Smoothness of the zero set of a bi-homogeneous polynomial in $\mathbb{P}^{1}\times\mathbb{P}^{1}$. Let $C\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ be the zero set of the bi-homogeneous polynomial $f(x_{0},x_{1},y_{0},y_{1}) = (x_{0}^{3} + x_{1}^{3})y_{0}^{2} + (x_{0}^{3}-x_{1}^{3})y_{1}^{2}$. I want to show that this variety is smooth. 
Definition: $C$ is smooth at a point $P$ if there is an open subvariety $U$ of $C$ containing $P$ and an isomorphism $\varphi:U\rightarrow Z(f_{1},...,f_{n-d})\subset\mathbb{A}^{n}$ for some $d\leqslant n$ and $f_{1},...,f_{n-d}$, such that the rank of the $n-d$ by $n$ matrix $\left(\frac{\partial f_{i}}{\partial x_{j}}(\varphi(P)\right)_{i,j}$  equals $n-d$. 
My usual strategy is to find an affine open cover of $C$ and then check smoothness on these affine opens. In this case I guess that I need to look for sets of the form $C\cap(U\times V)$ where $U,V$ are affine opens of $\mathbb{P}^{1}$ such that we can cover $C$ with open subvarietes of this form. What I could do is consider the cover consisting of $C\cap(D(x_{i})\times D(y_{j}))$ for $0\leqslant i,j\leqslant 1$, where $D(x_{i})$ is the set of points in $\mathbb{P}^{1}$ with $x_{i}\neq 0$. But then Im stuck with finding the isomorphisms we need. 

By the comment of KReiser we see that $D(x_{0})\times D(y_{0})\cong\mathbb{A}^{2}$ by the map $([1:x_{1}],[1:y_{1}])\mapsto(x_{1},y_{1})$. Consequently we also see that $C\cap(D(x_{0})\times D(y_{0}))\cong Z_{\mathbb{A}^{2}}((1+x_{1}^{3})+(1-x_{1}^{3})y_{1}^{2})$ by the same map. 
Consequently, we have to check for every $P\in Z_{\mathbb{A}^{2}}((1+x_{1}^{3})+(1-x_{1}^{3})y_{1}^{2})$ that $\left(3x_{1}^{2}(1-y_{1}^{2}), 2y_{1}(1-x_{1}^{3})\right)\neq (0,0)$. 
If $3x_{1}^{2}(1-y_{1}^{2})= 0$, then $x_{1} = 0$ or $y_{1}^{2} = 1$. But since $P\in Z_{\mathbb{A}^{2}}((1+x_{1}^{3})+(1-x_{1}^{3})y_{1}^{2})$ we find that $y_{1}^{2}\neq 1$. If $x_{1}=0$, then since $P\in Z_{\mathbb{A}^{2}}((1+x_{1}^{3})+(1-x_{1}^{3})y_{1}^{2})$ we find that $y_{1}^{2}=-1$, and thus $2y_{1}(1-x_{1}^{3})\neq 0$. So $C$ is smooth at every $P\in C\cap(D(x_{0})\times D(y_{0}))$. 
The other three cases go similarly. 
