Minimal Free Resolutions of the Tangent Cones of Gorenstein Monomial Curves

TL;DR

This paper explicitly determines the minimal free resolutions and Betti sequences of tangent cones of certain Gorenstein monomial curves in affine 4-space, providing new insights into their algebraic structure.

Contribution

It presents the explicit minimal free resolution for non-complete intersection Gorenstein monomial curves with tangent cones having five generators, and classifies their Betti sequences.

Findings

Betti sequences are either (1,5,6,2) or (1,5,5,1)

Hilbert functions of tangent cones are computed

Resolutions are explicitly constructed for specific Gorenstein curves

Abstract

We study the minimal free resolution of the tangent cone of Gorenstein monomial curves in affine 4-space. We give the explicit minimal free resolution of the tangent cone of non-complete intersection Gorenstein monomial curve whose tangent cone has five minimal generators and show that the possible Betti sequences are $(1,5,6,2)$ and $(1,5,5,1)$. Also, we compute the Hilbert function of the tangent cone of these families as a result.