$\bar{\beta}_0,\ldots,\bar{\beta}_{i-1}$; this follows for example from [@Az]. In other words, for each $i = 1, \ldots, g$, we can find non-negative integers $b_{ij}$ for $0 \leq j < i$ such that $$\label{eq:ni-betai}
        n_i\bar{\beta}_i=b_{i0}\bar{\beta}_0+\cdots +b_{i(i-1)}\bar{\beta}_{i-1}.$$ If we require in addition that $b_{ij}<n_j$ for $j \neq 0$, then these integers are unique. For later purposes, we denote $n_0 := b_{10}$ and we list some other properties used in this article:

1.  for $i = 0,\ldots, g-1$, we have that $e_i = n_{i+1}\cdots n_g$;

2.  for $i = 0,\ldots, g-1$, we have that $n_j \mid {\bar{\beta}}_i$ for all $j > i$;

3.  for $i = 1,\ldots, g$, we have that $\gcd(\frac{{\bar{\beta}}_i}{e_i},n_i) = \gcd(\frac{{\bar{\beta}}_i}{e_i},\frac{e_{i-1}}{e_i}) = 1$, and, in particular, that $\gcd(n_0,n_1) = \gcd(\frac{{\bar{\beta}}_1}{e_1},n_1) = 1$; and

4.  for $i = 1,\ldots, g$, we have that $n_i{\bar{\beta}}_i < {\bar{\beta}}_{i+1}$.

In terms of the generators $({\bar{\beta}}_0,\ldots, {\bar{\beta}}_g)$, the curve we will consider is defined as the image of the monomial map $M:({\mathbb C},0) \rightarrow ({\mathbb C}^{g+1},0): t \mapsto (t^{\bar{\beta}_0},t^{\bar{\beta}_1},\ldots,t^{\bar{\beta}_g})$. We denote it by $Y$ and call it the *monomial curve associated with* ${\mathcal C}$. It is an irreducible (germ of a) curve with $\Gamma({\mathcal C})$ as semigroup and which is smooth outside the origin, see [@T1] 