  We prove that any non-commutative smooth projective variety with a Bridgeland
stability condition of dimension less than $\frac{6}{5}$ must be a smooth
projective curve. As a consequence, we deduce the non-existence of such
categories with dimension in the interval $(1,\frac{6}{5})$. Moreover, we prove
a sharp reconstruction result for smooth projective curves of higher genus
using the moduli space of stable objects in a category of dimension $1$, and
deduce a structural result for their semi-orthogonal decompositions.
