Minimal graded free resolutions for monomial curves in 𝔸�4 defined by almost arithmetic sequences

Abstract

Let m = (m0, m1, m2, n) be an almost arithmetic sequence, i.e., a sequence of positive integers with gcd(m0, m1, m2, n) = 1, such that m0 < m1 < m2 form an arithmetic progression, n is arbitrary and they minimally generate the numerical semigroup Γ =m0ℕ +m1ℕ +m2ℕ +nℕ. Let k be a field. The homogeneous coordinate ring k[Γ] of the affine monomial curve parametrically defined by X0 = tm0, X1 = tm1, X2 = tm2, Y = tn is a graded R-module, where R is the polynomial ring k[X0, X1, X2, Y] with the grading degXi: = mi, degY: = n. In this paper, we construct a minimal graded free resolution for k[Γ].