Examples of a K3 surface in a product of $\mathbb P^1$'s

Question

On the Wikipeida page for K3 surfaces, there are several examples listed of how to produce a K3 surface as a subvariety of projective space by taking polynomials of specified degrees. Namely, a K3 surface is cut out by

• a degree 4 polynomial in $\mathbb P^3$,
• a degree 2 and a degree 3 polynomial in $\mathbb P^4$, and
• three degree 2 polynomials in $\mathbb P^5$.

Are there similar examples of how to get a K3 surface as an explicit subvariety of $\mathbb P^1 \times \cdots \times \mathbb P^1$? Or is there a reason this is not possible?

Answer 1

Huybrechts' book (1.4.1, page 18) gives an example in $\mathbb P^1 \times \mathbb P^1 \times \mathbb P^1$:

Smooth hypersurfaces $X \subset \mathbb P^2 \times \mathbb P^1$ and $X \subset \mathbb P^1 \times \mathbb P^1 \times \mathbb P^1$ of type $(3, 2)$ and $(2, 2, 2)$, respectively, provide K3 surfaces.