Chapter 2 Flashcards

1
Q

co-ordinate

ring

A

K[V ] = K[x1, x2, . . . , xn]/I(V)

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2
Q

morphism ϕ : V → W

A

a function of the form ϕ(P) = (ϕ1(P), ϕ2(P), . . . , ϕn(P))

with ϕi ∈ K[V ] for each i, 1 ≤ i ≤ n.

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3
Q

The morphism ϕ : V → W is called an isomorphism

A

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

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4
Q

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

a bijection between
morphisms of affine algebraic varieties ϕ : V → W and ring homomorphisms
α : K[W] → K[V ] preserving K.

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5
Q

ϕ : V → W is an isomorphism of affine algebraic varieties if

and only if

A

ϕ∗ : K[W] → K[V ] is an isomorphism of rings

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6
Q

function field of V

A

the set of fractions f/g f, g ∈ K[V ], g doesn’t = 0

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7
Q

If for some point P ∈ V , f(P) doesn’t= 0 and g(P) = 0

A

then

ϕ is not defined at P

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8
Q

rational map ϕ : V → W

A

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

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9
Q

A rational map ϕ : V → W is called dominant

A

if and only if its

image is not contained in any proper subvariety of W.

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10
Q

A rational map ϕ : V→ W is called a birational equivalence

A

if and only if there exists a rational map ψ : W → V such that ψ ◦ ϕ = id_v
and ϕ◦ψ = id_w

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