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 Playing with plane curves, Plane conics, Cubics and the group law, The
category of affine varieties, Affine varieties and the Nullstellensatz,
Functions on varieties, Projective and biration algeometry, Tangent space and
non singularity and dimension.
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.
The contents of this book include: Introduction, Algebraic
structures, Subalgebras, direct products, homomorphisms, Equations and
solutions, Algebraic sets and radicals, Equationally Noetherian algebras,
Coordinate algebras, Main problems of universal algebraic geometry, Properties
of coordinate algebras, Coordinate algebras of irreducible algebraic sets, When
all algebraic sets are irreducible, The intervention of model theory,
Geometrical equivalence, Unifying theorems, Appearances of constants, Coordinate
algebras with constants, Equational domains, Types of equational compactness,
Advances of algebraic geometry and further reading.
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.
An
introduction to both the geometry and the arithmetic of abelian varieties. It
includes a discussion of the theorems of Honda and Tate concerning abelian
varieties over finite fields and the paper of Faltings in which he proves
Mordell's Conjecture. Warning: These notes are less polished than the others.
This note covers the
following topics: The Pre-cursor of Bezout’s Theorem: High School Algebra, The
Projective Plane and Homogenization, Bezout’s Theorem and Some Examples.
Author(s): Stephanie
Fitchett, Florida Atlantic University Honors College