i = 0, \ldots, g.$$ For varying $v$ in $({\mathbb C},0)$, these equations define a family of germs of curves in $({\mathbb C}^{g+1} \times {\mathbb C},0)$, which is *equisingular* for instance in the sense that $\Gamma({\mathcal C})$ is the semigroup of all curves in the family. We denote this family by $(\chi,0)$ and we let $\eta:(\chi,0)\rightarrow ({\mathbb C},0)$ be the restriction of the projection onto the second factor $({\mathbb C}^{g+1}\times {\mathbb C},0)\rightarrow ({\mathbb C},0)$. The generic fiber $\eta^{-1}(v)$ for $v \neq 0$ is isomorphic to ${\mathcal C}$, and the special fiber $\eta^{-1}(0)$ is defined in $({\mathbb C}^{g+1},0)$ by the equations $x_i^{n_i} - c_ix_0^{b_{i0}} \cdots x_{i-1}^{b_{i(i-1)}} = 0$ for $i = 1, \ldots, g$. The coefficients $c_i$ are needed to see that any irreducible plane branch is a (equisingular) deformation of a such a curve. However, for simplicity, we will assume that every $c_i = 1$, which is always possible after a suitable change of coordinates. This leads to the monomial curve $Y$ associated with ${\mathcal C}$.

Clearly, we can also consider the global curve in ${\mathbb C}^{g+1}$ defined by the above binomial equations; from now on, we define *a (space) monomial curve* $Y \subset {\mathbb C}^{g+1}$ as the complete intersection 