Zariski closure of the curve $P_\alpha=0$ in the chart $(1/X,Y)$ of $\mathbb P^1\times\mathbb P^1$

Let $m\ge1$ be an integer and let $\alpha\in\mathbb C$ be arbitrary. Let $P_\alpha(X,Y)=X^m(\alpha+XY)+Y^m(\bar\alpha+XY)$ be the polynomial of equation (6). Near $X=\infty$, use the chart of $\mathbb P^1\times\mathbb P^1$ with coordinates $(u,Y)$, where $u=1/X$, and put $M_\alpha(u,Y)=\alpha u+Y+\bar\alpha\,u^{m+1}Y^m+u^mY^{m+1}$, which equals $u^{m+1}P_\alpha(1/u,Y)$. Let $\operatorname{Spec}\mathbb C[u,Y]$ be the prime spectrum of the polynomial ring in the chart coordinates, with its Zariski topology; for $w\in\mathbb C^2$ let $\mathfrak m_w$ be the maximal ideal of the polynomials vanishing at $w$, and let $V(M_\alpha)$ be the set of prime ideals containing $M_\alpha$.

Then the closure of the points of the affine curve $P_\alpha=0$ with $X\ne0$, written in the chart coordinates, is the whole zero locus of $M_\alpha$:

This is one of the chart computations behind $V_\alpha$, the closure of the curve (6) in $\mathbb P^1\times\mathbb P^1$, which the note introduces before Lemma 4. In the chart around the boundary point $(\infty,0)$, the closure of the affine curve is cut out by $M_\alpha$, the local equation used at that point in the proof of Lemma 4. Gluing the charts into a global statement is a separate step.

Formalization Note: points of $\mathbb C^2$ are sent into the prime spectrum by $w\mapsto\mathfrak m_w$, the kernel of evaluation at $w$; the closure and the zero locus are taken in Mathlib's prime spectrum with its Zariski topology, non-closed points included.