Tangent cones of monomial curves obtained by numerical duplication

Abstract

Given a numerical semigroup ring $R=k\llbracket S\rrbracket$, an ideal E of S and an odd element $b \in S$, the numerical duplication $S \bowtie ^b E$ is a numerical semigroup, whose associated ring $k\llbracket S \bowtie ^b E\rrbracket$ shares many properties with the Nagata’s idealization and the amalgamated duplication of R along the monomial ideal $I=(t^e \mid e\in E)$. In this paper we study the associated graded ring of the numerical duplication characterizing when it is Cohen–Macaulay, Gorenstein or complete intersection. We also study when it is a homogeneous numerical semigroup, a property that is related to the fact that a ring has the same Betti numbers of its associated graded ring. On the way we also characterize when $\mathrm{gr}_{\mathfrak {m}}(I)$ is Cohen–Macaulay and when $\mathrm{gr}_{\mathfrak {m}}(\omega _R)$ is a canonical module of $\mathrm{gr}_{\mathfrak {m}}(R)$ in terms of numerical semigroup’s properties, where $\omega _R$ is a canonical module of R.