The fact that the divisor "$X\,$" is not ample 
can also be seen directly from the definition.
For each point $p$ of $Y$ other than the image of the divisor "$X\,$", 
a section of $mX$ has no poles on $X \times p$,
and is thus constant on $X \times p$.  Therefore there is no $m$
for which the linear series $\Gamma(mX)$ separates the points of $X \times Y$.
(In fact the map from $X \times Y$ to projective space
associated with $\Gamma(mX)$ always factors through the projection
$X \times Y \rightarrow Y$.

P.S. How did this question from 2012 make it to the top page?