Application of Global Bertini Theorems

Let k be an infinite field of arbitrary characteristic, (A, M, K) a k-algebra of essentially finite type, with K/k separable and P a local property. We say that LBk(P) holds if: for the generic α = (₁,..., an) ∈ kn ⇒ P(A/xαA) ⊆ P(A) ∩ V(xα) ∩ Up (xα = Σ αixi (x₁,..., xn) = M, Up non empty open subset of SpecA and P (A) = {P ∈ SpecA|ap is P}). We show that: LBk(P) holds ⇒ LBk (GP) holds for the corresponding geometric property (in particular for P = regular, normal, reduced, Rs, LBk(GP) holds). As an application we obtain a Bertini Theorem for hypersurface section of a variety X ⊆ $P_k^n$ concerning the geometric properties.