Let $X=\mathbb{P}^n$ for some $n \ge 1$ and $Y$ be a closed subscheme in $X$. We know that if $Y$ is a complete intersection then the natural map from $H^0(X,\mathcal{O}_X(n))$ to $H^0(Y,\mathcal{O}_Y(n))$ is surjective for all $n \in \mathbb{Z}$ (see Ex. III.$5.5$ of Algebraic Geometry by Hartshorne). Is it possible to get a more general statement, for example in the case $Y$ is local complete intersection?
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$\begingroup$ If $Y$ is a cone, then all these maps are surjective. If the Zariski tangent space at at least one point of $Y$ has dimension $n$, then $H^0(\mathcal O_{\mathbb P^n}(1))$ surjects on $H^0(\mathcal O_Y(1))$ (I am not sure about $\mathcal O(j)$ with $j>1$). I would not say these results are general enough. $\endgroup$– Serge LvovskiSep 6, 2013 at 7:37
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A local complete intersection will definitely not be good enough. You should check out the exercises on projective normality in Hartshorne (I believe chapter II section 5). This surjectivity is basically whether or not $Y$ is projectively normally embedded into $X$ (at least in the case that $Y$ itself is normal).
EDIT: Of course, the surjectivity holds for all sufficiently high d-uple embeddings. But I don't think you can anything more based on local properties of $X$. Castelnuovo-Mumford regularity can be used to detect for which $n$ the map can be guaranteed to be surjective (see Lazarsfeld's book on positivity in algebraic geometry).
answered Sep 5, 2013 at 17:39
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$\begingroup$ @Schwede: Thanks for your response. I have seen this result i.e., when $Y$ is normal. I am mainly interested in the case when $Y$ is not normal. Is there any partial result to my question that you are aware of? For example imposing more condition on $Y$. $\endgroup$– ChenSep 5, 2013 at 17:47
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$\begingroup$ The surjectivity you ask for is not a local condition on $Y$, it depends completely on how $Y$ embeds into $X$. It holds for all sufficiently high d-uple embeddings of course. You can use Castelnuovo-Mumford regularity to prove some surjectivities like it $\endgroup$ Sep 5, 2013 at 17:54