Equations defining recursive extensions as set theoretic complete intersections

TL;DR

This paper presents an inductive method to construct infinitely many set-theoretic complete intersection monomial curves in projective space, providing explicit equations and illustrating with various examples.

Contribution

It introduces a constructive, inductive approach to generate monomial curves as set-theoretic complete intersections with explicit defining equations.

Findings

Infinite families of monomial curves are constructed as set-theoretic complete intersections.

Explicit binomial and polynomial equations are provided for each constructed curve.

The method is effective and illustrated with multiple examples.

Abstract

Based on the fact that projective monomial curves in the plane are complete intersections, we give an effective inductive method for creating infinitely many monomial curves in the projective $n$-space that are set theoretic complete intersections. We illustrate our main result by giving different infinite families of examples. Our proof is constructive and provides one binomial and $(n-2)$ polynomial explicit equations for the hypersurfaces cutting out the curve in question.