opaque IsSmoothSubvariety (k : Type u_1) [Field k] (V : Type u_2) [AddCommGroup V] [Module k V] : Set (Projectivization k V) → Prop

Opaque predicate asserting that a subset of ℙ(V) is a smooth (closed) subvariety.

noncomputable def projectiveHyperplane (k : Type u_1) [Field k] (V : Type u_2) [AddCommGroup V] [Module k V] : Projectivization k (Module.Dual k V) → Set (Projectivization k V)

The hyperplane in ℙ(V) cut out by a point H ∈ ℙ(V*), namely the zero locus of a representative linear form.

opaque IsNonemptyZariskiOpen (k : Type u_1) [Field k] (V : Type u_2) [AddCommGroup V] [Module k V] : Set (Projectivization k (Module.Dual k V)) → Prop

Opaque predicate asserting that a subset of the dual projective space ℙ(V*) is a nonempty Zariski-open subset.

opaque IsProperZariskiClosed (k : Type u_1) [Field k] (V : Type u_2) [AddCommGroup V] [Module k V] : Set (Projectivization k (Module.Dual k V)) → Prop

Opaque predicate asserting that a subset of the dual projective space ℙ(V*) is a proper Zariski-closed subset (i.e., closed and not the whole space).

theorem bertini_bad_locus_in_proper_closed (k : Type u_1) [Field k] (V : Type u_2) [AddCommGroup V] [Module k V] [FiniteDimensional k V] (X : Set (Projectivization k V)) (hX : IsSmoothSubvariety k V X) : ∃ (W : Set (Projectivization k (Module.Dual k V))), IsProperZariskiClosed k V W ∧ {H : Projectivization k (Module.Dual k V) | ¬IsSmoothSubvariety k V (X ∩ projectiveHyperplane k V H)} ⊆ W

Bertini "bad locus" lemma: For a smooth projective subvariety X, the set of hyperplanes H for which X ∩ H is not smooth is contained in a proper closed subset of the dual projective space.

theorem proper_closed_complement_nonempty_open (k : Type u_1) [Field k] (V : Type u_2) [AddCommGroup V] [Module k V] [FiniteDimensional k V] (W : Set (Projectivization k (Module.Dual k V))) (hW : IsProperZariskiClosed k V W) : IsNonemptyZariskiOpen k V Wᶜ

The complement of a proper Zariski-closed subset of ℙ(V*) is a nonempty Zariski open.

theorem bertini_generic_hyperplane_smooth_proper (k : Type u_1) [Field k] (V : Type u_2) [AddCommGroup V] [Module k V] [FiniteDimensional k V] (X : Set (Projectivization k V)) (hX : IsSmoothSubvariety k V X) : ∃ (U : Set (Projectivization k (Module.Dual k V))), IsNonemptyZariskiOpen k V U ∧ ∀ H ∈ U, IsSmoothSubvariety k V (X ∩ projectiveHyperplane k V H)

Bertini for proper hyperplane sections (Theorem 22.1): For a smooth projective subvariety X ⊆ ℙ(V), there is a nonempty Zariski-open set of hyperplanes H ∈ ℙ(V*) such that X ∩ H is smooth.