(Bridgeland stability conditions)How to get the heart of a bounded t-structure on $D^b(P^1 \times P^2)$? I have already known how to get the heart of a bounded t-structure on $D^b(P^n)$ by Macri`s paper,
https://arxiv.org/abs/math/0411613.
However I cannot purpose analogously on $D^b(P^1 \times P^2)$.
How to get the heart of a bounded t-structure on $D^b(P^1 \times P^2)$?
 A: I assume you are looking for an analogous result of your previous question.
In this case I think you could use Corollary 2.7 of Orlov's paper "PROJECTIVE BUNDLES, MONOIDAL TRANSFORMATIONS, AND DERIVED CATEGORIES OF COHERENT SHEAVES"

If there exists a complete exceptional set in the derived category D^{b}(M) then the derived category D^{b}(E) also posseses a complete exceptional set

Here M is a smooth projective variety, E is a rank r vector bundle over M and $p:\mathbb{P}(E)\to M$  is the associated projective bundle.
If the complete exceptional collection over $D^{b}(M)$ is $\{E_{0},\dots,E_{n}\}$ then the complete exceptional collection over $E$ is given by $\{p^{\ast}E_0\otimes \mathcal{O}_{E}(-r+1),\dots p^{\ast}E_n\otimes \mathcal{O}_{E}(-r+1),\dots, p^{\ast}E_{0},\dots, p^{\ast}E_{n}\}$.
In our particular case $M=\mathbb{P}^{2}$ and $E$ produces the trivial projective bundle $\mathbb{P}^{1}\times \mathbb{P}^{2}$ over $\mathbb{P}^{2}$, for example we can consider E to be the rank 2 vector bundle $\mathcal{O}_{\mathbb{P}^{2}}\oplus \mathcal{O}_{\mathbb{P}^{2}}$ over $\mathbb{P}^{2}$ (so $\mathbb{P}(E)=\mathbb{P}^{1}\times\mathbb{P}^{2}$ ), the complete exceptional collection over $\mathbb{P}^{2}$ we can take to be $\{ \mathcal{O}_{\mathbb{P}^{2}},\mathcal{O}_{\mathbb{P}^{2}}(1),\mathcal{O}_{\mathbb{P}^{2}}(2) \}$ by Beilinson's result and so by Orlov's result you have a complete exceptional collection on $D^{b}(\mathbb{P}^{1}\times \mathbb{P}^{2})$ given by $\{ p^{\ast}\mathcal{O}_{\mathbb{P}^{2}} \otimes \mathcal{O}_{\mathbb{P}^{1}\times  \mathbb{P}^{2}}(-1),p^{\ast}\mathcal{O}_{\mathbb{P}^{2}}(1) \otimes\mathcal{O}_{\mathbb{P}^{1}\times \mathbb{P}^{2}}(-1),p^{\ast}\mathcal{O}_{\mathbb{P}^{2}}(2)\otimes \mathcal{O}_{\mathbb{P}^{1}\times \mathbb{P}^{2} }(-1), p^{\ast}\mathcal{O}_{\mathbb{P}^{2}} ,p^{\ast}\mathcal{O}_{\mathbb{P}^{2}}(1) ,p^{\ast}\mathcal{O}_{\mathbb{P}^{2}}(2)  \}$.
Now it would be good if we could use Macri's result about Ext-exceptional collections. We can simply shift this in an adequate way.
Denote the exceptional collection above as $\{E_{0},\dots,E_{5}\}$, then the collection $\{ E_{0}[5],E_{1}[4],\dots,E_{5}\}$ is Ext-exceptional because $Hom^{-k}(E_{i}[-i+5],E_{j}[-j+5])=Hom(E_{i}[-i+5],E_{j}[-j+5-k])=0$ since $-i+5=-j+5-k$ if and only if $i>j$ and $\{E_{0},\dots, E_{5}\}$ is exceptional.
So using Macri's result we have that $\langle E_0[5],E_{1}[4], \dots, E_{5} \rangle$ is the heart of a bounded t-structure on $D^{b}(\mathbb{P}^{1}\times \mathbb{P}^{2})$
