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2021-NCTS-AG-LEARNING-SEMINAR DEFORMATION THEORY AND RATIONAL CURVES ON ALGEBRAIC VARIETIES
1. General rules
Here are general rules about our learning seminar:
• Each meeting is 60 to 90 minutes talk plus 30 minutes discuss. This can be flexible.
• Each talk can take up to 90 minutes. If the content of a talk is too much, then it can be split into two 50-minute talks, possibly with two speakers.
2. Tentative Seminar Plan
2.1. Part I - Abstract deformation theory. Reference: [Har10]. Pre-requisite: We should assume everyone is familiar with the definition and basic properties of the Hilbert scheme (but not with its construction). If this is not the case, we should recall them during the first talk.
Talk 1[Iacopo, 2/26, 3:30 pm] (Chapter I) First order deformations of line bundles, subvarieties, coherent sheaves. Construction and properties of the Ti functors. Infinitesimal lifting property, smoothness of a morphism and vanishing of Ti>0. Deformation of rings via T1. Example: deformation of cones.
• Flatness and details of constructing Ti functors.
• General computation of Ti: infinitesimal lifting property. (Remark: Possibly review/merge the difference between [Har10] and [Kol96].)
• Theorem 4.11, 4.13: local complete intersection.
• (Optional) Theorem 5.3, 5.4: can be combined in the section on deformation of singular schemes.
Talk 2[Chi-Kang,3/5, 10am] (Chapter II, Sec. 6-9) Obstructions for higher-order deformations of line bundles, subvarieties, coherent sheaves. Deformation of singular schemes.
• Section 6 maybe be reduced from Section 7.
• CM in codimension 2 : Main theorem is Corollary 8.13.
• (Optional) Gorenstein in codimension 3: Can replace it with other interesting examples, such as:
– Sec. 29: Cor. 29.10.
– Rigidity of isolated singularities in codimension three, cf. [Sch71].
– Smoothable quotient surface singularities: T-singularities, cf. [KSB88].
– Deformation of (weak) singular Fano (CY) threefolds, cf. Namikawa’s or Sano’s work.
Talk 3[Ting-yu, 3/12, 3:30 am] (Chapter II, Sec. 10-13) Obstruction for deformations of schemes, abstract obstruction theory, bound for the local dimension of the Hilbert scheme.
• Section 2. 10: Possibly review/merge the difference between [Har10] and [Kol96].
• Corollary 11.2: Important for Bend and Break, cf. [Deb01, Sec. 2.4, Theorem 2.6].
• (Optional) Section 13.
Talk 4[Bin, 3/19, 3:30 pm ] (Chapter III, Sec. 14-17) Introduction to formal moduli. Functors of Artin rings, mini/uni/versal families. Schlessinger criterion and application: pro-representability of Hilbert scheme and/or Picard scheme.
• Theorem 16.2 + Pro-representability of Pic.
• (Optional): Deligne, Rel ́evement des surfaces k3 en caract ́eristique nulle. LNM. ([Har10, Ref. 19])
Talk 5[Hsin-Ku, 3/26, 3:30 pm] (Chapter III, Sec. 18-20) Conditions for existence of miniversal family/pro-representalbilty of deformation functor of a scheme/sheaf. Abstract deformations vs. embedded deformations.
• Chapter 18, Ex. 18.4.2.
• (Optional) Section 20.
• Preparation on formal scheme and formal moduli.
Talk 6[Iacopo, 4/9, 3:30 pm] (Chapter III, Sec. 21-22) Brief introduction to formal schemes and Grothendieck’s existence theorem. Algebraization of formal moduli. Lifting from characteristic p.
• Grothendieck’s theorem(= Theorem 21.2) is an important ingredient, cf. [FGI+05] and [Ser06].
• We shall span time explaining all the proofs of Artin: Theorem 21.3, 21.4.
• p-Lifitability: Cor. 22.2. Theorem 22.4. An example of nonliftable variety [FGI+05, 8.6]. (Maybe more to say?)
2.2. Part II - Deformation theory and DGLA. We will mainly follow Manetti’s note.
Talk 1[Chi, 4/16, 3:30 pm] (Sec. 1 + Sec. 3) DGLA and Deformation functors associated to a DGLA.
Talk 2[Chi/Ray, 4/23, 3:30 pm] (Sec. 4 + Sec. 5) Kuranishi and gauge equivalence.
Talk 3[Ray, 4/30, 10 am] (Optional: Applications)
• A working example: An algebraic proof of BTT.
• (Optional) Any deformation problem has a DGLA, cf. Bourbaki talk of B. T ̈oen.
2.3. Part III - Deformation of rational curves. The outline of the talks is based of Debarre’s book, [Deb01], with possible references to Kolla rs book, [Kol96]. The talks will focus around chapters 3-5 of [Deb01], culminating in proving an effective bound for the volume of a smooth Fano variety based on dimensions which implies the boundedness of family of smooth fano varieties.
Talk 1[David, 5/7, 3:30 pm] (Bend-Break Lemma) Covers Chapter 3 (with possible reference to chapter 2).
• Prop. 3.3, Thm. 3.4, the generalizations of Bend Break Lemma and Theorem 3.22.
• Relative BB: This is important in deformation of RC varieties.
• (Optional) 3.6, 3.7.
Talk 2[David, 5/14, 3:30 pm] (Unirational, Rationally Connected, (Very) Free Rational Curves) Covers Chapter 4.1 - 4.3. Introduce the different aspects of
rational curves and compare them.
• Corollary 4.11 relating free rational curves to uniruledness and corollary 4.17 of very free rational curves to rational connectedness.
• Possibly include a short survey on rationality problem. E.g. ZAG talk of J. Nicaise.
Talk 3[Chih-Wei, 5/21, 3:30 pm] (Rational Chain Connectedness) Covers Chapter 4.5-4.7. Introduce Rational Chain Connectedness.
• Corollary 4.28: in characteristic zero that for smooth varieties it is equivalent to rational connectedness.
• The following topics are selective and even optional. A sketch of the proof shall be enough:
– Hacon-Mckernan criterion for RCC, cf. [HM07].
– Strongly RC: connecting points by very free rational curves in non-singular locus, cf. [Xu12].
– Char= p > 0 case: Seperably RC, cf. [Kol96].
Talk 4’s cover Chapter 5.2 and 5.4. Also one of speakers shall discuss the maximally rational quotient.
Talk 4-1[Ray, 5/28, 3:30 pm] (Quotients by Algebraic Relations I) Covers Chapter 5.2.
Talk 4-2[Alfred, 6/4, 3:30 pm] (Quotients by Algebraic Relations II) Covers Chapter 5.4.
Talk 5[Chi-Kang, 6/11, 3:30 pm] (Structure of MRC)
• Regularity of the MRC fibration, cf. [Cam04a, Cam04b].
• The base of the MRC fibration is not uniruled, cf. [GHS03].
• Two speakers for this part?
Talk 6[Jheng-Jie, 6/18, 3:30 pm] (The Case of Smooth Fano Varieties) Covers Chapter 5.5 - 5.9 (with maybe supplements from Koll ́ar, [Kol96, V.2].)
• Fano varieties are rationally connected.
• The boundedness of families of smooth Fano varieties.
• Possibly cover chapter 5.3 to focus on the Picard rank 1 case.
• Explain Koll ́ar-Matsusaka’s result on bounding coefficients of the Hilbert polynomials.
• (Optional) Section 5.8: There are some progress on Fano volume by Li on K-semistable case, or in Picard one smooth Fano fourfold by JM Hwang.
2.4. Part IV - Other related work. The following two talks are related to deformation theory, but we will have these talks only if there are volunteers.
Talk 1[Jheng-Jie, 6/25, 3:30 pm] (Misc. Flexible Talk - Rational Curves on K3 Surfaces) Following the Appendix of Mori and Mukai, [MM83]. Applying deformation theory with the moduli of K3 surfaces to show that you can find a rational curve on an algebraic K3 surface.
Talk 2[YP, 7/2, 3:30 pm] (Misc. Flexible Talk - Clemens Conjecture on Rational Curves on Quintic Threefolds) Go through the differential aspects (e.g. differential equations) of deformation theory of Clemens, [Cle83], with an application by [Kat86], to show that the number of rational curves of degree d ≤ 7 is finite for general quintic threefolds.
References
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[Cam04a]
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Frédéric Campana. Orbifolds, special varieties and classification theory. Annales de l’Institut Fourier, 54(3):499–630, 2004.
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[Cam04b]
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Frédéric Campana. Orbifolds, special varieties and classification theory: an appendix. Annales de l’Institut Fourier, 54(3):631–665, 2004.
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[Cle83]
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Herbert Clemens. Homological equivalence, modulo algebraic equivalence, is not finitely generated. Inst. Hautes Études Sci. Publ. Math., (58):19–38 (1984), 1983.
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[Deb01]
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Olivier Debarre. Higher-dimensional algebraic geometry. Universitext. Springer-Verlag, New York, 2001.
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[FGI+05]
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Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and Angelo Vistoli. Fundamental algebraic geometry, volume 123 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005. Grothendieck’s FGA explained.
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[GHS03]
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Tom Graber, Joe Harris, and Jason Starr. Families of rationally connected varieties. J. Amer. Math. Soc., 16(1):57–67, 2003.
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[Har10]
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Robin Hartshorne. Deformation theory, volume 257 of Graduate Texts in Mathematics. Springer, New York, 2010.
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[HM07]
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Christopher D. Hacon and James Mckernan. On Shokurov’s rational connectedness conjecture. Duke Math. J., 138(1):119–136, 2007.
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[Kat86]
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Sheldon Katz. On the finiteness of rational curves on quintic threefolds. Compositio Math., 60(2):151–162, 1986.
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[Kol96]
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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.
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[KSB88]
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J. Kollár and N. I. Shepherd-Barron. Threefolds and deformations of surface singularities. Invent. Math., 91(2):299–338, 1988.
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[MM83]
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Shigefumi Mori and Shigeru Mukai. The uniruledness of the moduli space of curves of genus 11. In Algebraic geometry (Tokyo/Kyoto, 1982), volume 1016 of Lecture Notes in Math., pages 334–353. Springer, Berlin, 1983.
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[Sch71]
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Michael Schlessinger. Rigidity of quotient singularities. Invent. Math., 14:17–26, 1971.
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[Ser06]
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Edoardo Sernesi. Deformations of algebraic schemes, volume 334 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2006.
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[Xu12]
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Chenyang Xu. Strong rational connectedness of surfaces. J. Reine Angew. Math., 665:189–205, 2012.
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