Let $Y$ be an irreducible complex projective variety of $\mathrm{dim}(Y) \geq 1$ and $X$ an irreducible complex projective variety. Assume that $M \in \mathrm{Pic}(Y)$ is a torsion element, say of order $e\neq 1$, and let $M'$ be a very ample line bundle on $X$. Consider the line bundle $L' \otimes L$ on $X \times Y$, where $L':=\pi_1^{*}M'$ and $L=\pi_2^{*}M$ and $\pi_1, \pi_2$ are the projections.
What can we say about $H^0(X \times Y ,(L'\otimes L)^m)$)?
We know that $L'$ is globally generated and that $L$ is torsion and non-trivial, as it should be true that $X \times Y \to Y$ is an algebraic fibre space, and so $\mathrm{Pic}(Y)\subset \mathrm{Pic}(X \times Y)$. This means that if $m$ is a multiple of $e$, than $(L' \otimes L)^m \cong (L')^m$ has global sections. It seems to me that for $m$ that is not a multiple of $e$ we shouldn't have global sections. Is this true? If yes how to prove it? At least it should be true that we cannot have global sections of the type $s \otimes t$ (for $m$ that isn't a multiple of $e$), as $L$ is torsion.
Moreover, I think that we can just consider $M'$ to be globally generated.
Best Answer
Use Künneth's Theorem. $$H^0(X\times Y, M'\boxtimes M)\simeq H^0(X,M')\otimes H^0(Y,M).$$ The $\boxtimes$ notation is the tensor product of pullbacks from each projection.
It works more generally over separated schemes over fields with $M'$ and $M$ quasicoherent sheaves, and I'm sure it is even more general than that.