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This queston can be regarded as a variant of this one:

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 $s=0$?

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By $\O_X$-module do you mean $O_X$? – kennytm Sep 2 '10 at 17:17
Yes, I am sorry, latex is almost never displayed correctly on this computer so I make lots of mistakes ): – Jan Sep 2 '10 at 17:56
How is LaTeX rendered incorrectly? Maybe you can create a bug report in meta. – kennytm Sep 2 '10 at 18:01
up vote 2 down vote accepted

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.

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