Nearly Gorenstein vs Almost Gorenstein Affine Monomial Curves

Abstract

We extend some results on almost Gorenstein affine monomial curves to the nearly Gorenstein case. In particular, we prove that the Cohen–Macaulay type of a nearly Gorenstein monomial curve in $${\mathbb {A}}^4$$ A 4 is at most 3, answering a question of Stamate in this particular case. Moreover, we prove that, if $${\mathcal {C}}$$ C is a nearly Gorenstein affine monomial curve that is not Gorenstein and $$n_1, \dots , n_{\nu }$$ n 1 , ⋯ , n ν are the minimal generators of the associated numerical semigroup, the elements of $$\{n_1, \dots , \widehat{n_i}, \dots , n_{\nu }\}$$ { n 1 , ⋯ , n i ^ , ⋯ , n ν } are relatively coprime for every i.