On the Rich Landscape of Complete Intersection Monomial Curves

TL;DR

This survey explores the properties, characterizations, and recent advances related to complete intersection monomial curves, emphasizing their algebraic and combinatorial aspects, and connecting them to numerical semigroups and knot theory.

Contribution

It bridges algebraic and combinatorial perspectives on complete intersection monomial curves and discusses recent interdisciplinary connections and deformation theory advances.

Findings

Deep algebraic and combinatorial characterizations of monomial curves

Recent links between numerical semigroups and Alexander polynomials of knots

Advances in deformation theory of these curves

Abstract

The aim of this survey is to explore complete intersection monomial curves from a contemporary perspective. The main goal is to help readers understand the intricate connections within the field and its potential applications. The properties of any monomial curve singularity will be first reviewed, highlighting the interaction between combinatorial and algebraic properties. Next, we will revisit the two main characterizations of complete intersection monomial curves. One is based on deep algebraic properties given by Herzog and Kunz in 1971, while the other is based on a combinatorial approach given by Delorme in 1976. Our aim is to bridge the gap between these perspectives present in the current literature. Then, we will focus on recent advances that show an intriguing connection between numerical semigroups and Alexander polynomials of knots. Finally, we will revisit the deformation theory of these curves.