On the defining equations of the tangent cone of a numerical semigroup ring (Q406344)

Let \(H\) be a numerical semigroup generated by \(a_1< a_2 < \dots < a_r\) and, for any non negative integer \(k\), let \(H_{k}\) be the semigroup generated by the \(k\)-shifted sequence \(a_1 +k < a_2+k < \dots < a_r+k\).NEWLINENEWLINEEvery numerical semigroup is finitely generated and it admits a uniquely minimal system of generators with cardinality \(\mu(H)\).NEWLINENEWLINEThe difference between the largest and the smallest element in such minimal system of generators is called width of the numerical semigroup \(H\), \(\mathrm{wd}(H)\).NEWLINENEWLINEFor any field \(K\), one can define the algebra homomorphism NEWLINE\[NEWLINE\varphi: K [x_1, x_2, \dots, x_r] \rightarrow K[t],NEWLINE\]NEWLINE where \(x_i\) is sent to \(t^{a_i}\).NEWLINENEWLINEThe image of such homomorphism is called the semigroup ring \(K[H]\) associated to \(H\). In other words, \(K[H]\) is the \(K\)-subalgebra of \(K[t]\) generated by \(t^{a_1}, t^{a_2}, \dots, t^{a_r}\). If \(H\) is minimally generated by \(a_1< a_2 < \dots < a_r\), then the kernel of \(\varphi\) only depends on \(H\) and for this reason we denoted it by \(I_H\).NEWLINENEWLINEThe associated graded ring of \(K[H]\) with respect to the maximal ideal \(m=(t^{a_1}, t^{a_2}, \dots, t^{a_r})\) is denoted by \(\mathrm{gr}_m K[H]\). In geometric terms, \(\mathrm{gr}_m K[H]\) is the coordinate ring of the tangent cone of \(K[H]\) and one can also see that \(\mathrm{gr}_m K[H]=K [x_1, x_2, \dots, x_r]/I^*_H\), where \(I^*_H\) is the ideal of initial forms of polynomials in \(I_H\).NEWLINENEWLINEWe denote by \(\mu(I_H)\), and respectively \(\mu(I^*_H)\), the cardinality of a minimal system of generators for \(I_H\), respectively \(I^*_H\).NEWLINENEWLINEFor general \(r\), \(\mu(I_H)\) may be arbitrarily large, but \textit{T. Vu} [J. Algebra 418, 66--90 (2014; Zbl 1317.13037)] has showed that there exists an upper bound for \(\mu(I_{H_k})\) and this is independent of \(k\).NEWLINENEWLINEOne of the main results of the paper is the following:NEWLINENEWLINE{ Theorem 1.4.} For any numerical semigroup \(H\), there exists a natural number \(k_0\) such that for all \(k\geq k_0\), the ideal \(I_{H_k}\) is minimally generated by a standard basis, \(\beta_i(I_{H_k})=\beta_i(I_{H_k}^*)\) for all \(i\) and \(\mathrm{gr}_m K[H_k]\) is Cohen-Macaulay.NEWLINENEWLINEAn easy consequence of this result is this corollary:NEWLINENEWLINE{ Corollary 1.6.} Let \(\mathcal{H}_w\) be the set of all numerical semigroups with width less or equal to \(w\) (\(\geq 2\)). Then for any integer \(i\geq 0\) there exists an integer \(b\) such that \(\beta_i(I_H^*)\leq b\) for all \(H\in \mathcal{H}_w\).NEWLINENEWLINEThis result implies that there exists a global upper bound for \(\mu(I_H^*)\) for all numerical semigroup with a fixed width.NEWLINENEWLINEEven if an explicit bound is not provided, because of computational evidence, the authors conjecture that such upper bound is \({\mathrm{wd}(H)+1 \choose 2}\).NEWLINENEWLINE{ Conjecture 2.1.} Let \(H\) be a numerical semigroup, then \(\mu(I_H^*)\leq {\mathrm{wd}(H)+1 \choose 2}\). Moreover if \(\mu(H)\geq 2\), then the equality holds if and only if there exist integers \(w\) and \(k\) greater than \(1\) such that \(H\) is generated by \(kw+1, kw+2, \dots, (k+1)w+1\).NEWLINENEWLINEThey support mathematically this conjectural upper bound. Indeed, in Section 2, they show that this is the case for any numerical semigroup with \( \mu(I_H^*)\leq \mu(I_{\tilde{H}}^*)\), where \(\tilde{H}\) is the semigroup generated by all integers in the interval spanned by the smallest and the largest generator of \(H\).NEWLINENEWLINEMoreover, Proposition 2.10 says that the conjecture holds for numerical semigroups generated by an arithmetic sequence.NEWLINENEWLINEWe also stress that the conjecture implies that \(\mu(I_H)\leq {\mathrm{wd}(H)+1 \choose 2}\).NEWLINENEWLINEFinally, in Section 3, the authors study several examples of semigroups satisfying the conjecture: Sally semigroups, Bresinsky semigroups, Frobenius semigroups and a variation of Shibuta semigroups.