Defines an equivalence relation
Course:- Mathematics
Reference No.:- EM132137778

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Mathematics- Algebraic Geometry Problem

Let K denotes an algebraically closed field and let P1 be constructed as in Example 5.5(a) in Gathmanns notes, i.e. P1 is the gluing of X1 = A1 and X2 = A1 along  the open subsets U12 = A1\{0} ⊆ X1 and U21 = A1\{0} ⊆ X2, where U12 and U21 is identified by the isomorphism

U12 → U21,

t ι→ t-1

We let i1 : X1 → P1 and i2 : X2 → P1 denote the associated morphisms.

1) Show that the map

π : A2\{0} → P1


is well defined and surjective.

For elements (a, b), (c, d) ∈ A2\{0} we write  (a, b) ∼ (c, d) when (a, b) and (c, d) are linearly dependent as elements in the K-vector space K2.

2) Show that ∼ defines an equivalence relation on A2\{0}.

3) Show that π induces a bijective map

π- = (A2\{0})/∼ → P1

[(a, b)] ι→ π(a, b).

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