def Lec8Blowup.BlowupEquations (k : Type u_1) [Field k] (n : ℕ) (x y : Fin n → k) : Prop

The defining equations x_i y_j = x_j y_i for the blowup of affine space at the origin (Lec 8/9, Def 20).

def Lec8Blowup.BlowupAffineOrigin (k : Type u_1) [Field k] (n : ℕ) : Set ((Fin n → k) × Projectivization k (Fin n → k))

The blowup of 𝔸ⁿ at the origin, as the incidence subset of 𝔸ⁿ × ℙⁿ⁻¹ consisting of pairs (x, ℓ) with x ∈ ℓ (Lec 8/9, Def 20).

def Lec8Blowup.blowupProjection (k : Type u_1) [Field k] (n : ℕ) (p : ↑(BlowupAffineOrigin k n)) : Fin n → k

Projection from the blowup Bl₀(𝔸ⁿ) to 𝔸ⁿ (Lec 8/9, Def 20).

theorem Lec8Blowup.Projectivization.submodule_eq_span_rep {k : Type u_1} [Field k] {n : ℕ} (v : Projectivization k (Fin n → k)) : v.submodule = k ∙ v.rep

A point of projective space is the span of its chosen representative.

theorem Lec8Blowup.blowupEquations_iff_mem_span {k : Type u_1} [Field k] {n : ℕ} {x y : Fin n → k} (hy : y ≠ 0) : BlowupEquations k n x y ↔ x ∈ k ∙ y

For y ≠ 0, the blowup equations x_i y_j = x_j y_i hold iff x lies on the line through y (Lec 8/9, Def 20).

theorem Lec8Blowup.mem_blowupAffineOrigin_iff (k : Type u_2) [Field k] (n : ℕ) (p : (Fin n → k) × Projectivization k (Fin n → k)) : p ∈ BlowupAffineOrigin k n ↔ BlowupEquations k n p.1 p.2.rep

A pair (x, ℓ) lies on the blowup iff x satisfies the blowup equations with any chosen representative of ℓ.

theorem Lec8Blowup.proj_unique_of_nonzero {k : Type u_1} [Field k] {n : ℕ} {x : Fin n → k} (hx : x ≠ 0) (l : Projectivization k (Fin n → k)) (hmem : x ∈ l.submodule) : l = Projectivization.mk k x hx

A nonzero vector x determines the unique line in ℙⁿ⁻¹ containing it.

theorem Lec8Blowup.zero_mem_blowupAffineOrigin {k : Type u_1} [Field k] {n : ℕ} (l : Projectivization k (Fin n → k)) : (0, l) ∈ BlowupAffineOrigin k n

For any line ℓ, the point (0, ℓ) lies on the blowup (the exceptional fiber).

theorem Lec8Blowup.nonzero_mem_blowupAffineOrigin {k : Type u_1} [Field k] {n : ℕ} {x : Fin n → k} (hx : x ≠ 0) : (x, Projectivization.mk k x hx) ∈ BlowupAffineOrigin k n

For x ≠ 0, the pair (x, [x]) lies on the blowup.

noncomputable def Lec8Blowup.exceptionalFiberEquiv {k : Type u_1} [Field k] {n : ℕ} : { p : ↑(BlowupAffineOrigin k n) // blowupProjection k n p = 0 } ≃ Projectivization k (Fin n → k)

The fiber of the blowup projection over 0 is canonically the projective space ℙⁿ⁻¹ (exceptional divisor).

noncomputable def Lec8Blowup.blowupAwayEquiv {k : Type u_1} [Field k] {n : ℕ} : { p : ↑(BlowupAffineOrigin k n) // (↑p).1 ≠ 0 } ≃ { x : Fin n → k // x ≠ 0 }

Away from the origin, the blowup projection is a bijection between the open locus of the blowup and the punctured affine space.