Abstract

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 LB_k(P) holds if: For the generic α = (α₁,…,α_n) in kⁿ ⇒ P(Ax_αA) ⊆ P(A) ∩ V (x_α) ∩ U_P (x_α = ∑ α_ix_i, ⟨x₁,…,x_n →= M, U_P non-empty open subset of SpecA and P(A) = {P in SpecA|A_p is P}). We show that: LB_K(P) holds ⇒ LB_K(GP) holds for the corresponding geometric property (in particular, for P = regular, normal, reduced, R_s, LB_K(GP) holds). As an appliance we obtain a Bertini Theorem for hypersurgace setions of a variety X ⊆ P_kⁿ concerning the geometric properties.

Authors