Dimension of two complete intersections Let $A$ be a variety over $\mathbb{C}$ of dimension $n$. Suppose $V,W$ are closed subvarieties of $A$ with dimensions $d,l$ respectively. Let $Z = V \cap W$ be a complete intersection of $V,W$. If $V' \subset V$ is a closed subvariety of dimension at most $d-2$, is it possible that $(V-V') \cap W = \emptyset $? 
I guess if $W$ is a hypersurface, then this should not happen because the dimension should only drop one by complete intersection, but don't know how to show the general case. 

Sure, it is possible for $Z$ to have dimension $\leq d - 2$ and in that case $V' = Z$ has the property that $(V \setminus V') \cap W = \emptyset$.
