# Global section of pull-back of structure sheaf of projective scheme

Let $X$ be a smooth projective variety and $Z_1, Z_2$ two smooth projective divisors in $X$. Is it true that the natural restriction morphism from $H^0(\mathcal{O}_X(-Z_1-Z_2))$ to $H^0(\mathcal{O}_X (-Z_1-Z_2) \otimes_{\mathcal{O}_X} \mathcal{O}_{Z_1})$ is surjective? In particular, if $h^0(\mathcal{O}_X(-Z_1-Z_2))=0$ does it imply that $h^0(\mathcal{O}_X(-Z_1-Z_2) \otimes_{\mathcal{O}_X} \mathcal{O}_{Z_1})=0$?

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Consider the case when $Z_1$ and $Z_2$ are effective (which is probably implicit in your question, but I'm not sure) and $Z_2$ is disjoint from $Z_1$ (which can certainly happen, e.g. imagine $Z_1$ and $Z_2$ are distinct fibres in a surface that is fibred by curves; or that $Z_1$ is an exceptional divisor in a blow up, and $Z_2$ is the preimage in the blow-up of a divisor that doesn't pass through the blown-up point; or you could just imagine the case when $Z_2 = 0$.).
Then $\mathcal O(-Z_1 - Z_2) \otimes \mathcal O_{Z_1} = \mathcal O(-Z_1)\otimes \mathcal O_{Z_1} = C_{X/Z_1}$, the conormal bundle to $Z_1$ in $X$. Supposing that at least one of $Z_1$ and $Z_2$ is non-zero, and that $X$ is connected, we have $H^0(\mathcal O(-Z_1 - Z_2) ) = 0$ (since this is the space of global sections of a proper ideal sheaf). Thus you are asking if the conormal bundle to $Z_1$ necessarily has trivial global sections.
The answer is: not necessarily. The adjunction formula shows that $C_{X/Z} = (K_X)_{|Z} \otimes K_Z^{-1}.$ Thus for example if $Z$ is an elliptic curve on a K3 surface $X$, then $K_X$ and $K_Z$ are both trivial, and so $C_{X/Z}$ is trivial, and hence has a one-dimensional (and so non-trivial) space of global sections.
Thanks for the answer. I have a follow up question. In the case $Z_1 \cup Z_2$ is complete intersection subscheme (in the same projective space which contains $X$) can we expect an affirmative answer? – Jana Dec 3 '13 at 12:38