Let $X$ be a smooth projective variety (over an algebraically closed field; it could be the field of complex numbers); $Z$ is its hyperplane section. When there exists an etale $U/X$ such that:


1. $Z$ can be lifted to $U$.
2. There exists a morphism from $U$ to a curve $C$ such that $Z$ is the preimage of some point of $C$.



For a hyperplane section Z of X, when there exists its etale X-neighbourhood such that Z is a fibre of its morphism to a curve
If $X$ is a curve, then just take linear projection from a disjoint codimension 2 linear subspace in the spanning hyperplane of $Z$. If the dimension of $X$ is $2$ or more, then there never exists such an étale neighborhood and morphism. For $Z$ of dimension $n-1 \geq 1$, for the normal sheaf $\mathcal{N}\_{Z/X}$, the top intersection number $c\_1(\mathcal{N}\_{Z/X} )^{n-1}$ equals the degree of $X$, which is positive. Since $\mathcal{N}\_{Z/X}$ equals $\mathcal{N}\_{Z/U}$, also $c\_1(\mathcal{N}\_{Z/U})^{n-1}$ is positive. However, for a fiber of a morphism, $\mathcal{N}\_{Z/U}$ is isomorphic to $\mathcal{O}\_Z$ so that $c\_1(\mathcal{N}\_{Z/U})$ equals $0$.


