Spanned vector bundles with high rank and no trivial factor on n-folds

Abstract

Let X be a smooth n-fold, \(n\ge 3\), and \(\mathcal {L}\) a spanned and ample line bundle on X with \(h^1(\mathcal {O}_X)=0\). Let \(\mathcal {E}\) be a spanned rank r vector bundle on X with \(\det (\mathcal {E})\cong \mathcal {L}\) and no trivial factor. Here we prove that \(r\le h^0(\mathcal {L})-1\) and prove the existence of some \(\mathcal {E}\) for all r with \(n\le r \le h^0(\mathcal {L})-1\). In the case \(n=3\) we show that if \(h^0(\mathcal {L}) -r\) is very small (in terms of the t-connectedness of \((X,\mathcal {L})\)), then \(\mathcal {E}\) is obtained by a standard procedure from \(\mathcal {L}\). If \(h^1(\mathcal {O}_X)>0\) we prove the upper bound for r substituting the integer \(h^0(\mathcal {L})-2\) with an integer \(\rho (\mathcal {L})\) with \(h^0(\mathcal {L})-2+ \max \{0,2h^1(\mathcal {O}_X)-h^1(\mathcal {L})\}\le \rho (\mathcal {L})\le h^0(\mathcal {L})-2+2h^1(\mathcal {O}_X)\).