TITLE: Intersections in $\mathbb{P}^1\times\mathbb{P}^1$
QUESTION [5 upvotes]: 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?

REPLY [11 votes]: 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.