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Mathematics- Algebraic Geometry Problem
Let K denotes an algebraically closed field and let P^{1} be constructed as in Example 5.5(a) in Gathmanns notes, i.e. P^{1} is the gluing of X_{1} = A^{1} and X_{2} = A^{1} along the open subsets U12 = A^{1}\{0} ⊆ X_{1} and U_{21} = A^{1}\{0} ⊆ X_{2}, where U12 and U21 is identified by the isomorphism
U_{12} → U_{21},
t ι→ t^{-1}
We let i_{1} : X_{1} → P_{1} and i_{2 }: X_{2} → P^{1} denote the associated morphisms.
1) Show that the map
π : A2\{0} → P^{1}
is well defined and surjective.
For elements (a, b), (c, d) ∈ A^{2}\{0} we write (a, b) ∼ (c, d) when (a, b) and (c, d) are linearly dependent as elements in the K-vector space K^{2}.
2) Show that ∼ defines an equivalence relation on A^{2}\{0}.
3) Show that π induces a bijective map
π^{-} = (A^{2}\{0})/∼ → P^{1}
[(a, b)] ι→ π(a, b).
Attachment:- Assignment File.rar
The problem is related to the objectivity of given function defined in problem. The study also shows the well-defined function. Also we derived the property call equivalence relation for given.