On monomial curves obtained by gluing

Abstract

We study arithmetic properties of tangent cones associated to large families of monomial curves obtained by gluing. In particular, we characterize their Cohen-Macaulay and Gorenstein properties and prove that they have non-decreasing Hilbert functions. The results come from a careful analysis of some special Apéry sets of the numerical semigroups obtained by gluing under a condition that we call specific gluing. As a consequence, we complete and extend the results proved by Arslan et al. (in Proc. Am. Math. Soc. 137:2225-2232, 2009) about nice gluings by using different techniques. Our results also allow to prove that for a given numerical semigroup with a non-decreasing Hilbert function and an integer $q>1$, extensions of it by $q$, except a finite number, have non-decreasing Hibert functions.