The blow-up of a coordinate subspace in Hauser's examples #
[Hau14, Example 4.34]: the blow-up of 𝔸²_k at the origin, I = (x, y), with its two charts
k[x, t₂] (t₂ = y/x) and k[t₁, y] (t₁ = x/y); and [Hau14, Example 4.47]: the blow-up of
𝔸³_k along the z-axis V(x, y), with its two chart transition maps. The examples
instantiate the model blow-up of Hironaka/Scheme/BlowUp/CoordinateSubspace/ (the blow-up of 𝔸ⁿ_k
along the coordinate subspace x_0 = ⋯ = x_{r−1} = 0: its charts, their transitions, smoothness,
and the exceptional divisor) at n = 2 = r and at n = 3, r = 2, over a field k, the
variables being x = x_0, y = x_1 (and z = x_2). The clause F ≅ ℙ¹_k of Example 4.34 is
instantiated from exists_iso_exceptional_proj at n = r = 2.
Hauser's Example 4.34: 𝔸² blown up at the origin (n = 2 = r) #
[Hau14, Example 4.34]: the exceptional curve F of the blow-up of 𝔸²_k at the origin is
ℙ¹_k. In general F ≅ ℙ¹_L = Proj 𝒪(L)[u₀, u₁] over the center L = Spec 𝒪(L),
𝒪(L) = k[x, y]/(x, y); here L is the origin, 𝒪(L) ≅ k (the quotient by all the variables is
the polynomial ring in no variables), so ℙ¹_L = ℙ¹_k.