The origin of a mixed chart lies in the Zariski closure of the chart curve off the axis $u=0$

Let $m\ge1$ be an integer, let $a,b\in\mathbb C$ be arbitrary, and let $Q_{a,b}(u,y)=a\,u+y+b\,u^{m+1}y^m+u^my^{m+1}\in\mathbb C[u,y]$. With $P_\alpha(X,Y)=X^m(\alpha+XY)+Y^m(\bar\alpha+XY)$, the choice $(a,b)=(\alpha,\bar\alpha)$ gives the polynomial equal to $u^{m+1}P_\alpha(1/u,y)$ for $u\ne0$, the equation of the curve $P_\alpha=0$ in the chart $u=1/X$ near the boundary point $(\infty,0)$ of $\mathbb P^1\times\mathbb P^1$; the choice $(a,b)=(\bar\alpha,\alpha)$ gives the analogous equation near $(0,\infty)$, with the two coordinates exchanged. In $\operatorname{Spec}\mathbb C[u,y]$ with its Zariski topology let $V(Q_{a,b})$ be the set of prime ideals containing $Q_{a,b}$, let $D(u)$ be the set of prime ideals not containing $u$, and let $\mathfrak m_{(0,0)}=(u,y)$ be the point of the origin.

Then

The origin of the chart represents a corner of $\mathbb P^1\times\mathbb P^1$, while $D(u)$ is the part of the chart where $X=1/u$ is finite. So the corner lies in the Zariski closure of the part of the chart curve over the affine plane, as needed for the description of the closure of the curve (6) in $\mathbb P^1\times\mathbb P^1$.

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