Computational Algebraic Geometry by Wolfram Decker

Computational Algebraic Geometry by Wolfram Decker

Computational Algebraic Geometry by Wolfram Decker

This
PDF book covers the following topics related to Algebraic Geometry : General
Remarks on Computer Algebra Systems, The Geometry–Algebra Dictionary, Affine
Algebraic Geometry, Ideals in Polynomial Rings, Affine Algebraic Sets, Hilbert’s
Nullstellensatz, Irreducible Algebraic Sets, Removing Algebraic Sets, Polynomial
Maps, The Geometry of Elimination, Noether Normalization and Dimension, Local
Studies, Projective Algebraic Geometry, The Projective Space, Projective
Algebraic Sets, Affine Charts and the Projective Closure, The Hilbert
Polynomial, Computing, Standard Bases and Singular, Applications, Ideal
Membership, Elimination, Radical Membership, Ideal Intersections, Ideal
Quotients, Kernel of a Ring Map, Integrality Criterion, Noether Normalization,
Subalgebra Membership, Homogenization, Dimension and the Hilbert Function,
Primary Decomposition and Radicals, Buchberger’s Algorithm and Field Extensions,
Sudoku, A Problem in Group Theory Solved by Computer Algebra, Finite Groups and
Thompson’s Theorem, Characterization of Finite Solvable Groups.

This note
covers introduction, Affine and projective space, Algebraic sets, Examples of
algebraic sets, The ideal of an algebraic set and the Hilbert Nullstellensatz,
The projective closure of an affine algebraic set, Irreducible components,
Dimension, Regular and rational functions, Regular and rational maps, Products
of quasi projective varieties, The dimension of an intersection, Complete
varieties and the tangent space.

This note
covers notation, What is algebraic geometry, Affine algebraic varieties,
Projective algebraic varieties, Sheaves, ringed spaces and affine algebraic
varieties, Algebraic varieties, Morphisms, Products, Dimension, The fibres of a
morphism, Sheaves of modules, Hilbert polynomials and bezouts theorem, Schemes,
Products of preschemes, Relative differentials, Cartier divisors, Rational
equivalence and the chow group, Proper push forward and flat pull back, Chern
classes of line bundles and chern classes of vector bundles.

The
contents of this book include: Course Introduction, Zariski topology, Affine
Varieties, Projective Varieties, Noether Normalization, Grassmannians, Finite
and Affine Morphisms, More on Finite Morphisms and Irreducible Varieties,
Function Field, Dominant Maps, Product of Varieties, Separateness, Sheaf
Functors and Quasi-coherent Sheaves, Quasi-coherent and Coherent Sheaves,
Invertible Sheaves, (Quasi)coherent sheaves on Projective Spaces, Divisors and
the Picard Group, Bezout’s Theorem, Abel-Jacobi Map, Elliptic Curves,
KSmoothness, Canonical Bundles, the Adjunction Formulaahler Differentials,
Cotangent Bundles of Grassmannians, Bertini’s Theorem, Coherent Sheves on
Curves, Derived Functors, Existence of Sheaf Cohomology, Birkhoff-Grothendieck,
Riemann-Roch, Serre Duality, Proof of Serre Duality.

This note covers the following
topics: Functors, Isomorphic and equivalent categories, Representable functors,
Some constructions in the light of representable functors, Schemes: Definition
and basic properties, Properties of morphisms of schemes, general techniques and
constructions.

This book is intended to give a
serious and reasonably complete introduction to algebraic geometry, not just for
experts in the field. Topics covered includes: Sheaves, Schemes, Morphisms of
schemes, Useful classes of morphisms of schemes, Closed embeddings and related
notions, Fibered products of schemes, and base change, Geometric properties:
Dimension and smoothness, Quasicoherent sheaves, Quasicoherent sheaves on
projective A-schemes, Differentials,Derived functors, Power series and the
Theorem on Formal Functions, Proof of Serre duality.

This book
explains the following topics: Systems of algebraic equations, Affine algebraic
sets, Morphisms of affine algebraic varieties, Irreducible algebraic sets and
rational functions, Projective algebraic varieties, Morphisms of projective
algebraic varieties, Quasi-projective algebraic sets, The image of a projective
algebraic set, Finite regular maps, Dimension, Lines on hypersurfaces, Tangent
space, Local parameters, Projective embeddings and Riemann-Roch Theorem.