Let $F$ be an algebraically closed field and $\mathbb{P}^1$ the projective line over $F$. Suppose $V\_1, V\_2$ are two 1-dimensional subvarieties of the 2-dimensional variety $\mathbb{P}^1\times\mathbb{P}^1$.  

Now $V\_1$ and $V\_2$ do not necessarily intersect, as the simple example $V\_i=\{x\_i\}\times\mathbb{P}^1$ with $x\_1\neq x\_2$ shows. But are there any conditions on $V\_1$ and $V\_2$ known, under which they do intersect?



Intersections in $\mathbb{P}^1\times\mathbb{P}^1$
The curve $V\_i$ is given by the vanishing of a polynomial $F\_i(x\_1,x\_2,y\_1,y\_2)$ that is homogeneous in $x\_1,x\_2$ of degree $d\_{i,1}$ and homogeneous in $y\_1,y\_2$ of degree $d\_{i,2}$. Then counting intersection points with multiplicities,
$$
V\_1 \cdot V\_2 = d\_{1,1}d\_{2,1} + d\_{1,2}d\_{2,2}.
$$
So $V\_1$ and $V\_2$ always intersect except in the case that they are both horizontal slices or they are both vertical slices, as in the example that you give.


