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.