Tangent cones of monomial curves obtained by numerical duplication

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.