On set-theoretic complete intersections for smooth curves in three-dimensional affine schemes

Abstract: We prove that every local complete intersection curve in $Spec(A)$, where $A$ is a commutative Noetherian ring of dimension three, is a set-theoretic complete intersection. An analogous result is established for local complete intersection surfaces when $A$ is a four-dimensional affine algebra over the algebraic closure of a finite field of $p$ elements. Furthermore, we show that any local complete intersection curve (respectively, surface) in $Spec(A)$, where $A$ has dimension three (respectively, four), having trivial conormal bundle is, in fact, a complete intersection.