On the Apery sets of monomial curves

TL;DR

This paper investigates the properties of Apéry sets of numerical semigroups associated with monomial curves, providing new characterizations, proofs of conjectures, and insights into Hilbert functions for various embedding dimensions.

Contribution

It offers new characterizations of Apéry tables for monomial curves, proves conjectures by Sapko and Shen, and establishes non-decreasing Hilbert functions in specific cases.

Findings

Shape characterization of Apéry tables for embedding dimension 3

Proof of non-decreasing Hilbert functions for certain monomial curves

Validation of conjectures by Sapko and Shen

Abstract

In this paper, we use the Ap\'ery table of the numerical semigroup associated to an affine monomial curve in order to characterize arithmetic properties and invariants of its tangent cone. In particular, we precise the shape of the Ap\'ery table of a numerical semigroup of embedding dimension 3, when the tangent cone of its monomial curve is Buchsbaum or 2-Buchsbaum, and give new proofs for two conjectures raised by V. Sapko (Commun. Algebra {29}:4759-4773, 2001) and Y. H. Shen (Commun. Algebra {39}:1922-1940, 2001). We also provide a new simple proof in the case of monomial curves for Sally's conjecture (Numbers of Generators of Ideals in Local Rings, 1978) that the Hilbert function of a one-dimensional Cohen-Macaulay ring with embedding dimension three is non-decreasing. Finally, we obtain that monomial curves of embedding dimension 4 whose tangent cones are Buchsbaum, and also monomial curves of any embedding dimensions whose numerical semigroups are balanced, have non-decreasing Hilbert functions. Numerous examples are provided to illustrate the results, most of them computed by using the NumericalSgps package of GAP (Delgado et al., NumericalSgps-a GAP package, 2006).