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Some Applications of Vector Bundles in Algebraic Geometry Daniele Faenzi

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Some Applications of Vector Bundles in Algebraic Geometry Daniele Faenzi Some applications of vector bundles in algebraic geometry Daniele Faenzi To cite this version: Daniele Faenzi. Some applications of vector bundles in algebraic geometry. Algebraic Geometry [math.AG]. Université de Pau et des Pays de l’Adour, 2013. tel-01062906v2 HAL Id: tel-01062906 https://tel.archives-ouvertes.fr/tel-01062906v2 Submitted on 15 May 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Habilitation à Diriger des Recherches Daniele FAENZI Some applications of vector bundles in algebraic geometry Mémoire de synthèse Exposé de recherche, travaux en cours ÉQUIPE D’ALGÈBRE ET GÉOMÉTRIE DE PAU LMA UMR CNRS 5142 UNIVERSITÉ DE PAU ET DES PAYS DE L’ADOUR Thématiques de recherche Mathématique Géométrie algébrique - Géométrie complexe - Singularités Composition du jury Enrique Artal Bartolo Jacky Cresson Olivier Debarre (rapporteur) Laurent Manivel Frank-Olaf Schreyer (rapporteur) Edoardo Sernesi (rapporteur) Jean Vallès HDR soutenue à Pau le 5 décembre 2013 To my family Contents Introduction. 1 Acknowledgements . 7 Chapter 0. Preliminaries and background. 1 I. Notation and conventions . 1 I.1. Polynomial rings . 1 I.2. Projective spaces and Grassmannians. 1 I.3. Varieties . 2 I.4. Divisors, polarized varieties . 3 II. Basic material on coherent sheaves . 3 II.1. Semistable sheaves and moduli spaces . 3 II.2. Cohomology and derived categories . 6 III. Projective and Grassmann bundles and cokernel sheaves . 7 III.1. Grassmann bundles and morphisms of vector bundles . 8 III.2. Grassmann bundles and degeneracy loci . 9 Chapter 1. Logarithmic vector fields along a divisor. 11 I. Logarithmic derivations and syzygies of the Jacobian ideal . 11 I.1. Logarithmic vector fields and differentials . 12 I.2. Dolgachev’s logarithmic forms and residue sequence. 14 I.3. Logarithmic vector fields and deformations . 15 I.4. Logarithmic vector fields and duality . 15 I.5. Logarithmic vector fields, Jacobian ring and primitive cohomology . 17 II. Free hypersurfaces . 17 II.1. Free hyperplane arrangements . 20 II.2. Free hypersurfaces with components of higher degree. 26 III. Polar map . 29 III.1. Polar map and logarithmic derivations . 29 III.2. Homaloidal polynomials . 30 IV. Torelli problem for hypersurfaces . 32 IV.1. Torelli theorems for hyperplane arrangement . 33 IV.2. Generic Torelli theorem for hypersurfaces . 37 V. Open questions . 42 V.1. Generalized Weyl arrangements. 42 V.2. Free divisors associated with fibrations . 44 Chapter 2. Cohen-Macaulay bundles . 47 I. ACM varieties and bundles . 47 iii I.1. ACM varieties. 47 I.2. ACM sheaves . 48 I.3. Ulrich sheaves . 49 II. CM type of varieties . 50 II.1. Varieties of finite CM type . 52 II.2. Some varieties of wild CM type . 54 III. Existence and classification problems for ACM bundles . 54 III.1. Hypersurfaces . 54 III.2. Fano threefolds. 58 III.3. Classification of rigid ACM bundles on Veronese varieties. 58 IV. A smooth projective surface of tame CM type . 62 IV.1. Segre-Veronese varieties. 63 IV.2. ACM line bundles . 64 IV.3. Computing resolutions of ACM bundles. 64 IV.4. Kronecker-Weierstrass canonical form for extension bundles . 69 V. Open questions . 74 V.1. ACM and Ulrich bundles . 74 V.2. Families of determinantal varieties . 74 V.3. Generalized Lax conjecture . 75 Chapter 3. Odd instantons on Fano threefolds . 77 I. Introduction to even and odd instantons. 77 I.1. Existence of instantons. 79 I.2. Instantons with small charge, Jacobians, periods . 81 I.3. Instantons on Fano threefolds with trivial Jacobian . 84 I.4. Summary of basic formulas for Fano threefolds . 86 II. Fano threefolds of genus 10 . 86 II.1. Basic features of Fano threefolds of genus 10 . 86 II.2. Rank-3 bundles and homological projective duality . 91 II.3. Hilbert schemes of curves of low degree . 96 II.4. Instanton bundles . .101 II.5. Lines and 3-Theta divisors of the Jacobian . 103 II.6. Bundles of rank 3 with canonical determinant . 110 III. Open questions . .112 III.1. Properties of the moduli space of instantons . 112 III.2. A conjecture of Kuznetsov . 113 III.3. A conjecture of Mukai . 113 III.4. Instantons and the non-commutative plane . 114 Bibliography . 115 Introduction Algebraic geometry studies algebraic varieties, which are the loci of points in space satisfying a set of polynomial equations in several variables. This is an an- cient and popular subject of mathematics, connected to many other areas such as algebraic topology, singularity theory, representation theory, combinatorics, commutative algebra, and perhaps even theoretical physics and computational complexity. Vector bundles can be though of as vector spaces varying continuously along a given variety. These geometric objects offer a valuable point of view on alge- braic geometry, and represent the technical core of this work. The first example of vector bundles that we will encounter are logarithmic n vector fields along a (reduced) divisor D in projective space P . This bundle, denoted by n log D , was originally introduced by Deligne P ( ) T − and Saito (cf. [96, 276]) to study the Hodge theory of the complement of D. It is actually not a bundle in general.
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