Vanishing criterion for sections of module on a product This queston can be regarded as a variant of this one:
Does a section that vanishes at every point vanish?
Let $X,Y$ be varieties, $M$ be an $\mathcal{O}_X$-Module and 
$$\pi:X\times Y\rightarrow X$$
be the projection. Let $s$ be a local section of $M$, which vanishes along each "horizontal line":
$$\forall y: i_y^* s=0$$ 
where denotes the map $i_y:x\mapsto (x,y)$. Does it follow, that $s=0$?

If you take $X$ to be a point, then this reduces to your previous question: you are just taking fibres at each point $y \in Y$, and so the answer is "no" for the same reason.
Just to see that this isn't just being pedantic, let me explain how to bootstrap the answer
to the previous question to give examples in this case which truly have a non-trivial product
structure.  For this, suppose that $Y = \mathbb A^1$ and that $X$ is the variety attached to
the ring $A$, so that $X \times Y$ corresponds to the ring $A[t]$.  Let $M$ be the module
$A[t]/t^2,$ and let $s$ be the section $t \bmod t^2$.  Then $s$ will vanish along each "horizontal line", but is non-zero.  (The case $X = $ a point gives the counterexample to
your previous question.)
Again, if $M$ is torsion free and finitely generated then you should be okay.
