Proyective Cohen-Macaulay monomial curves and their affine charts

TL;DR

This work analyzes when the Betti numbers $\beta_i(k[\mathcal{C}])$ of a projective monomial curve $\mathcal{C}$ and those of its affine chart $k[\mathcal{C}_1]$ coincide, introducing a Gröbner-free combinatorial criterion based on Apery-set posets. The main result shows that equality of all Betti numbers follows from an isomorphism between the posets $({\rm Ap}_1,\le_1)$ and $({\rm AP}_{\mathcal{S}},\le_{\mathcal{S}})$, and the authors derive infinite families of curves with this property while sharpening Vu's bound in the shifted family. They also develop a method to construct arithmetically Gorenstein projective curves from symmetric numerical semigroups, clarifying when Cohen–Macaulayness transfers to the projective case and exploring consequences for tangent cones. Together, the results connect combinatorial semigroup data with homological invariants of monomial curves, providing practical criteria for Betti-number equality and new avenues for Gorenstein constructions.

Abstract

In this paper, we explore when the Betti numbers of the coordinate rings of a projective monomial curve and one of its affine charts are identical. Given an infinite field $k$ and a sequence of relatively prime integers $a_0 = 0 < a_1 < \cdots < a_n = d$, we consider the projective monomial curve $\mathcal{C}\subset\mathbb{P}_k^{\,n}$ of degree $d$ parametrically defined by $x_i = u^{a_i}v^{d-a_i}$ for all $i \in \{0,\ldots,n\}$ and its coordinate ring $k[\mathcal{C}]$. The curve $\mathcal{C}_1 \subset \mathbb A_k^n$ with parametric equations $x_i = t^{a_i}$ for $i \in \{1,\ldots,n\}$ is an affine chart of $\mathcal{C}$ and we denote by $k[\mathcal{C}_1]$ its coordinate ring. The main contribution of this paper is the introduction of a novel (Gröbner-free) combinatorial criterion that provides a sufficient condition for the equality of the Betti numbers of $k[\mathcal{C}]$ and $k[\mathcal{C}_1]$. Leveraging this criterion, we identify infinite families of projective curves satisfying this property. Also, we use our results to study the so-called shifted family of monomial curves, i.e., the family of curves associated to the sequences $j+a_1 < \cdots < j+a_n$ for different values of $j \in \mathbb N$. In this context, Vu proved that for large enough values of $j$, one has an equality between the Betti numbers of the corresponding affine and projective curves. Using our results, we improve Vu's upper bound for the least value of $j$ such that this occurs.

Figures (2)

• Figure 1: The posets $({\rm Ap}_1,\leq_1)$ (in blue) and $({\rm AP}_\mathcal{S},\leq_\mathcal{S})$ (in black) for $\mathcal{S}_1 = \langle 5,11,13\rangle$.
• Figure 2: The posets $({\rm Ap}_1,\leq_1)$ (in blue) and $({\rm AP}_\mathcal{S},\leq_\mathcal{S})$ (in black) for $\mathcal{S}_1 = \langle 5,6,7,8,9,10\rangle$.

Theorems & Definitions (44)

• Example 1
• Example 2
• Proposition 1.1
• Proposition 1.2
• proof
• Definition 1.3
• Proposition 1.4
• proof
• Remark 1.5
• Corollary 1.6