Exercise 111.61.6. Let $k$ be an algebraically closed field. Consider the following types of surfaces

$S = C_1 \times C_2$ where $C_1$ and $C_2$ are smooth projective curves,

$S = C_1 \times C_2$ where $C_1$ and $C_2$ are smooth projective curves and the genus of $C_1$ is $> 0$,

$S \subset \mathbf{P}^3_ k$ is a hypersurface of degree $4$, and

$S \subset \mathbf{P}^3_ k$ is a smooth hypersurface of degree $4$.

For each type briefly indicate why or why not the class of surfaces of this type contains rational surfaces.

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