Projective monomial curves associated to numerical semigroups with multiplicity $e$, width $e-1$, and embedding dimension $e-2$
Om Prakash Bhardwaj, Trung Chau, Omkar Javadekar
TL;DR
The paper analyzes projective monomial curves linked to numerical semigroups with multiplicity $e$, width $e-1$, and embedding dimension $e-2$, focusing on the family $S(e,m,n)=\langle \{e,\dots,2e-1\}\setminus \{e+m,e+n\}\rangle$. Using Gröbner-basis techniques and the Cohen–Macaulay criterion for projective monomial curves, it establishes a complete CM classification: $\mathbb{K}[\overline{S(e,m,n)}]$ is CM precisely when $(m,n)\neq (e-4,e-3)$, and it determines Gorenstein cases as $(e,m,n)\in\{(4,1,2),(5,2,3)\}$, with the latter offering complete intersections. For the subfamily $m=1$, it provides an explicit generating set for the defining ideal and shows non-symmetry (hence non-Gorenstein) for $e\ge5$, except the small $e$-case. When CM, the authors compute the Castelnuovo–Mumford regularity of the coordinate rings and supply Macaulay2 checks to corroborate the theoretical results. Overall, the work delivers explicit Gröbner bases, CM/Gorenstein criteria, and regularity formulas for a broad class of projective monomial curves arising from Sally-type numerical semigroups.
Abstract
Numerical semigroups with multiplicity $e$, width $e-1$, and embedding dimension $e-2$ are of the form $$S(e,m,n) = \langle \{e, e+1, \ldots, 2e-1\} \setminus \{e+m, e+n\} \rangle,$$ for some $1 \leq m < n \leq e-2$. Inspired by the work of Sally, Herzog and Stamate studied the special case $S(e,2,3)$, which they called the ``Sally numerical semigroups''. Recently, Dubey et. al. computed a minimal generating set of the defining ideal of the numerical semigroups $S(e,m,n)$ for $m \geq 2$. In this article, we first obtain an analog for the numerical semigroups $S(e,1,n)$, and then shift our focus to the projective monomial curves in $\mathbb{P}^{e-2}$ defined by the semigroups $S(e,m,n)$. We obtain a Gröbner basis for the defining ideal of the projective monomial curves associated to the semigroups $S(e,m,n)$. Moreover, we provide characterizations of Cohen--Macaulay and Gorenstein properties of these curves. Specifically, we prove that these are Cohen--Macaulay if and only if $(m,n) \neq (e-4,e-3)$, and Gorenstein if and only if $(e,m,n)\in \{ (4,1,2), (5,2,3)\}$. Furthermore, when these curves are Cohen--Macaulay, we compute the Castelnuovo--Mumford regularity of their coordinate ring.