Explicit minimal resolution for certain monomial curves

TL;DR

This paper constructs an explicit minimal free resolution for the coordinate ring of certain monomial curves associated with arithmetic sequence-generated numerical semigroups, aiding the study of their smoothability.

Contribution

It provides an explicit, independent construction of the minimal free resolution for these monomial curves, including detailed descriptions of all maps involved.

Findings

Explicit minimal free resolution constructed

Resolution applicable to algebroid monomial curves from arithmetic sequences

Facilitates analysis of smoothability for these curves

Abstract

With a view to study problems of smoothability, we construct a minimal free resolution for the coordinate ring of an algebroid monomial curve associated to an $AS$ numerical semigroup (i.e. generated by an arithmetic sequence), obtained independently of the result of \cite{gss2} and equipped with the explicit description of all the involved maps.