Reference no: EM132137778
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).
Attachment:- Assignment File.rar
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