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&#39; of E of rank n&#39; those of maximal degree d&#39;. Then s_n&#39;(E)= n&#39;d-nd&#39;\le n&#39;(n-n&#39;)g. If E is stable s_n&#39;(E)>0 while if E is generic s_n&#39;(E)\ge n&#39;(n-n&#39;)(g-1) . The following statement was conjectured by Lange: If 0<s\le n&#39;(n-n&#39;)(g-1), then there exist stable vector bundles with s_n&#39;(E)=s. We prove this result for the generic curve. We also clarify what happens in the interval n&#39;(n-n&#39;)(g-1)<s\le n&#39;(n-n&#39;)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.