A note on the \(k\)-very ampleness of semi-stable bundles on an algebraic curve (Q1306498)

A vector bundle \(E\) on a smooth projective curve \(X\) is called \(k\)-very ample if for any effective divisor \(D\) on \(X\) of degree \(k+1\) the restriction map \(H^0(X,E)\to H^0(D,E\mid D)\) is surjective. The Clifford index of a semi-stable vector bundle \(E\) on an algebraic curve \(X\) of genus \(g\geq 2\) is defined by \(\text{Cliff} (E)=(\deg E-2(h^0 (X,E)-\text{rk} E)/\text{rk} E\). If \(g\geq 4\), the \(r\)-th Clifford index of \(X\) itself is defined by \[ \text{Cliff}_r (X)=\inf\{\text{Cliff}(E) \mid E\text{ semistable, }1\leq \text{rk} E \leq r,\;h^0(X,E)\geq 2,\;h^1(X,E)\geq 2\}. \] So \(\text{Cliff}_1(X)\) coincides with the usual Clifford index. The main result of the paper is the following theorem: If \(E\) is a semi-stable vector bundle of rank \(r\) with \(h^0(X,E)\geq r+1\) and \(h^1(X,E)\neq 0\), then \(E\) is \(k\)-very ample if only \[ \deg E/\text{rk} E\geq 2g+k+1- {1\over r}-{2\over r}h^1(X,E)- \text{Cliff}_r(E). \] As a consequence one obtains that the canonical line bundle \(\omega_X\) is \(\text{Cliff}_1(X)\)-very ample.