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 $h^0(\mathcal {L})-2+ \max \{0,2h^1(\mathcal {O}_X)-h^1(\mathcal {L})\}\le \rho (\mathcal {L})\le h^0(\mathcal {L})-2+2h^1(\mathcal {O}_X)$.