Bounds for syzygies of monomial curves (Q6583743)

Let \(\Gamma \subseteq \mathbb{N}\) be a numerical semigroup generated by \(a_0 < \cdots < a_n\). By the result of \textit{T. Vu} [J. Algebra 418, 66--90 (2014; Zbl 1317.13037)], the Betti numbers of the semigroup ring \(\mathrm{k}[\Gamma] = \mathrm{k}[t^\gamma \mid \gamma \in \Gamma]\) is bounded by a function on the width of the sequence \(\mathbf{a} = (a_0, \ldots, a_n)\), \(w(\mathbf{a}) = a_n - a_0\). \textit{J. Herzog} and \textit{D. I. Stamate} [J. Algebra 418, 8--28 (2014; Zbl 1333.13023)] conjectured that the number of generators of the defining equation of the tangent cone of \(\mathrm{k}[\Gamma]\) is bounded by \(\binom{w(\mathbf{a})+1}{2}\). Recently, the conjecture has been proved for monomial space curves [\textit{N. P. H. Lan}, \textit{N.C. Tu}, and \textit{T. Vu}, ``Betti numbers of the tangent cones of monomial space curves'', Acta Math Vietnam (2024; doi:10.1007/s40306-024-00546-4)]. Besides that, the conjecture is open. Even a weaker version of the conjecture that bounds the Betti numbers of the semigroup ring \(\mathrm{k}[\Gamma]\) itself is not known for monomial curves of embedding dimension \(\ge 4\).\N\NIn the current paper under review, the authors provided explicit bounds for the Betti numbers of \(\mathrm{k}[\Gamma]\). Though the bound is exponential in terms of the width, it is the first explicit bound purely in terms of the width. The authors also proved the Herzog-Stamate bound for monomial curves of embedding dimension \(4\) whose width is at least \(40\). The technique of proof is interesting. The ideas can be summarized as follows. First, we can reduce to bound the Betti numbers of the initial ideal. Now, the initial ideal of \(\mathrm{k}[\Gamma]\) is the pullback of the initial ideal of the interval completion of \(\Gamma\), which is easy to describe. Hence, we get a degree-wise bound on the Hilbert function of the initial ideal. The Betti numbers of the initial ideals then are bounded by the Betti numbers of the lex segment ideal with the same Hilbert function. Analyzing the property of the lex segment ideals gives the desired conclusion.