Chapter 2 Flashcards
co-ordinate
ring
K[V ] = K[x1, x2, . . . , xn]/I(V)
morphism ϕ : V → W
a function of the form ϕ(P) = (ϕ1(P), ϕ2(P), . . . , ϕn(P))
with ϕi ∈ K[V ] for each i, 1 ≤ i ≤ n.
The morphism ϕ : V → W is called an isomorphism
m if and only
if it has an inverse morphism, i. e., if there exists a morphism ψ : W → V
such that ψ ◦ ϕ = idV and ϕ ◦ ψ = idW
1 Let V ⊆ A^m(K) and W ⊆ A^n(K) be affine algebraic varieties.
Let y1, y2, . . . , yn be co-ordinates on A^n. For a morphism ϕ : V → W, define ϕ∗
: K[W] → K[V ], ϕ∗ (f) = f ◦ ϕ.∗ : ϕ → ϕ∗
a bijection between
morphisms of affine algebraic varieties ϕ : V → W and ring homomorphisms
α : K[W] → K[V ] preserving K.
ϕ : V → W is an isomorphism of affine algebraic varieties if
and only if
ϕ∗ : K[W] → K[V ] is an isomorphism of rings
function field of V
the set of fractions f/g f, g ∈ K[V ], g doesn’t = 0
If for some point P ∈ V , f(P) doesn’t= 0 and g(P) = 0
then
ϕ is not defined at P
rational map ϕ : V → W
is a function defined
on a non-empty subset of V given by ϕ(P) = (ϕ1(P), ϕ2(P), . . . , ϕn(P)) with ϕi ∈ K(V ) for each i, 1 ≤ i ≤ n
A rational map ϕ : V → W is called dominant
if and only if its
image is not contained in any proper subvariety of W.
A rational map ϕ : V→ W is called a birational equivalence
if and only if there exists a rational map ψ : W → V such that ψ ◦ ϕ = id_v
and ϕ◦ψ = id_w