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Helices and vector bundles : seminaire Rudakov /
This volume is devoted to the use of helices as a method for studying exceptional vector bundles, an important and natural concept in algebraic geometry. The work arises out of a series of seminars organised in Moscow by A.N. Rudakov. The first article sets up the general machinery, and later ones e...
| Clasificación: | Libro Electrónico |
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| Otros Autores: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés Ruso |
| Publicado: |
Cambridge ; New York :
Cambridge University Press,
1990.
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| Colección: | London Mathematical Society lecture note series ;
148. |
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Title; Copyright; Contents; Exceptional Collections, Mutations and Helixes; 1. Helices.; 2. Notation.; 3. Exceptional Objects.; 4. Mutations.; References; Construction of Bundles on an Elliptic Curve; 1. Properties of indecomposable bundles over an elliptic curve.; 2. Mutations of indecomposable and simple bundles.; 3. The Theorems of Atiyah.; 4. The Coprimality Equation and the Classification of Simple Bundles.; References; Computing Invariants of Exceptional Bundles on a Quadric; Introduction.; 1. Basic Properties of Exceptional Bundles on P x P.
- 2. Constructing Short Exact Sequences from Exceptional Bundles. 3. Restricting Exceptional Bundles to Elliptic Curves.; References; Exceptional Bundles of Small Rank on P1 x P1; References; On the Functors Ext Applied to Exceptional Bundles on P2; 1. A Graphic Representation of Helices; 1.1. The Markov Tree.; 1.2. Graphs from Helices.; 1.3. Convenient Foundations.; 1.4. Definition.; 1.5. The Classification of Exceptional Bundles on P2.; 1.6. Proposition.; 2. Proof of the Theorem.; 2.1. Lemma.; 2.2. Lemma.; References; Homogeneous Bundles; References
- Exceptional Objects and Mutations in Derived CategoriesIntroduction.; Two further important remarks.; 1. Basic Concepts.; 1.1. The Functor RHom(*, *).; 1.2. Exceptional Objects.; 1.3. Complexes and their Associated Total Complexes.; 1.4. Orthogonal decomposition of Functors.; 2. Mutations of Pairs.; 2.1. Canonical Morphisms.; 2.2. Definition of Mutations.; 2.3. Properties of Mutations.; 2.4. Exceptional Pairs.; 2.5. Mutations of Sheaves.; 3. Exceptional Collections and Orthogonal Decompositions.; 3.1. Definition.; 3.2. Projections.; 3.3. Calculating Projections.
- 3.4. Connection with Helix Theory. References; Helixes, Representations of Quivers and Koszul Algebras; 1. Introduction, ; 2. Mutations.; 3. The Adjoint Functor.; 4. Helices.; 5. Quivers.; 6. Functors in the Category Dh(mod-A) related to an Exceptional Collection.; 7. Koszul Algebras.; 8. Self-consistent Algebras and Mutations of Strongly Exceptional Collections.; References; Exceptional Collections on Ruled Surfaces; 1. The Geometry of the Base.; 2. Cohomology of Invertible Sheaves.; 3. Exceptional Sheaves and Exceptional Bundles.; 4. Regular Exceptional Collections.
- 5. Helices on Ruled Surfaces.References; Exceptional Bundles on K3 Surfaces; 1. Exceptional Bundles and the Mukai Lattice.; 2. Exceptional Pairs and Canonical Maps.; 3. Exceptional Bundles on the Dual Plane.; 4. Exceptional Bundles on a Quartic.; References; Stability of Exceptional Bundles on Three Dimensional Projective Space; References; A Symmetric Helix on the Plücker Quadric; 1. General Notions.; 1.1. Definition.; 1.2. Definition.; 1.3. Definition.; 1.4. Mutations of elementary collections.; 1.5. Remark.; 1.6. Definition.; 1.7. Proposition.; 2. Exceptional Collections on G.