I am trying to work out example 5.6 in Bridgeland's paper "Stability conditions on triangulated categories".
http://annals.math.princeton.edu/wp-content/uploads/annals-v166-n2-p01.pdf
A standard example is the category $\mathcal{Coh}(X)$ of coherent sheaves on smooth projective curve with stability condition given by $$Z(E)=-\operatorname{deg}(E)+i\operatorname{rk}(E)$$
Image of such is discrete in $\mathbb{C}$ and so it is of finite length, i.e. it's heart is artinian and noetherian abelian category.
Now one can modify this stability condition by fixing slope $\zeta = \tan(\pi \alpha)$ for some $\alpha\in(0,1/2]$ and taking
$$Z_\zeta=i(\zeta\operatorname{deg}(E)+\operatorname{rk}(E))$$
It is easy to see that it's heart is $\mathcal{A}_\zeta=\mathcal{P}(\zeta,\zeta+1]$, where $\mathcal{P}(a,b]$ is the category, generated by all stable elements of $Z$ with phase on the interval. Note that $Z_{\zeta=0}$ does not make sense, since skyscraper sheaves are mapped to zero and there is no HN filtration for them.
In Bridgeland's paper there is statement that for irrational $\zeta$ corresponding stability condition is not of finite length in general. Is this true for the case $X=\mathbb{P}_1$? Then all stable objects for $Z$ are shifts of line bundles and skyscraper sheaves of degree 1. Then for nonzero $\zeta$ stable objects of $Z_\zeta$ are those that lie above the line with slope $\zeta$ for standard stability condition $Z$. It is clear from picture that their image under $Z_\zeta$ is bounded from below, so it should follow that corresponding heart is of finite length $\mathcal{A}_\zeta$. Am I missing something, or such conditions are indeed included in spaces of stability conditions for $\mathbb{P}_1$?
If so, how does this intuition break down for curves of higher genus (maybe need to restrict attention to numeric stability conitions here)?
Also any comments on how shall I think of hearts which are not of finite length are very welcome!
