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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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1 Answer 1

up vote 3 down vote accepted

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.

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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
@Jana: Dear Jana, I'm not sure off the top of my head. There are theorems of Zariski and others saying that in some situations the restriction map on global sections is surjective, but I would have to look things up to make precise statements. My memory is that projective normality can be relevant here. A good reference is the old survey on cohomology of sheaves by Zariski himself (published in the Bulletin of the AMS in the mid 50's), which does a good job of explaining how to connect coherent sheaf techniques to concrete geometric questions. The lemma of Enriques--Severi--Zariski ... –  Matt E Dec 3 '13 at 17:55
... as stated in Hartshorne Ch. III is one of the relevant technical tools. I hope this is some help. Regards, –  Matt E Dec 3 '13 at 17:56
Thanks. It was a very helpful answer and comment. –  Jana Dec 3 '13 at 23:33

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