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I have two simple guesses/hopes about the following facts. Let $X$ be an algebraic projective variety and $i: X \to \mathbb P(V)$. Let $L = i^*(\mathcal O_P(1))$ be the pullback of the tautologial line bundle over $\mathbb P(V)$.

1) $i^*(\mathcal O_P(N)) \simeq L^N$ for every $N \in \mathbb Z$.

2) Every $s \in H^0(X,L^N)$ extends to some $s^* \in H^0(X,\mathcal O_P(N))$, i.e. every section of $L^N$ can be seen as the restriction of a homogeneus polynomial of degree $N$ in the coordinates of $\mathbb P(V)$.

They are not true in general, but are they true in this special case?

Maffred
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  • is true for all $N$, if $X$ is projectively normal ($S/I(X)$ normal, where $\mathrm{proj}(S) = \mathbb{P}(V)$, so $S$ is the coordinate-ring of $\mathbb{P}(V)$). For general $X$, it holds for $N \gg 0$, because of the cohomology of $0 \to \mathcal{I}X(N) \to \mathcal{O}{P}(N) \to i_* \mathcal{O}_X(N) \to 0$.
    • – Jürgen Böhm Apr 08 '17 at 15:42
  • @JürgenBöhm Thanks! If I'm not wrong partial flag varieties with canonical Plucker-Veronese-Segre embedding are projectively normal right? – Maffred Apr 08 '17 at 16:19
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    I do not know it myself, but a quick look at Google gave: Allen Knutson (http://mathoverflow.net/users/391/allen-knutson), Defining Equations of a Flag Variety, URL (version: 2013-05-04): http://mathoverflow.net/q/129643 So it seems the answer to your question is "yes". – Jürgen Böhm Apr 08 '17 at 16:43
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    Isn't $\mathcal{O}(-1)$ the tautological line bundle? This post seems to indicate that it is $\mathcal{O}(-1)$, too. – Viktor Vaughn Apr 08 '17 at 19:57
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    @Quasicoherent I'm glad to totally agree with you! – Maffred Apr 08 '17 at 19:59