A characterization of the hereditary categories derived equivalent to some category of coherent sheaves on a weighted projective line

Let $\mathcal{H}$ be a connected hereditary abelian category over an algebraically closed field $k$, with finite dimensional homomorphism and extension spaces. There are two main known types of such categories: those derived equivalent to $\operatorname{mod}\lambda$ for some finite dimensional hereditary $k$-algebra $\lambda$ and those derived equivalent to some category $\mathrm{coh}\,\mathbb{X}$ of coherent sheaves on a weighted projective line $\mathbb{X}$ in the sense of Geigle and Lenzing (1987). The aim of this paper is to give a characterization of the second class in terms of some properties known to hold for these hereditary categories.