The restriction of $\varphi_{p,H}$ to the principal open subset can be written in coordinates by choosing basis of $N_{1}$ and of $N_{2}$. Let $n_{i}$ be the rank of $N_{i}$. The chosen basis determine isomorphisms $X_{\Sigma_{i},0}\simeq\mathbb{G}_{m}^{n_{i}}$, which give coordinates ${\boldsymbol{x}}=(x_{1},\dots,x_{n_{1}})$ and ${\boldsymbol{t}}=(t_{1},\dots,t_{n_{2}})$ for $X_{\Sigma_{1},0}$ and $X_{\Sigma_{2},0}$, respectively. We write the the linear map $H$ with respect to these basis as a matrix, and we denote its rows by $a_{i}$, $i=1,\dots,n_{2}$. Write $p=(p_{1},\dots,p_{n_{2}})$. In these coordinates, the morphism $\varphi_{p,H}$ is given by