Projective Cohen-Macaulay monomial curves and their affine charts

Abstract

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 $$a_0 = 0< a_1< \cdots < a_n = d$$ a 0 = 0 < a 1 < ⋯ < a n = d , we consider the projective monomial curve $$\mathcal {C}\subset \mathbb {P}_k^{\,n}$$ C ⊂ P k n of degree d parametrically defined by $$x_i = u^{a_i}v^{d-a_i}$$ x i = u a i v d - a i for all $$i \in \{0,\ldots ,n\}$$ i ∈ { 0 , … , n } and its coordinate ring $$k[\mathcal {C}]$$ k [ C ] . The curve $$\mathcal {C}_1 \subset \mathbb {A}_k^n$$ C 1 ⊂ A k n with parametric equations $$x_i = t^{a_i}$$ x i = t a i for $$i \in \{1,\ldots ,n\}$$ i ∈ { 1 , … , n } is an affine chart of $$\mathcal {C}$$ C and we denote by $$k[\mathcal {C}_1]$$ k [ 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 $$k[\mathcal {C}]$$ k [ C ] and $$k[\mathcal {C}_1]$$ k [ 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 $$j+a_1< \cdots < j+a_n$$ j +