The corner $(\infty,\infty)$ lies in the Zariski closure of complex points of its chart curve with $uv\ne0$

Let $m\ge1$ be an integer and let $\alpha\in\mathbb C$ be arbitrary. Let $P_\alpha(X,Y)=X^m(\alpha+XY)+Y^m(\bar\alpha+XY)$ and $R_\alpha(u,v)=u^m+v^m+uv\,(\bar\alpha u^m+\alpha v^m)\in\mathbb C[u,v]$. For $uv\ne0$ one has $R_\alpha(u,v)=u^{m+1}v^{m+1}P_\alpha(1/u,1/v)$, so $R_\alpha$ is the equation of the curve $P_\alpha=0$ in the chart $u=1/X$, $v=1/Y$ around the point $(\infty,\infty)$ of $\mathbb P^1\times\mathbb P^1$. For $w\in\mathbb C^2$ let $\mathfrak m_w$ be the maximal ideal of the polynomials vanishing at $w$, a point of $\operatorname{Spec}\mathbb C[u,v]$, and let $V(R_\alpha)$ be the set of prime ideals containing $R_\alpha$.

Then, with the closure taken in the Zariski topology,

The origin of this chart represents the point $(\infty,\infty)$ of $\mathbb P^1\times\mathbb P^1$, while at the approximating points both $X=1/w_0$ and $Y=1/w_1$ are finite. So $(\infty,\infty)$ lies in the Zariski closure of points of the affine curve $P_\alpha=0$, part of the description of the closure of the curve (6) in $\mathbb P^1\times\mathbb P^1$; the gluing of the charts is a separate step.

Formalization Note: points of $\operatorname{Spec}\mathbb C[u,v]$ are elements of PrimeSpectrum (MvPolynomial (Fin 2) ℂ) with Mathlib's Zariski topology, $u$ and $v$ are the variables X 0 and X 1, and $\mathfrak m_w$ is the kernel of evaluation at $w$.