global sections of some sheaves

Let $Y$ be a connected, nonsingular, positive dimensional subvariety of $\mathbb{P}^n_k$ over an algebraically closed field $k$, and let $\mathcal{I}$ be the ideal sheaf of $Y$.

Why are the followings true?

1) $\Gamma(Y,\mathcal{I}^r/\mathcal{I}^{r+1})=0$ for any $r\ge 1$. 2) $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^r)=k$ for any $r\ge 1$.

Could someone explains this for me, thanks.

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2 Answers

Put $\mathcal N=(\mathcal I/\mathcal I^2)^\vee$, the normal bundle to $Y$ in $X=\mathbb P^n$. The tangent bundle sequence $0\to T_Y\to (T_X)\vert_Y\to \mathcal N\to 0$ and the ampleness of $T_X$ shows that $\mathcal N$ is ample. Now $\mathcal I^r/\mathcal I^{r+1}$ is the dual of the $r$th symmetric power of $\mathcal N$, so cannot have sections.

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Note that Kodaira's vanishing theorem fails in positive characteristics (counterexamples are given by Mumford with normal surfaces and Raynaud with smooth surfaces). So this nice proof (and also that of J.C. Ottem) works only in characteristic 0. –  Qing Liu Mar 2 '11 at 13:35
Liu: Unless there is something I missed in the proof, no H^1 is used here; only the fact that the dual of an ample line bundle has no sections. Am I wrong? –  ACL Mar 2 '11 at 17:49
You are right. I am sorry ! –  Qing Liu Mar 2 '11 at 18:53

Write for simplicity $X=\mathbb{P}^n$.

The easiest way of showing 1) is probably by noting that $\mathcal I / \mathcal I^2$ injects as a subbundle of $\mathcal O_Y(-1)^{n+1}$ (this follows from combining the conormal sequence with the Euler sequence) and so none of it's symmetric powers $S^r(\mathcal I / \mathcal I^2)=\mathcal I^r / \mathcal I^{r+1}$ can have any global sections. Now taking the cohomology sequence of $$0 \to \mathcal{I}^r/\mathcal{I}^{r+1}\to\mathcal{O}_X/\mathcal{I}^{r+1} \to\mathcal{O}_X/\mathcal{I}^{r} \to 0,$$shows that $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^{r+1})$ injects into $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^{r})$, and so by induction on $r$, we get 2).

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Sorry, but how do you know $rY$ has no global functions? Connectivity does not imply this, because the scheme is non-reduced, so it may have global nilpotent functions. –  t3suji Mar 1 '11 at 14:13
@t3suji: JC is saying that $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^{r+1})$ injects into $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^{r})$ for all $r$, so by iterating this you get that $\Gamma(Y,\mathcal{O}_X/\mathcal{I}^{r})$ injects into $\Gamma(Y,\mathcal{O}_X/\mathcal{I})$. There are no nilpotents there. –  Sándor Kovács Mar 1 '11 at 17:54
ps: In other words, the nilpotents would have to show up in $\Gamma(Y,\mathcal{I}^r/\mathcal{I}^{r+1})$ for some $r$. –  Sándor Kovács Mar 1 '11 at 17:55
@Sandor Kovacs. Of course it is true that 1) and 2) are equivalent. I was objecting to the original answer (which has been edited away): the idea was to get 2) from connectivity and then derive 1). –  t3suji Mar 1 '11 at 18:55
I edited the answer above to make it more precise. Thanks to t3suji and Sandor for their comments. –  J.C. Ottem Mar 1 '11 at 19:46