On Lange's Conjecture

Abstract

Let C be an algebraic curve of genus g. Let E be a vector bundle of rank n and degree d. Consider among all subbundles F' of E of rank n' those of maximal degree d'. Then s_n'(E)= n'd-nd'\le n'(n-n')g. If E is stable s_n'(E)>0 while if E is generic s_n'(E)\ge n'(n-n')(g-1) . The following statement was conjectured by Lange: If 0<s\le n'(n-n')(g-1), then there exist stable vector bundles with s_n'(E)=s. We prove this result for the generic curve. We also clarify what happens in the interval n'(n-n')(g-1)<s\le n'(n-n')g Our method uses a degeneration argument to a reducible curve. A similar result has been obtained by L.Bambrila-Paz and H.Lange using a different method.