def Lec22.veroneseMonomialCount (n d : ℕ) : ℕ

The number of monomials of degree d in n + 1 variables, equal to choose (n + d) d.

def Lec22.veroneseTargetDim (n d : ℕ) : ℕ

The dimension N of the projective target of the degree-d Veronese embedding of ℙⁿ, equal to choose (n + d) d - 1.

def Lec22.IsSmooth_Hypersurface (k : Type u_1) [CommRing k] (n : ℕ) (f : MvPolynomial (Fin (n + 1)) k) : Prop

Smoothness of the projective hypersurface V(f) ⊆ ℙⁿ: at every nonzero point where f vanishes, some partial derivative of f is nonzero (Jacobian criterion).

structure Lec22.SmoothProjVariety (k : Type u_1) [CommRing k] (n : ℕ) : Type u_1

Combinatorial proxy for a smooth projective subvariety of ℙⁿ: a set of defining polynomials, the nonzero points on the variety, and the smoothness/dimension data.

• definingPolys : Set (MvPolynomial (Fin (n + 1)) k)
• points : Set (Fin (n + 1) → k)
• points_nonzero (x : Fin (n + 1) → k) : x ∈ self.points → ∃ (i : Fin (n + 1)), x i ≠ 0
• points_vanish (x : Fin (n + 1) → k) : x ∈ self.points → ∀ f ∈ self.definingPolys, (MvPolynomial.eval x) f = 0
• dim : ℕ
• smooth (x : Fin (n + 1) → k) : x ∈ self.points → ∀ g ∈ self.definingPolys, ∃ (i : Fin (n + 1)), (MvPolynomial.eval x) ((MvPolynomial.pderiv i) g) ≠ 0

def Lec22.IsSmooth_Intersection (k : Type u_1) [CommRing k] (n : ℕ) (f : MvPolynomial (Fin (n + 1)) k) (X : SmoothProjVariety k n) : Prop

The hypersurface V(f) cuts X transversally: at every point of X where f vanishes, some partial derivative of f is nonzero.

noncomputable def Lec22.zariskiTopologyHypersurfaces (k : Type u_1) [Field k] (n d : ℕ) : TopologicalSpace (MvPolynomial (Fin (n + 1)) k)

The Zariski topology on the parameter space of degree-d hypersurfaces in ℙⁿ, used to formulate "generic" properties.

def Lec22.IsGenericProperty (k : Type u_1) [Field k] (n d : ℕ) (P : MvPolynomial (Fin (n + 1)) k → Prop) : Prop

A property P of degree-d hypersurfaces holds generically if there is a nonempty Zariski-open (and dense) subset of the parameter space on which P holds.

theorem Lec22.bertini_hyperplane_smooth (k : Type u_1) [Field k] [IsAlgClosed k] (N : ℕ) (hN : 0 < N) (Y : SmoothProjVariety k N) : IsGenericProperty k N 1 fun (ℓ : MvPolynomial (Fin (N + 1)) k) => IsSmooth_Intersection k N ℓ Y

Bertini for hyperplane sections: For a smooth projective variety Y ⊆ ℙᴺ over an algebraically closed field, the generic hyperplane section is smooth.

noncomputable def Lec22.veronese_preserves_smoothness (k : Type u_1) [Field k] (n d : ℕ) (hd : 0 < d) (X : SmoothProjVariety k n) : SmoothProjVariety k (veroneseTargetDim n d)

The degree-d Veronese embedding ℙⁿ ↪ ℙᴺ carries a smooth projective variety to a smooth projective variety, allowing degree-d hypersurfaces in ℙⁿ to be related to hyperplane sections in ℙᴺ.

theorem Lec22.veronese_transfers_genericity (k : Type u_1) [Field k] [IsAlgClosed k] (n d : ℕ) (hn : 0 < n) (hd : 0 < d) (X : SmoothProjVariety k n) (Y : SmoothProjVariety k (veroneseTargetDim n d)) (hY : Y = veronese_preserves_smoothness k n d hd X) (hBertini : IsGenericProperty k (veroneseTargetDim n d) 1 fun (ℓ : MvPolynomial (Fin (veroneseTargetDim n d + 1)) k) => IsSmooth_Intersection k (veroneseTargetDim n d) ℓ Y) : IsGenericProperty k n d fun (f : MvPolynomial (Fin (n + 1)) k) => IsSmooth_Intersection k n f X

Genericity of smooth hyperplane sections of the Veronese image transfers back to genericity of smooth degree-d hypersurfaces of the original variety.

def Lec22.smoothProjSpace (k : Type u_1) [CommRing k] (n : ℕ) : SmoothProjVariety k n

The projective space ℙⁿ itself, viewed as the trivial smooth projective variety with no defining equations.

theorem Lec22.veroneseTargetDim_pos (n d : ℕ) (hn : 0 < n) (hd : 0 < d) : 0 < veroneseTargetDim n d

For positive n and d, the Veronese target dimension is strictly positive.

theorem Lec22.corollary28_generic_smooth_hypersurface (k : Type u_1) [Field k] [IsAlgClosed k] (n d : ℕ) (hn : 0 < n) (hd : 0 < d) : IsGenericProperty k n d (IsSmooth_Hypersurface k n)

Corollary 28 (generic smooth hypersurface): Over an algebraically closed field, a generic degree-d hypersurface in ℙⁿ is smooth.

theorem Lec22.corollary28_generic_smooth_intersection (k : Type u_1) [Field k] [IsAlgClosed k] (n d : ℕ) (hn : 0 < n) (hd : 0 < d) (X : SmoothProjVariety k n) : IsGenericProperty k n d fun (f : MvPolynomial (Fin (n + 1)) k) => IsSmooth_Intersection k n f X

Corollary 28 (generic smooth intersection): Over an algebraically closed field, for any smooth projective variety X ⊆ ℙⁿ, a generic degree-d hypersurface cuts X smoothly.