# Varieties
# Quasi-Projective Varieties
# Quasi-Projective Varieties
A quasi-projective variety is a locally closed subset of , considered witht the Zariski topology induced from . A locally closed set of any topological space is a closed subset of an open subspace, that is, an intersection of an open set and a closed set.
Examples
The class of quasi-projective varieties includes all projective varieties, all affine varieties, and all Zariski open subsets of these.
# Morphisms
Let and be quasi-projective varieties. A morphism is a map such that for each , there exist homogeneous polynomials in variables such that the map
is well-defined at and agress with on some non-empty open set containing .
# Birational Geometry
# Projective Morphisms
A morphism of varieties is called a projective morphism if is a closed subvariety of a product variety and is the restricion of the projection onto the first coordinate.
Remark
Projective morphisms have the property that the preimage of any point is a projective variety. A projective morphism is a proper morphism in the Euclidean topology.
# Birational Morphisms
A morphism of quasi-projective varieties is called a birational morphism if its restriction to some dense open set is an isomorphism onto some dense open subset .
Example
The blow-up of a point in -dimensional affine space is an example of a birational morphism.
# Rational Maps
Let be a quasi-projective variety, and let and be dense open subsets of . Suppose we are given two morphisms and of quasi-projective varieties. We say that and are equivalent if the mappings and coincide on the intersection .
A rational map is an equivalence class of morphisms defined on dense open sets of as above.
Example
Projecting from a point in projective space is an example of a rational map. Let be a fixed hyperplane in and let be any point not on . The projection from onto the hyperplane is the rational map
The rational map is a well-defined morphism everywhere outside .
# Birational Equivalence
Let and be irreducible algebraic varieties. They are called birationally equivalent if there a re mutually inverse rational maps
Remark
This
Schemes →