Let $X$ be a K3 surface obtained as a double covering of $\mathbb{P}^1 \times \mathbb{P}^1$ branching along a $(4,4)$-divisor. I think the natural line bundle $\pi^\*\mathcal{O}\_{\mathbb{P}^1\times \mathbb{P}^1}(1,1)$ is an ample line bundle on $X$. How can one prove that $\pi^\*\mathcal{O}\_{\mathbb{P}^1\times \mathbb{P}^1}(1,1)$ is ample? What is the degree of $X$ with respect to this ample line bundle?



An ample line bundle on a K3 surface
The ampleness follows from Nakai--Moichezon criterion. The degree is twice the degree of $O(1,1)$ on $P^1\times P^1$, so is $4$.


