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) ∈ 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 ∈ 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.

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) ∈ 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 ∈ 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.

Let $k$ be an infinite field of arbitrary characteristic, $\left(A,M,K\right)$ a $k$-algebra of essentially finite type, with $K∕k$ separable and $P$ a local property. We say that $L{B}_k\left(P\right)$ holds if: For the generic $\alpha =\left({\alpha }_1,\dots ,{\alpha }_n\right)\in k^n\Rightarrow P\left(A{x}_{\alpha }A\right)\subseteq P\left(A\right)\cap V\left(x_{\alpha }\right)\cap U_P$ ($x_{\alpha }=\sum {\alpha }_i{x}_i,$ $⟨x_1,\dots ,x_n\to =M,$ $U_P$ non-empty open subset of $SpecA$ and $P\left(A\right)=\left\{P\in SpecA|A_p\text{is}P\right\}$). We show that: $L{B}_K\left(P\right)$ holds $\Rightarrow L{B}_K\left(GP\right)$ holds for the corresponding geometric property (in particular, for $P=$ regular, normal, reduced, $R_s,$ $L{B}_K\left(GP\right)$ holds). As an appliance we obtain a Bertini Theorem for hypersurface setions of a variety $X\subseteq P_k^n$ concerning the geometric properties.

Let $k$ be an infinite field of arbitrary characteristic, $\left(A,M,K\right)$ a $k$-algebra of essentially finite type, with $K∕k$ separable and $P$ a local property. We say that $L{B}_k\left(P\right)$ holds if: For the generic $\alpha =\left({\alpha }_1,\dots ,{\alpha }_n\right)\in k^n\Rightarrow P\left(A{x}_{\alpha }A\right)\subseteq P\left(A\right)\cap V\left(x_{\alpha }\right)\cap U_P$ ($x_{\alpha }=\sum {\alpha }_i{x}_i,$ $⟨x_1,\dots ,x_n\to =M,$ $U_P$ non-empty open subset of $SpecA$ and $P\left(A\right)=\left\{P\in SpecA|A_p\text{is}P\right\}$). We show that: $L{B}_K\left(P\right)$ holds $\Rightarrow L{B}_K\left(GP\right)$ holds for the corresponding geometric property (in particular, for $P=$ regular, normal, reduced, $R_s,$ $L{B}_K\left(GP\right)$ holds). As an appliance we obtain a Bertini Theorem for hypersurface setions of a variety $X\subseteq P_k^n$ concerning the geometric properties.

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