Zero Rank Asymptotic Bridgeland Stability

In this paper we examine the conditions that an object $E$ with ${\text{ch}}_0(E)=0$ has to satisfy in order for it to be asymptotically (semi)stable with regard to Weak or Bridgeland stability conditions. This notion turned out to be equivalent to sheaf Gieseker-Simpson (semi)stability or a dual of it, depending on the curve considered.