Decomposable subbundles of polystable vector bundles on projective curves (Q1817891)

Let \(X\) be a smooth projective curve of genus \(g\geq 4\). Here we show the existence for several numerical variants \(x>0\), \(\deg (E_i)\), \(\text{rank} (E_i)\), \(1\leq i\leq x\), \(\deg (F)\), \(\text{rank} (F)\) of semistable vector bundles \(E_i\), \(1\leq i\leq x\), \(F\) on \(X\) such that \(E:= \bigoplus_{1\leq i\leq x}E_i\) is a saturated subbundle of \(F\) and \(F/E\) is semistable. If \(X\) is either bielliptic or with general moduli we may find stable vector bundles \(E_i\), \(1\leq i\leq x\), and \(F\) with \(F/E\) stable.