to plane branches. To construct these ideals, we start with a germ ${\mathcal C}:=\{f=0\}\subset ({\mathbb C}^2,0)$ of a complex plane curve defined by an irreducible series $f\in{\mathbb C}[[x_0,x_1]]$ with $f(0) = 0$. The semigroup $\Gamma({\mathcal C})$ of ${\mathcal C}$ is the image of the associated valuation $$\nu_{{\mathbb C}}: \frac{{\mathbb C}[[x_0,x_1]]}{(f)}\setminus \{0\} \rightarrow {\mathbb N}: h \mapsto \dim_{{\mathbb C}}\frac{{\mathbb C}[[x_0,x_1]]}{(f,h)}.$$ This semigroup is finitely generated and has a unique minimal generating set $({\bar{\beta}}_0,\ldots,{\bar{\beta}}_g)$. Define $Y$ as the image of the monomial map $M:({\mathbb C},0) \rightarrow ({\mathbb C}^{g+1},0): t\mapsto (t^{\bar{\beta}_0},t^{\bar{\beta}_1},\ldots,t^{\bar{\beta}_g}).$ This is an irreducible curve which is smooth outside the origin and whose semigroup is the plane semigroup $\Gamma({\mathcal C})$. Furthermore, it is the special fiber of an equisingular family $\eta:(\chi,0) \subset ({\mathbb C}^{g+1},0)\rightarrow ({\mathbb C},0)$ with generic fiber isomorphic to ${\mathcal C}$. The ideal ${\mathcal I}\subset {\mathbb C}[x_0,\ldots,x_g]$ that defines $Y$ in ${\mathbb C}^{g+1}$ is generated by binomial equations of the form $$\left\{ 
    \begin{array}{r c l l}
    f_1 := x_1^{n_1} & - & x_0^{n_0}  &  = 0 \\
    f_2 := x_2^{n_2}  &- & x_0^{b_{20}}x_1^{b_{21}} &= 0  \\
    & \vdots & &\\
    f_g := x_g^{n_g} &- & x_0^{b_{g0}}x_1^{b_{g1}}\cdots x_{g-1}^{b_{g(g-1)}} & = 0.\\
     \end{array}
    \right.$$ Here, $n_i > 1$ and $b_{ij} \geq 0$ 