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Let $S$ be a smoooth projective surface and $Z\subset S$ a zero-dimensional subscheme of $S$ corresponding to a bunch of distinct points $x_1,\dots,x_r\in S$ (i.e. each counted with multiplicity 1). Let $L$ be a line bundle on $S$, say very ample for the seek of simplicity.

My question is: how should I be thinking of the sheaf $L\otimes\mathcal{I}_Z$, where $\mathcal{I}_Z$ is the ideal sheaf of $Z$ in $S$ ? For example, what is the geometric interpretation of the global sections $H^0(L\otimes\mathcal{I}_Z)$?

If we were on a curve then I would see it as the subspace of divisors in $|L|$ passing through each of the $x_i$ (with multiplicity 1). But on a surface, I am not sure if this is still the correct interpretation.

Any shared insight and/or concrete example would be greatly appreciated.

Possible second question: how about if we add more multiplicities?

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There is an exact sequence $$ 0 \to I_Z \to O_S \to O_Z \to 0, $$ where $O_Z = \bigoplus O_{X_i}$. Tensoring it by $L$, one gets $$ 0 \to L \otimes I_Z \to L \to L \otimes O_Z \to 0. $$ Next, $$ L \otimes O_Z = \bigoplus L \otimes O_{x_i} \cong \bigoplus O_{x_i}. $$ Finally, the cohomology exact sequence $$ 0 \to H^0(S,L \otimes I_Z) \to H^0(S,L) \to \bigoplus H^0(S,O_{x_i}) $$ (and the fact that its right arrow is the evaluation of sections at points $x_i$) shows that $H^0(S,L \otimes I_Z)$ is the space of sections of $L$ that vanish at all points $x_i$.

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  • $\begingroup$ Dear Sasha, thanks for the very neat answer! Everything is clear in what you wrote, except the identity $L\otimes \mathcal{O}_p \simeq \mathcal{O}_p$. Can you teach me the algebra here? $\endgroup$ Commented Jan 25, 2018 at 8:05
  • $\begingroup$ @HeitorFontana: $L$ is locally trivial, choose a trivialization of $L$ in a neighborhood of $p$, and you will get the isomorphism. $\endgroup$
    – Sasha
    Commented Jan 26, 2018 at 12:02

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