On the Betti numbers of the tangent cones for Gorenstein Monomial Curves

TL;DR

This paper investigates the Betti numbers of tangent cones of Gorenstein monomial curves in affine 4-space, identifying specific Betti sequences for certain Cohen-Macaulay tangent cones.

Contribution

It characterizes the possible Betti sequences of tangent cones for non-complete intersection Gorenstein monomial curves with Cohen-Macaulay tangent cones.

Findings

Betti sequences are (1,5,5,1), (1,5,6,2), and (1,6,8,3).

Focus on tangent cones of Gorenstein monomial curves in affine 4-space.

Results apply to non-complete intersection cases with Cohen-Macaulay tangent cones.

Abstract

The aim of the article is to study the Betti numbers of the tangent cone of Gorenstein monomial curves in affine 4-space. If $C_S$ is a non-complete intersection Gorenstein monomial curve whose tangent cone is Cohen-Macaulay, we show that the possible Betti sequences are (1,5,5,1), (1,5,6,2) and (1,6,8,3).