Let $X$ be a canonically polarized smooth projective geometrically connected variety over $k$ with Hilbert polynomial $h$.

What is the Hilbert polynomial of $X\times_k \mathbf{P}^1_k$? How does it depend on $h$?

Example. Let $g\geq 2$ be an integer and let $X$ be a genus $g$ curve. Then the Hilbert polynomial of $X\times_k \mathbf{P}^1_k$ should depend only on $g$.

Probably one should use some "canonical" embedding of $\mathbf{P}^n \times\mathbf{P}^1$ into $\mathbf{P}^N$...