Let $X$ be a smooth projective variety, $V, W$ closed subschemes in $X$ such that $V \cap W$ is finitely many points. Let $\mathcal{L}$ be a line bundle on $X$. Is there any relation between $h^0(\mathcal{L} \otimes\_{\mathcal{O}\_X} \mathcal{O}\_{V.W})$ and the intersection multiplicity of V.W (like is the former bounded by the later)? Is so, can we say anything similar if $\mathcal{L}$ is a locally free sheaf not necessarily of rank $1$?



Intersection multiplicty and global sections
If $V$ and $W$ are Cohen--Macaulay, then $V \cap W$ is a finite number of points implies $Tor\_{>0}(O\_V,O\_W) = 0$. This means that $V\cdot W = \ell(O\_V \otimes O\_W) = \ell(L \otimes O\_V \otimes O\_W)$ which is equal to the $h^0$ you are interested in. 


