Equations defining tangent cones of Gorenstein monomial curves

TL;DR

This paper investigates the algebraic structure of tangent cones of Gorenstein monomial curves in four-dimensional space, focusing on their minimal generators and Cohen-Macaulay properties at the origin.

Contribution

It provides new insights into the minimal generating sets of tangent cones for Gorenstein monomial curves, especially those with Cohen-Macaulay tangent cones.

Findings

Characterization of tangent cone generators for Gorenstein monomial curves

Conditions for Cohen-Macaulay tangent cones at the origin

Explicit equations defining tangent cones

Abstract

Let $C$ be a Gorenstein non complete intersection monomial curve in the 4-dimensional affine space. In this paper we study the minimal number of generators of the tangent cone of $C$. Special attention will be paid to the case where $C$ has Cohen-Macaulay tangent cone at the origin.