An answer for a Mistretta-Stoppino's conjecture

We study the relation between linear stability of generated linear series on smooth curves and slope stability of their associated syzygy bundles. Motivated by conjectures of Mistretta and Stoppino, we establish new cases in which linear stability implies slope stability, focusing first on generated linear series over general curves and then on curves lying on polarized K3 surfaces. In the case of general curves, we use Brill-Noether-theoretic arguments to relate the numerical conditions on the linear series to the semi-stability of the syzygy bundle. For curves on K3 surfaces, we combine Lazarsfeld-Mukai bundles with Bridgeland stability conditions and restriction techniques to obtain slope-stability results under explicit degree bounds. These results provide further evidence for the expected equivalence between linear stability of linear series and slope stability of syzygy bundles.