# Quadric surface: how to calculate the degree of this morphism?

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!

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.