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

TL;DR

This paper proves that in certain three- and four-dimensional affine schemes, local complete intersection curves and surfaces are set-theoretic complete intersections, especially when they have trivial conormal bundles.

Contribution

It establishes that all local complete intersection curves in three-dimensional schemes are set-theoretic complete intersections, extending to surfaces in four-dimensional schemes over algebraically closed finite fields.

Findings

Every local complete intersection curve in a 3-dimensional Noetherian ring is a set-theoretic complete intersection.

Local complete intersection surfaces in 4-dimensional affine algebras over algebraic closures of finite fields are also set-theoretic complete intersections.

Curves and surfaces with trivial conormal bundle in these schemes are actual complete intersections.

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.