This course provides an introduction to the language of schemes,
properties of morphisms, and sheaf cohomology. Covered topics are: Basics of
category theory, Sheaves, Abelian sheaves, Schemes, Morphisms of schemes,
Sheaves of modules, More properties of morphisms, Projective morphisms,
Projective morphisms, Flat morphisms and descent, Differentials Divisors,
Divisors on curves, Homological algebra, Sheaf cohomology, Cohomology of
quasicoherent sheaves, Cohomology of projective spaces, Hilbert polynomials,
GAGA, Serre duality for projective space, Dualizing sheaves and RiemannRoch,
CohenMacaulay schemes and Serre duality, Higher RiemannRoch and Etale
cohomology.
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.
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
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.
The material
presented here consists of a more or less self contained advanced course in
complex algebraic geometry presupposing only some familiarity with the theory of
algebraic curves or Riemann surfaces. But the goal, is to understand the
Enriques classification of surfaces from the point of view of Mori theory.