Algebraic Geometry I Lecture Notes Roman Bezrukavnikov

Algebraic Geometry I Lecture Notes Roman Bezrukavnikov

Algebraic Geometry I Lecture Notes Roman Bezrukavnikov

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: 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.

Combinatorics
and Algebraic Geometry have classically enjoyed a fruitful interplay. The aim of
this series of lectures is to introduce recent development in this research
area. The topics involve classical algebraic varieties endowed with a rich
combinatorial structure, such as toric and tropical varieties.

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

This book explains the following topics: Etale
Morphisms, Etale Fundamental Group, The Local Ring for the Etale Topology,
Sheaves for the Etale Topology, Direct and Inverse Images of Sheaves, Cohomology:
Definition and the Basic Properties, Cohomology of Curves, Cohomological
Dimension, Purity; the Gysin Sequence, The Proper Base Change Theorem,
Cohomology Groups with Compact Support, The Smooth Base Change Theorem, The
Comparison Theorem, The Kunneth Formula, Proof of the Weil Conjectures, The Weil
Conjectures, The Geometry of Lefschetz Pencils and Cohomology of Lefschetz
Pencils.

This book explains
the following topics: Polarity, Conics, Plane cubics, Determinantal equations,
Theta characteristics, Plane Quartics, Planar Cremona transformations, Del Pezzo
surfaces, Cubic surfaces, Geometry of Lines.

These notes are an introduction to the theory of algebraic varieties. In
contrast to most such accounts they study abstract algebraic varieties, and not
just subvarieties of affine and projective space. This approach leads more
naturally into scheme theory.

This book explains the following topics: What is algebraic geometry,
Functions, morphisms, and varieties, Projective varieties, Dimension, Schemes,
Morphisms and locally ringed spaces, Schemes and prevarieties, Projective
schemes, First applications of scheme theory, Hilbert polynomials.