class Formalization.Lec8CompleteVarieties.IsComplete {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) : Prop

A morphism f : X ⟶ Y is complete if it is separated and universally closed (Lec 8, Def 19).

• isSeparated : AlgebraicGeometry.IsSeparated f
• universallyClosed : AlgebraicGeometry.UniversallyClosed f

instance Formalization.Lec8CompleteVarieties.isComplete_of_isProper {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsProper f] : IsComplete f

A proper morphism is complete.

theorem Formalization.Lec8CompleteVarieties.lemma19_ii_image_complete {X Z S : AlgebraicGeometry.Scheme} (f : X ⟶ Z) (hX : X ⟶ S) (hZ : Z ⟶ S) [hXp : AlgebraicGeometry.IsProper hX] [AlgebraicGeometry.IsSeparated hZ] (hcomm : CategoryTheory.CategoryStruct.comp f hZ = hX) : AlgebraicGeometry.IsClosedImmersion (AlgebraicGeometry.Scheme.Hom.imageι f) ∧ AlgebraicGeometry.IsProper (AlgebraicGeometry.Scheme.Hom.toImage f)

Lec 8, Lem 19(ii): if X is proper over S and Z is separated over S, then the image of f : X ⟶ Z is a closed subscheme and X ↠ im(f) is proper.

def Formalization.Lec8CompleteVarieties.IsBirational (X Y : AlgebraicGeometry.Scheme) : Prop

Two schemes are birational if they share a common dense open subscheme (Lec 9 / Chow's lemma setting).

def Formalization.Lec8CompleteVarieties.IsProjective (X : AlgebraicGeometry.Scheme) : Prop

A scheme X is projective if it admits a closed immersion into some Proj 𝒜 for a graded ring 𝒜 with finite-type degree-zero piece (Lec 8/9, used for Chow's lemma).

theorem Formalization.Lec8CompleteVarieties.exists_projective_completion (U : AlgebraicGeometry.Scheme) [AlgebraicGeometry.IsAffine U] : ∃ (Y : AlgebraicGeometry.Scheme), IsProjective Y ∧ ∃ (ι : U ⟶ Y), AlgebraicGeometry.IsOpenImmersion ι ∧ Dense (Set.range ⇑ι)

An affine scheme U admits an open immersion with dense image into some projective scheme (input to Chow's lemma).

theorem Formalization.Lec8CompleteVarieties.closed_subscheme_projective {X Y : AlgebraicGeometry.Scheme} (i : X ⟶ Y) (hi : AlgebraicGeometry.IsClosedImmersion i) (hY : IsProjective Y) : IsProjective X

A closed subscheme of a projective scheme is projective.

theorem Formalization.Lec8CompleteVarieties.product_projective_schemes (Y₁ Y₂ : AlgebraicGeometry.Scheme) (h₁ : IsProjective Y₁) (h₂ : IsProjective Y₂) : ∃ (P : AlgebraicGeometry.Scheme), IsProjective P

The product of two projective schemes embeds into a projective scheme (via Segre); used in Chow's lemma.

theorem Formalization.Lec8CompleteVarieties.graph_closed_of_separated {U Y S : AlgebraicGeometry.Scheme} (Δ : U ⟶ Y) (fU : U ⟶ S) (fY : Y ⟶ S) [AlgebraicGeometry.IsSeparated fY] (hcomp : CategoryTheory.CategoryStruct.comp Δ fY = fU) : ∃ (Γ : AlgebraicGeometry.Scheme) (incl : Γ ⟶ U), AlgebraicGeometry.IsClosedImmersion incl ∧ AlgebraicGeometry.IsOpenImmersion incl ∧ Dense (Set.range ⇑incl)

For fY : Y ⟶ S separated, the graph of a morphism into Y from an S-scheme is closed, giving a dense locally closed subscheme of U (input to Chow's lemma).

theorem Formalization.Lec8CompleteVarieties.closure_birational {S X : AlgebraicGeometry.Scheme} (f : X ⟶ S) [AlgebraicGeometry.IsProper f] [AlgebraicGeometry.IsIntegral X] : ∃ (X_tilde : AlgebraicGeometry.Scheme), IsBirational X X_tilde

Given a proper morphism of an integral scheme, there exists a birational model X̃ (closure of the graph step in Chow's lemma).

theorem Formalization.Lec8CompleteVarieties.second_proj_closed_immersion {S X : AlgebraicGeometry.Scheme} (f : X ⟶ S) [AlgebraicGeometry.IsProper f] [AlgebraicGeometry.IsIntegral X] : ∃ (X_tilde : AlgebraicGeometry.Scheme) (Y : AlgebraicGeometry.Scheme) (_ : IsProjective Y) (g : X_tilde ⟶ Y), AlgebraicGeometry.IsClosedImmersion g ∧ IsBirational X X_tilde

In the Chow's lemma setup, the second projection from the graph closure to a projective scheme is a closed immersion.

theorem Formalization.Lec8CompleteVarieties.chows_lemma {S X : AlgebraicGeometry.Scheme} (f : X ⟶ S) [AlgebraicGeometry.IsProper f] [AlgebraicGeometry.IsIntegral X] : ∃ (X' : AlgebraicGeometry.Scheme) (_ : IsProjective X'), IsBirational X X'

Lec 9, Lem 20 (Chow's lemma): every proper integral scheme over S admits a projective birational model.