Suppose $Q=\mathbb P^1\times \mathbb P^1$ (corresponding to the quadric surface $V_+(xy-zw)$ in $\mathbb P^3$). Denote the projection of $Q$ to its two factors by $p$ and $q$.
For two positive integers $a,b>0$, let's consider $a$-uple embedding and $b$-uple embedding:$\mathbb P^1 \to \mathbb P^a$ and $\mathbb P^1\to \mathbb P^b$. These morphisms gives a closed immersion:
$$Q\to \mathbb P^a\times \mathbb P^b.$$
Then we consider the Segre embedding:$\mathbb P^a\times \mathbb P^b\to \mathbb P^N$, thus $Q$ is embedded in $\mathbb P^N$. Using Bertini's theorem we can find a hyperplane $H$ in $\mathbb P^N$ such that $Q\cap H$ is a nonsingular irreducible curve, which corresponds to a divisor of $Q$ of form $(a,b)$. Then we consider the composition of maps:
$$H\cap Q\to Q\stackrel{p}{\to} \mathbb P^1.$$
This is a map of nonsingular curves. Can we prove the degree of this map is $a$? (I want to prove it strictly, without cheating.) Could you provide some help? Thanks!
Best Answer
The equation of $H$ restricts to a homogeneous polynomial $P(x_0,x_1,y_0,y_1)$ on $\Bbb P^1\times\Bbb P^1$ of bidegree $(a,b)$. Fixing a point on the second factor and computing the preimage amounts to plugging in values in $k$ for $y_0$ and $y_1$, which by the hypothesis that $a,b>0$ and smoothness gives that $P$ is a nonzero homogeneous polynomial of degree $a$. So the fiber over any point is a closed subscheme of $\Bbb P^1$ cut out by a nonzero homogeneous polynomial of degree $a$ and we are done.