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 $1$?
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$\begingroup$ What is $\mathcal{O}_{V.W}$? $\endgroup$– SashaCommented Feb 23, 2015 at 18:55
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$\begingroup$ @Sasha: It is the cokernel of the natural morphism $\mathcal{I}_V+\mathcal{I}_W \to \mathcal{O}_X$, where $\mathcal{I}_V, \mathcal{I}_W$ are the ideal sheaves of $V$ and $W$ respectively. Another definition, that one can use is $\mathcal{O}_V \otimes_{\mathcal{O}_X} \mathcal{O}_W$. $\endgroup$– KaliCommented Feb 23, 2015 at 19:39
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$\begingroup$ So, this is just the structure sheaf of the scheme-theoretical intersection. $\endgroup$– SashaCommented Feb 23, 2015 at 20:46
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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.
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$\begingroup$ Thank you very much. This is exactly the setting I am interested in. $\endgroup$– KaliCommented Feb 23, 2015 at 22:10
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$\begingroup$ Non-example: Let $V$ be a union of two planes at a point inside $\mathbb P^4$ (so not C-M), and $W$ a plane going through that point. Then the intersection multiplicity is of course $2$, but the scheme-theoretic intersection is a fat point of length $3$. To see the problem, let $W = W_1 \cap W_2$ be an intersection of hyperplanes. Then $V\cap W_1$ is a union of two lines and an embedded point, and $(V\cap W_1)\cap W_2$ has a point for each line (that's good) and also the embedded point (that's bad, and in particular, not transverse). $\endgroup$ Commented Feb 24, 2015 at 5:35