions 2.11, 2.12]. Give examples of singularities. The most classical ones are cones and quotients. We may choose to focus on cone singularities over various types of varieties (curves): in [Kol13] there are very general examples, we should reduce them to simpler ones.
• Talk 4 (Singularities of the minimal model program II (2h) Jason). The goal of the talk is to give more examples of singularites. Start with surface singularities of Du-Val type (equiv. canonical, equiv. ADE) see [Rei87] and [KM98, Theorem 4.20]. It could be nice to explain Artin contraction theorem [Rei93][Section 4.15], indeed it shows how rational singularities arise (compare it to Castelnuovo’s contraction thm). In the second part treat some examples from [Kol13, Sections 3.1, 3.2] and Reid–Thai criterion. [Kol13, Thm 3.21]
• Talk 5 (Singularities of the MMP — properties (2h) Marta). This talk should give some interesting technical properties of singularities of pairs. See [KM98, Example 2.1] (together with [Uen75, Proposition 16.17]): explain why we need to introduce singularities when looking for a minimal model. In fact, this is a smooth threefold with no smooth minimal model. (See also Singular minimal model.) See also [KM98, Corollary 2.31, 2.32, 2.33, 2.35].
• Talk 6 (Flips (2h) Franco). It should be a talk that puts emphasis on examples.
  1. Motivation: why we need flips (sketch of MMP algorithm 2.14 [KM98])
  2. Definition of flip 2.8 [KM98]
  3. Example 2.7 [KM98]
  4. Mukai flop and why it is a flip, see also [Huy03][Example 21.7].
  5. State [CJ07, Theorem 5.1.1] (this is used in [BCHM10], so we need to
  state the result).
• Talk 7 (Base point free, non-vanishing, rationality, cone theorem (2h) Anne Lonjou). Discuss the following fundamental theorems and some ideas of the proofs.
  1. Non-vanishing [KM98].
  2. Base point free theorem with its proof [KM98] and state [Kol93, Theorem 1.1].
  3. Rationality [KM98].
  4. Cone and Contraction Theorem [KM98] –only statement since we have seen it already in the smooth case, but do highlight the new elements. It is not clear how much of the proof we should see on the blackboard, it is the task of the speaker to decide! See talks of 19th and 26th February 2021 for inspiration.
• Talk 8 (Adjunction). Treat the adjunction for surfaces [Kol13, Proposition 2.35] with some examples. Some suggestions: the nodal curve and the cusp curve in the plane or the ruling of a cone. Then do the general case that is found in [Kol13, 4.5], note that this version is very general and should be reduced to a more down to earth statement (discuss with Marta for suggestions if needed). State inversion of adjunction [Kol13, Theorem 4.9 (1) and (2)].
• Talk 9 (Nakayama–Zariski decomposition). Follow the corresponding chapter in [Nak04].
  1. Surface case with examples (e.g. blow-up of the plane) 2.3.E, 2.3.19,
  2.3.21, p.170 [Laz04]
  2. Section 3.3 [BCHM10]
• Talk 10 (Negativity lemma). This is a very important tool in birational geometry.
  1. Hodge index theorem (for surfaces) V1.9 [Har13]
  2. Negativity lemma 3.39 [KM98]
  3. Lemma 3.6.2 [BCHM10]
• Talk 11 (Kawamata-Viehweg vanishing (2h)). [BCHM10] uses a version of Kawamata-Viehweg vanishing, which is a generalization of Kodaira vanishing.
  Explain Kodaira’s vanishing theorem [KM98, Theorem 2.47]. At the end state also [KM98, Theorems 2.64 and 2.70]

2. The Minimal Model Program for varieties of general type [BCHM10]

Remark 1. For the sake of the reading group, in order to not make notation too heavy, we can state everything in the ABSOLUTE setting, i.e. MMP for varieties X → Spec(C) (we do not consider the relative setting of relative MMP X → U).
Also, for the sake of the reading group state all results with R-divisors but do all the proofs for Q-divisors. Actually all the statements for R-divisors are firstly proved for Q-divisors and then an approximation process is applied, see for example [BCHM10, beginning of proof of Theorem 3.9.1]. Warning: there are some theorems for which it is necessary to consider to R-divisors, the theorems
involving polytopes and finiteness of models. Make sure to not mess up!

 

• Talk 12 (Introduction –Vanja). Present and motivate the statements of the main results in [BCHM10, THMs 1.1 and 1.2] following the introduction of the paper. On the way, define the different types of models that are needed ([BCHM10, Definition 3.6.6]) and the different polytopes ([BCHM10, Definition 1.1.4]) we will encounter (if needed). Inspiration can be taken from talks of Sep 10th 2021 and end of talk of Sep 24th 2021 from Moraga’s learning seminar.
• Talk 13 (Outline of the proof –Marta). Give a presentation of the sketch of the proof as in [BCHM10, Section 2.1]. This should be a talk that gives an outline of the main ideas. It’s a very important talk, so that later we can fill in the details, but keeping an eye on the big picture. Inspiration can be taken from talk of Sep 17th 2021 from Moraga’s learning seminar. Remark: this is probably the most difficult talk because it involves understanding the main points of the proof without having seen the whole proof, the speaker should be a bit familiar with the material already.
• Talk 14 (MMP with scaling and Shokurov’s polytopes). This talk should follow [BCHM10, Sections 3.10, 3.11]. The main results on which the speaker should focus on are: [BCHM10, Remark 3.10.9] which outlines what is the MMP with scaling, and [BCHM10, Theorem 3.11.1 and its corollaries]. Inspiration can be taken from talk of Oct 1st 2021 from Moraga’s learning seminar.
• Talk 15 (Special finiteness). The goal of this talk is to prove finiteness of weak log canonical models around a fixed reduced divisor ([BCHM10, Lemma 4.4]).
  Inspiration can be taken from talk of Oct 8th 2021 from Moraga’s learning seminar.
• Talk 16 (Existence of log terminal models). The goal of this talk is to prove the existence of log terminal models ([BCHM10, Lemma 5.6]). Inspiration can be taken from talk of Oct 8th 2021 from Moraga’s learning seminar.
• Talk 17 (Non-vanishing). The goal of this talk is to prove the non-vanishing theorem ([BCHM10, Lemma 6.6]). Inspiration can be taken from talk of Oct 15th 2021 from Moraga’s learning seminar.
• Talk 18 (Finiteness of models). The goal of this talk is to prove finiteness of weak log canonical models ([BCHM10, Lemmas 7.2/ 7.3]). Inspiration can be taken from talk of Oct 22nd 2021 from Moraga’s learning seminar.
• Talk 19 (Conclusions). This talk should cover [BCHM10, Sections 8, 9] and will conclude the proof of the main theorems. Make sure to summarise clearly the results we are using and how, maybe reviewing the sketch of proof to summarise the path we have followed. Finish by presenting some of the open conjectures in [BCHM10, Section 2.2]. Remark: probably this is also a difficult talk and would need a speaker with some familiarity with the material.

3. Applications and further techniques

The next talks can be given at any point during the reading group to motivate. The topics can follow interests of people.

• Talk 20 (Birational geometry). Talk about some of the corollaries according to the interest of the speakers. Avoid [BCHM10, Corollary 1.2.1, Corollary 1.3.1], they will be treated in a later talk. Note: the most used corollary in birational geometry is [BCHM10, Corollary 1.4.3], which can be compared with the existence of terminalization, Q-factorialization, dlt modifications, ... Other fundamental corollaries: existence of flips [BCHM10, Corollary 1.4.1], inversion of adjunction [BCHM10, Corollary 1.4.5], finite generation [BCHM10, Corollary 1.1.2], geography of models [BCHM10, Corollary 1.1.5]. 
• Talk 21 (Fano varieties and moduli spaces of curves). In the first part of this talk we will present the notion of Mori dream spaces ([HK00], [BCHM10, Corollary 1.3.1]) and prove that we can run an MMP terminating with a Mori fibrespace whenever the canonical divisor is not pseudoeffective ([BCHM10, Corollary 1.3.2]). According to the interest of the speaker or if there is time, one can also talk about moduli spaces of curves M_g,n and present [BCHM10, Corollary 1.2.1].
• Talk 22 (Another approach to finite generation). Siu’s paper on finite generation with analytic methods [Siu06] 

References

 

• [BCHM10] Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan. Existence of minimal models for varieties of log general type. J. Amer. Math. Soc., 23(2):405–468, 2010.
• [Bea78] Arnaud Beauville. Surfaces algébriques complexes. SMF, 1978.
• [CJ07] Christopher Hacon and James McKernan. Flips for 3-folds and 4-folds. Oxford University Press, 2007.
• [Deb01] Olivier Debarre. Higher-dimensional algebraic geometry. Universitext. Springer-Verlag, New York, 2001.
• [Har13] Robin Hartshorne. Algebraic geometry, volume 52. Springer Science & Business Media, 2013.
• [HK00] Yi Hu and Sean Keel. Mori dream spaces and GIT. volume 48, pages 331–348. 2000. Dedicated to William Fulton on the occasion of his 60th birthday.
• [Huy03] Daniel Huybrechts. Compact hyperkähler manifolds. In Calabi-Yau manifolds and related geometries (Nordfjordeid, 2001), Universitext, pages 161–225. Springer, Berlin, 2003.
• [HX15] Christopher D. Hacon and Chenyang Xu. On the three dimensional minimal model program in positive characteristic. J. Amer. Math. Soc., 28(3):711–744, 2015.
• [KM98] János Kollár and Shigefumi Mori. Birational geometry of algebraic varieties, volume 134 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original.
• [Kol93] János Kollár. Effective base point freeness. Math. Ann., 296(4):595–605, 1993.
• [Kol96] János Kollár. Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics. Springer-Verlag, Berlin, 1996.
• [Kol13] János Kollár. Singularities of the minimal model program, volume 200 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2013. With a collaboration of Sándor Kovács.
• [Laz04] Robert Lazarsfeld. Positivity in algebraic geometry. I, volume 48 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series.
• [Nak04] Noboru Nakayama. Zariski-decomposition and abundance, volume 14 of MSJ Memoirs. Mathematical Society of Japan, Tokyo, 2004.
• [PST17] Zsolt Patakfalvi, Karl Schwede, and Kevin Tucker. Positive characteristic algebraic geometry. In Surveys on recent developments in algebraic geometry, volume 95 of Proc. Sympos. Pure Math., pages 33–80. Amer. Math. Soc., Providence, RI, 2017.
• [Rei87] Miles Reid. Young person’s guide to canonical singularities. In Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), volume 46 of Proc. Sympos. Pure Math., pages 345–414. Amer. Math. Soc., Providence, RI, 1987.
• [Rei93] Miles Reid. Chapters on algebraic surfaces. in Complex algebraic geometry, IAS/Park city, 1993.
• [Siu06] Yum-Tong Siu. A general non-vanishing theorem and an analytic proof of the finite generation of the canonical ring, 2006.
• [Tan14] Hiromu Tanaka. Minimal models and abundance for positive characteristic log surfaces. Nagoya Math. J., 216:1–70, 2014.
• [Uen75] Kenji Ueno. Classification theory of algebraic varieties and compact complex spaces. Lecture Notes in Mathematics, Vol. 439. SpringerVerlag, Berlin-New York, 1975. Notes written in collaboration with P. Cherenack.