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Topics in Algebraic Geometry II. Rationality of Algebraic Varieties

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Math 732: Topics in Algebraic Geometry II Rationality of Algebraic Varieties Mircea Mustat¸˘a∗ Winter 2017 Course Description A fundamental problem in algebraic geometry is to determine which varieties are rational, that is, birational to the projective space. Several important developments in the field have been motivated by this question. The main goal of the course is to describe two recent directions of study in this area. One approach goes back to Iskovskikh and Manin, who proved that smooth, 3-dimensional quartic hypersurfaces are not rational. This relies on ideas and methods from higher-dimensional birational geometry. The second approach, based on recent work of Claire Voisin and many other people, relies on a systematic use of decomposition of the diagonal and invariants such as Chow groups, Brauer groups, etc. to prove irrationality. Both directions will give us motivation to introduce and discuss some important concepts and results in algebraic geometry. The following is a rough outline of the course: Introduction 1. Rationality in dimension 2 (Castelnuovo's criterion). Part I 2. Introduction to Chow groups and intersection theory. 3. Decomposition of the diagonal and non-stable-rationality. 4. Examples of classes on non-stably-rational varieties. Part II 5. Introduction to singularities of pairs and vanishing theorems. 6. Birational rigidity of Fano hypersurfaces of index 1. Mircea Mustat¸˘a'sown lecture notes are available at: http://www-personal.umich.edu/~mmustata/lectures_rationality.html Contents 1 January 5 5 1.1 The L¨urothproblem and its solution for curves..........................5 1.2 Birational invariance of plurigenera.................................7 2 January 10 9 2.1 Birational invariance of plurigenera (continued)..........................9 2.2 Other properties close to rationality................................ 11 2.2.1 Stably rational varieties................................... 11 2.2.2 Uniruled varieties....................................... 11 2.3 Cubic surfaces............................................ 12 ∗Notes were taken by Takumi Murayama, who is responsible for any and all errors. Please e-mail [email protected] with any corrections. Compiled on April 29, 2017. 1 3 January 12 14 3.1 Cubic surfaces (continued)...................................... 14 3.2 Some rational cubic hypersurfaces................................. 17 4 January 17 18 4.1 Some rational cubic hypersurfaces (continued)........................... 18 4.2 Unirationality of cubic hypersurfaces................................ 21 4.2.1 Review of projections.................................... 22 5 January 19 23 5.1 Unirationality of cubic hypersurfaces (continued)......................... 23 5.1.1 Projections.......................................... 23 5.1.2 Quadric bundles....................................... 26 5.1.3 Proof of Theorem 5.1.................................... 27 6 January 24 28 6.1 Unirationality of cubic hypersurfaces (continued)......................... 28 6.1.1 Proof of Theorem 5.1 (continued).............................. 28 6.2 More on incidence correspondences................................. 29 7 January 26 32 7.1 Comments on Theorem 5.1..................................... 32 7.2 Varieties containing linear subspaces................................ 33 7.3 Unirationality of hypersurfaces of higher degree.......................... 36 8 January 31 36 8.1 General statements about (uni)rationality............................. 36 8.2 Unirationality of hypersurfaces of higher degree (continued)................... 37 9 February 2 40 9.1 Overview of topology of algebraic varieties............................. 40 9.1.1 Singular (co)homology.................................... 40 9.1.2 Complex algebraic varieties................................. 43 10 February 14 45 10.1 Overview of topology of algebraic varieties (continued)...................... 45 10.2 Rationality and unirationality for surfaces............................. 45 11 February 16 49 11.1 Rationality and unirationality for surfaces (continued)...................... 49 12 February 21 53 12.1 Shioda's examples of unirational but not rational surfaces.................... 53 12.2 More topology............................................ 55 13 February 23 58 13.1 Some cohomology computations................................... 58 13.2 An invariant that detects non-stable-rationality.......................... 59 3 13.3 How to produce examples with H (X; Z)tors 6= 0......................... 60 14 March 7 61 14.1 The Artin{Mumford example.................................... 61 14.1.1 The complete linear system of quadric hypersurfaces in P3 ............... 61 14.1.2 Geometry of the map p in (14:1).............................. 63 14.1.3 Introduction to cyclic covers................................. 64 2 15 March 9 65 15.1 The Artin{Mumford example (continued)............................. 65 16 March 21 68 16.1 The Artin{Mumford example (continued)............................. 69 16.2 Introduction to ´etalecohomology.................................. 71 16.2.1 Presheaves and sheaves in the ´etaletopology....................... 71 17 March 23 73 17.1 Introduction to ´etalecohomology (continued)........................... 73 17.1.1 Some statements from descent theory........................... 73 17.1.2 Presheaves and sheaves in the ´etalecohomology (continued)............... 74 17.2 Brauer group............................................. 75 17.2.1 Connection with singular cohomology........................... 75 17.2.2 Brauer groups and Pn-fibrations.............................. 76 18 March 28 77 18.1 Brauer groups (continued)...................................... 77 18.1.1 Brauer groups and Pn-fibrations (continued)....................... 77 18.2 Introduction to first cohomology of non-abelian sheaves..................... 78 19 March 30 81 19.1 Introduction to Chow groups.................................... 81 19.1.1 Operations on Chow groups................................. 82 19.1.2 Chern classes......................................... 85 20 April 4 86 20.1 Introduction to Chow groups (continued)............................. 86 20.1.1 Refined Gysin maps..................................... 86 20.1.2 Pullback by lci morphisms.................................. 88 20.1.3 Connection with singular cohomology........................... 89 20.2 Correspondences........................................... 90 21 April 6 91 21.1 Correspondences (continued).................................... 91 21.1.1 Actions of correspondences on Chow groups........................ 91 21.1.2 CH0 is a birational invariant................................ 92 21.1.3 Actions of correspondences on singular cohomology.................... 93 21.1.4 The specialization map................................... 94 22 April 11 95 22.1 Decomposition of the diagonal................................... 95 22.1.1 The Bloch{Srinivas theorem and stable rationality.................... 96 23 April 13 98 23.1 Decomposition of the diagonal (continued)............................. 98 23.1.1 CH 0-triviality for morphisms................................ 101 24 April 18 102 24.1 Decomposition of the diagonal (continued)............................. 102 24.1.1 A criterion for universal CH 0-triviality for morphisms.................. 102 24.1.2 Behavior of decomposition of the diagonal in families................... 103 3 25 April 25 106 25.1 Non-stable-rationality of quartic threefolds............................. 106 25.1.1 Rationality and base change................................. 106 25.1.2 Proof of Theorem 25.1.................................... 107 25.1.3 The geometry of quartic symmetroids........................... 108 25.1.4 The construction for Theorem 25.2............................. 110 References 111 4 1 January 5 The theory of rationality centers around the question: Which varieties are rational, i.e., birational to the projective space Pn? This theory is very interesting because it uses vastly different methods. On one hand, showing something is rational usually requires a lot of explicit geometric constructions that are classical in flavor. On the other hand, showing something is not rational requires more complicated invariants like Chow groups, etc., many of them arising from birational geometry. We first give a brief outline of the course: Introduction: Some definitions and easy properties; examples In particular, we will study surfaces (Castelnuovo's rationality criterion), cubic three-folds and quartic three-folds, and the unirationality of hypersurfaces of small degree. Note that for these last examples, it is already hard to show they are not rational. These examples will build intuition and help you develop tools to study more complicated cases. Part I: Stable rationality We will review some things about Chow groups and intersection theory, and then study the theory of \decomposition of the diagonal" and how it relates to stable rationality, following ideas of Voisin and others. Part II: Birational rigidity of certain Fano varieties A key example here is that of hypersurfaces of degree n in Pn, which are \extremal" Fano varieties. Birationally rigid varieties are those for which birational self-maps can all be extended to automorphisms; for the aforementioned hypersurfaces, the group of birational self-maps will be finite, and so they cannot be rational. The main ideas in this theory come from Mori theory, and so we will need to talk about invariants of singularities and vanishing
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