Linear stability and rank two Clifford indices of algebraic curves with applications

We prove that any vector bundle computing the rank-two Clifford index of a smooth projective algebraic curve is linearly semistable. We also identify conditions under which such bundles become linearly stable, thereby addressing a question posed by A. Castorena, G. H. Hitching and E. Luna in the rank-two case. Furthermore, we demostrate that in certain special cases, this property is equivalent to the (semi)stability of the associated Lazarsfeld-Mukai bundles. This yields a positive answer, in specific cases, to a generalized version of a conjecture proposed by Mistretta and Stoppino. We also study the moduli space $S_0(n,d,5)$ of generated $\alpha$-stable coherent systems of type $(n,d,5)$ for small values of $\alpha$ and $n=2,3$. We show that a general element of an irreducible component of $X \subseteq S_0(2,d,5)$ or $X \subseteq S_0(3,d,5)$ is linearly stable whenever $2\delta_2 \leq d \leq \frac{3g}{2}$. As an application of this, we prove that Butler's conjecture holds non-trivially for coherent systems of type $(2,d,5)$ within the given range for $d$.