Lectures on Algebraic Geometry II [electronic resource] : Basic Concepts, Coherent Cohomology, Curves and their Jacobians /
Harder, Günter.author.
In this second volume of "Lectures on Algebraic Geometry", the author starts with some foundational concepts in the theory of schemes and gives a somewhat casual introduction into commutative algebra. After that he proves the finiteness results for coherent cohomology and discusses important applications of these finiteness results. In the two last chapters, curves and their Jacobians are treated and some outlook into further directions of research is given. The first volume is not necessarily a prerequisite for the second volume if the reader accepts the concepts on sheaf cohomology. On the other hand, the concepts and results in the second volume have been historically inspired by the theory of Riemann surfaces. There is a deep connection between these two volumes, in spirit they form a unity. Basic concepts of the Theory of Schemes - Some Commutative Algebra - Projective Schemes - Curves and the Theorem of Riemann-Roch - The Picard functor for curves and Jacobians. Prof. Dr. Günter Harder, Department of Mathematics, University of Bonn, and Max-Planck-Institute for Mathematics, Bonn, Germany.
Description based upon print version of record.
Preface; Contents; Introduction; 6 Basic Concepts of the Theory of Schemes; 6.1 Affine Schemes; 6.1.1 Localization; 6.1.2 The Spectrum of a Ring; 6.1.3 The Zariski Topology on Spec(A); 6.1.4 The Structure Sheaf on Spec(A); 6.1.5 Quasicoherent Sheaves; 6.1.6 Schemes as Locally Ringed Spaces; Closed Subschemes; Sections; A remark; 6.2 Schemes; 6.2.1 The Definition of a Scheme; The gluing; Closed subschemes again; Annihilators, supports and intersections; 6.2.2 Functorial properties; Affine morphisms; Sections again; 6.2.3 Construction of Quasi-coherent Sheaves; Vector bundles
Vector Bundles Attached to Locally Free Modules6.2.4 Vector bundles and GLn-torsors.; 6.2.5 Schemes over a base scheme S.; Some notions of finiteness; Fibered products; Base change; 6.2.6 Points, T-valued Points and Geometric Points; Closed Points and Geometric Points on varieties; 6.2.7 Flat Morphisms; The Concept of Flatness; Representability of functors; 6.2.8 Theory of descend; Effectiveness for affine descend data; 6.2.9 Galois descend; A geometric interpretation; Descend for general schemes of finite type; 6.2.10 Forms of schemes; 6.2.11 An outlook to more general concepts
7.4.4 Formal Schemes and Infinitesimal Schemes7.5 Smooth Points; 7.5.1 Generic Smoothness; The singular locus; 7.5.2 Relative Differentials; 7.5.3 Examples; 7.5.4 Normal schemes and smoothness in codimension one; Regular local rings; 7.5.5 Vector fields, derivations and infinitesimal automorphisms; Automorphisms; 7.5.6 Group schemes; 7.5.7 The groups schemes Ga,Gm and μn; 7.5.8 Actions of group schemes; 8 Projective Schemes; 8.1 Geometric Constructions; 8.1.1 The Projective Space pnA; Homogenous coordinates; 8.1.2 Closed subschemes; 8.1.3 Projective Morphisms and Projective Schemes
Locally Free Sheaves on pnOpn (d) as Sheaf of Meromorphic Functions; The Relative Differentials and the Tangent Bundle of pnS; 8.1.4 Separated and Proper Morphisms; 8.1.5 The Valuative Criteria; The Valuative Criterion for the Projective Space; 8.1.6 The Construction Proj(R); A special case of a finiteness result; 8.1.7 Ample and Very Ample Sheaves; 8.2 Cohomology of Quasicoherent Sheaves; 8.2.1 Čech cohomology; 8.2.2 The Künneth-formulae; 8.2.3 The cohomology of the sheaves Opn (r); 8.3 Cohomology of Coherent Sheaves; 8.3.1 The coherence theorem for proper morphisms
Digression: Blowing up and contracting
Includes bibliographical references and index.
In this second volume of "Lectures on Algebraic Geometry", the author starts with some foundational concepts in the theory of schemes and gives a somewhat casual introduction into commutative algebra. After that he proves the finiteness results for coherent cohomology and discusses important applications of these finiteness results. In the two last chapters, curves and their Jacobians are treated and some outlook into further directions of research is given. The first volume is not necessarily a prerequisite for the second volume if the reader accepts the concepts on sheaf cohomology. On the other hand, the concepts and results in the second volume have been historically inspired by the theory of Riemann surfaces. There is a deep connection between these two volumes, in spirit they form a unity. Basic concepts of the Theory of Schemes - Some Commutative Algebra - Projective Schemes - Curves and the Theorem of Riemann-Roch - The Picard functor for curves and Jacobians. Prof. Dr. Günter Harder, Department of Mathematics, University of Bonn, and Max-Planck-Institute for Mathematics, Bonn, Germany.
Algebraic geometry.
Algebra.
Algebraic Geometry.
Algebra.