# Intersections in $\mathbb{P}^1\times\mathbb{P}^1$

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?

• Look for "Picard group" on the web. The question is not appropriate for this site. – abx Jul 25 '18 at 17:34
• I Disagree with abx. All algebraic geometers know this, but not all professional mathematicians. Algebraic geometers should want our field to be useful to the rest of mathematics, yes? – David E Speyer Jul 25 '18 at 17:57
• @abx Whatever happened to MO being a place where researchers could get help with questions they suspect have answers well-known to experts, but where they don't have said experts at hand? (+1 to David Speyer) – Yemon Choi Jul 26 '18 at 16:19

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.