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Let $X = \mathbb{P}^n_k$ be a projective space over an algebraically closed field $k$ and $x$ be a closed point.

Given an integer $m$ and a positive integer $r$.

What are the global sections of $\mathcal{O}_X(m) \otimes \mathcal{I}_x^r$ where $\mathcal{I}_x$ is the ideal sheaf of $x$ ?

Or what is the dimension of $\Gamma(X,\mathcal{O}_X(m) \otimes \mathcal{I}_x^r)$ ?

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Did you try to compute this for yourself? – Jason Starr May 3 '13 at 16:40
I computed for n = 2, m = 1, r = 3 and x = [1,0,0]. What I get is h^0(X,O(1) \otimes I^3) = 0 and h^0(X,O(1)) = 3 , h^0(X,O(1) \otimes O_X/I^3) = 3. From the exact sequence 0 \to O(1) \otimes O_X/I^3 \to O(1) \to O(1)\otimes I^3 \to 0, we have a long exact sequence and get H^0(X,O(1)) \to H^0(X,O(1) \otimes I^3) is surjective. But from what I recently read, it is wrong. So I don't know where I go wrong. – Philip May 4 '13 at 7:25
You have incorrectly computed $h^0(\mathbb{P}^2,\mathcal{O}(1)/\mathcal{I}_x^3\mathcal{O}(1))$. The dimension should be $6$, not $3$. – Jason Starr May 4 '13 at 20:07
The cohomology group corresponds to the vector space of homogeneous polynomials in $x_0,\ldots,x_n$ vanishing at $p$ with multiplicitly $r$. That is, the Taylor expansion of $f$ near $p$ has the form $f=f_r+\ldots+f_m$ where $\deg f_j=j$. Using this description you can show that $ \dim \Gamma(\mathbb P^n,I_x^r(m))=\max\left(0,{m+n \choose n}-{r+n-1 \choose n}\right) $ (I might have the binomial coefficients wrong..) – J.C. Ottem May 5 '13 at 17:03
Thanks. Jason Starr and J.C. Ottem I found where I misunderstood. – Philip May 6 '13 at 12:57

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