Bisecant and trisecant curves on ruled surfaces (Q1693078)

Let \(C\) be a smooth projective curve of genus \(g>0\). For any rank \(2\) vector bundle on \(C\) set \(S:= \mathbb {P}(C)\) and let \(l\) be the Segre invariant of \(E\), i.e. \(\deg (E)-2\deg (L)\), where \(L\) is a line subbundle of \(E\) with maximal degree, i.e. \(l= -C_0^2\), where \(C_0\) is a minimal section of the ruling of \(S\); call \(f\) the numerical class of a fiber of the ruling of \(S\). Let \(V_l\), \(l>0\), be the moduli space of bundles with Segre invariant \(l\). This paper studies in the case \(l>0\), i.e. \(E\) stable, the existence of bisecant and trisecant for \(S\), i.e. integral curves \(D\) numerically equivalent to \(kC_0+bf\), \(k=2,3\); let \(\delta _k(E)\) be the minimum of the integers \(b\) for which an integral \(D\) exists. They prove that \(\delta _2(E) = \lfloor (2g/3) -l\rfloor\) for a general \(E\in V_l\) when \(2g/3 < l\leq g\). They prove that \(\delta _3(E) = \lfloor -3g/4\rfloor\) (resp. \(\delta _3(E) = \lfloor -3(g-2)/4\rfloor\) when \(E\) is a general stable bundle and \(d-g\) is even (resp. odd). They obtain that for \(k=2,3\) and a general \(E\) the bundle \(\mathrm{Sym}^k(E)\) has the general Lange-stability; they also state a conjecture for the case \(k\geq 4\).