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Tangent Cones of Bresinsky and Arslan Curves

Ranjana Mehta, Joydip Saha

TL;DR

This work computes Apery sets for Bresinsky and Arslan numerical semigroups and uses the resulting Apery tables to obtain explicit descriptions of the tangent cones at the origin and their Hilbert series. It establishes that both families yield Cohen–Macaulay tangent cones by exhibiting a free $F(I)$-module structure with precise shifts, and it provides explicit decompositions and Hilbert series formulas for these tangent cones. Building on prior Gröbner-basis analyses, the paper gives a detailed,-table-driven account of the tangent cone structure for these affine monomial curves, including concrete examples. The results enhance understanding of the invariants and algebraic properties of tangent cones in the Bresinsky and Arslan families, with implications for related numerical semigroup and monomial-curve theories.

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.

Tangent Cones of Bresinsky and Arslan Curves

TL;DR

This work computes Apery sets for Bresinsky and Arslan numerical semigroups and uses the resulting Apery tables to obtain explicit descriptions of the tangent cones at the origin and their Hilbert series. It establishes that both families yield Cohen–Macaulay tangent cones by exhibiting a free -module structure with precise shifts, and it provides explicit decompositions and Hilbert series formulas for these tangent cones. Building on prior Gröbner-basis analyses, the paper gives a detailed,-table-driven account of the tangent cone structure for these affine monomial curves, including concrete examples. The results enhance understanding of the invariants and algebraic properties of tangent cones in the Bresinsky and Arslan families, with implications for related numerical semigroup and monomial-curve theories.

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.

Paper Structure

This paper contains 4 sections, 17 theorems, 33 equations, 1 table.

Table of Contents

  1. introduction
  2. Preliminaries and Notations
  3. Tangent cone of Bresinsky Curves
  4. Tangent cone of Arslan curves

Key Result

Theorem 2.1

(Cortadellas, Zarzuela.) With the above notations, where $G_{\mathfrak{m}_{\Gamma}}$ is the tangent cone of $\Gamma$ and $F_{\Gamma}=F((t^{a_{1}}))$ is the fiber cone.

Theorems & Definitions (21)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Example 3.5
  • Corollary 3.6
  • Corollary 3.7
  • Corollary 3.8
  • ...and 11 more