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This is most likely a stupid question, but I am curious. It is well known that $\mathbb{P}^n$ has a semi-orthogonal decomposition

$$D^b(X)=\langle \mathcal{O},\dots,\mathcal{O}(n)\rangle$$

Suppose that for any reason I want to consider it as embedded in a bigger projective space via a Veronese map (or even a more elaborated, non complete, very ample linear system). Can I find a similar nice description in terms of (restriction) of tautological sheaves? My guess is that the answer is no. Another case that comes to my mind could be $\mathbb{P}^1 \times \mathbb{P}^1$ embedded as some kind of scroll $S(n,n)$ instead of as a smooth quadric surface.

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    $\begingroup$ For $X = P^1$ any nontrivial s.o.d. has form $D(P^1) = \langle O(i),O(i+1) \rangle$ for some $i$. Clearly, for emebdding of degree higher than 1 you cannot get both $O(i)$ and $O(i+1)$ as restrictions. $\endgroup$
    – Sasha
    Commented Feb 9, 2014 at 18:35

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