Helices and vector bundles : seminaire Rudakov / A.N. Rudakov [and others] ; translated by A.D. King, P. Kobak, and A. Maciocia.

Format:

Book

Author/Creator:

Rudakov, A. N., author.

Contributor:

King, A. D., translator.

Kobak, P, translator.

Maciocia, A., translator.

Series:

London Mathematical Society lecture note series ; 148.

London Mathematical Society lecture note series ; 148

Language:

English

Subjects (All):

Vector bundles.

Helices (Algebraic topology).

Physical Description:

1 online resource (143 pages) : digital, PDF file(s).

Other Title:

Helices & Vector Bundles

Place of Publication:

Cambridge : Cambridge University Press, 1990.

Language Note:

English

Summary:

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 explore its use in various contexts. As to be expected, the approach is concrete; the theory is considered for quadrics, ruled surfaces, K3 surfaces and P3(C).

Contents:

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.

2.1. Examples of Exceptional Bundles on G.

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and index.

ISBN:

1-139-88456-5

1-107-36653-4

1-107-37122-8

1-107-36162-1

1-107-36991-6

1-299-40429-4

1-107-36407-8

0-511-89248-9

0-511-72152-8