1 1 Ju l 2 00 3 The Theory of Tight Closure from the Viewpoint of Vector Bundles

Contents Introduction 3 1. Foundations 13 1.1. A survey about the theory of tight closure 13 1.2. Solid closure and forcing algebras 23 1.3. Cohomological dimension 25 1.4. Vector bundles, locally free sheaves and projective bundles 28 2. Geometric interpretation of tight closure via bundles 30 2.1. Relation bundles 30 2.2. Affine-linear bundles arising from forcing algebras 32 2.3. Cohomology classes, torsors and their geometric realizations 33 2.4. Projective bundles arising from forcing data 37 2.5. The graded case: relation bundles on projective varieties 39 2.6. The forcing sequence on projective varieties 42 2.7. Ample and basepoint free forcing divisors 46 2.8. The two-dimensional situation 49 3. The two-dimensional parameter case 53 3.1. Ruled surfaces and forcing sections 53 3.2. Examples 55 3.3. Examples over the complex numbers C 57 3.4. Plus closure in positive characteristic 59 4. Slope criteria for affineness and non-affineness 61 4.1. Slope of bundles over a curve 61 4.2. Ampleness criteria for vector bundles over projective curves 62 4.3. Criteria for affineness 65 4.4. Criteria for P(G ′) − P(G) not to be affine 67 4.5. Starting an algorithm 70 5. Inclusion and exclusion bounds for tight closure 72 5.1. Inclusion bounds 72 5.2. Exclusion bounds for tight closure 74 5.3. Vanishing theorems for tight closure 77 5.4. Computing the tight closure of three elements 80 5.5. Slope bounds for indecomposable bundles 82 5.6. The degree of relations 85 5.7. Correlations and big forcing divisors 90 6. Tight closure and plus closure in cones over elliptic curves 95 6.1. Vector bundles over elliptic curves 95 6.2. A numerical criterion for subbundles to have affine complement 96 6.3. Numerical criteria for tight closure and plus closure for cones over elliptic curves 98 1 2 Bibliography 101 Introduction In this habilitation thesis we link together two mathematical subjects which are unrelated so far: the theory of tight closure on one hand, and the theory of vector bundles on the other. The aim is to translate algebraic problems in tight closure theory into problems about vector bundles and projective bundles in order to attack them with the help of the geometric tools then available. A vector bundle is a geometric object over a base space which is locally isomorphic to a standard bundle (like R n , C n , A n k , depending on the category). Vector bundles, and also their relatives, locally free sheaves, projective …