(Bridgeland stability conditions) How can I get the heart of a bounded t-structure on $D^b(P^3)$? In the article, Bayer, Arend; Macrì, Emanuele; Toda, Yukinobu, Bridgeland stability conditions on threefolds. I Bogomolov-Gieseker type inequalities, J. Algebr. Geom. 23, No. 1, 117-163 (2014). ZBL1306.14005, I cannot understand something.
Above Lemma 8.2.3, the authors wrote that “by a classical result of Beilinson, on $D^b(P^3)$ we have a bounded t-structure with heart given by $C\mathrel{:=}\langle O_{P^3}(-1)[3], O_{P^3}[2], O_{P^3}(1)[1],  O_{P^3}(2)\rangle$.”
Q: How can I get the heart as above?
 A: I believe this is Lemma 3.14 of Macri's paper "Some examples of spaces of stability conditions on derived categories" combined with Beilinson's classic theorem.
There he shows that given an Ext-exceptional collection $\{ E_{1},\dots , E_{n} \} $ on a triangulated category, then $\langle E_{1},\dots, E_{n} \rangle$ is the heart of a bounded t-structure on T.
An Ext-exceptional collection is an exceptional collection such that $Hom^{\leq 0}(E_i,E_j)=0$ for every $i\not= j$. Beilinson's theorem gives a complete exceptonal collection of $D^{b}(\mathbb{P}^{n})$ as $\{ \mathcal{O}_{\mathbb{P}^{n}} , \dots , \mathcal{O}_{\mathbb{P}^{n}}(n) \}$ . Any twist also produces such a collection so in particular twisting by -1.
What remains to show is that $\{ \mathcal{O}_{\mathbb{P}^{3}}(-1)[3],\mathcal{O}_{\mathbb{P}^{3}}[2],\mathcal{O}_{\mathbb{P}^{3}}(1)[1],\mathcal{O}_{\mathbb{P}^{3}}(2) \}$ is Ext-exceptional.
We have $Hom^{-i}(\mathcal{O}_{\mathbb{P}^{3}}(-1)[3],\mathcal{O}_{\mathbb{P}^{3}}[2])=Hom(\mathcal{O}_{\mathbb{P}^{3}}(-1)[3],\mathcal{O}_{\mathbb{P}^{n}}[2-i])=0$, similarly $Hom^{-i}(\mathcal{O}_{\mathbb{P}^{3}}(2),\mathcal{O}_{\mathbb{P}^{3}}(1)[1])=Hom(\mathcal{O}_{\mathbb{P}^{3}}(2),\mathcal{O}_{\mathbb{P}^{3}}(1)[1-i])=0$, in the first case because the shift wont let the complexes match correctly, and in the second case because the collection is exceptional. The rest of the cases are analogous.
I hope I didnt mess up any grading.
