Exploring the interplay of semistable vector bundles and their restrictions on reducible curves

Abstract Let C be a comb-like curve over ℂ and let E be a vector bundle of rank n on C . In this paper, we investigate the criteria for the semistability of the restriction of E onto the components of C when E is given to be semistable with respect to a polarization w . As an application, assuming that each irreducible component of C is general in its moduli space, we investigate the w -semistability of kernel bundles on such curves, extending the results (completely for rank two and partially for higher rank) known in the case of a reducible nodal curve with two smooth components, but here using different techniques.