Tangent Cones of Bresinsky and Arslan Curves

TL;DR

This paper investigates the tangent cones of Bresinsky and Arslan curves using Apery tables, computes their Hilbert series, and proves their Cohen-Macaulay property.

Contribution

It introduces a method to analyze tangent cones of these curves via Apery tables and establishes their Cohen-Macaulayness.

Findings

Tangent cones are Cohen-Macaulay for both curves.

Hilbert series of the tangent cones are explicitly calculated.

Apery tables effectively describe the tangent cones.

Abstract

In this paper, we study the Apery tables for the numerical semigroups given by Bresinsky and Arslan. Using the Apery tables we write the tangent cones of the Bresinsky and Arsalan curves at the origin. Further, we calculate Hilbert series of the tangent cone of the Bresinsky and Arslan curves. We prove that both classes of the curve have Cohen- Macaulay tangent cone.