LAUSR

Let X be a smooth n-fold, $$n\ge 3$$n≥3, and $$\mathcal {L}$$L a spanned and ample line bundle on X with $$h^1(\mathcal {O}_X)=0$$h1(OX)=0. Let $$\mathcal {E}$$E be a spanned rank r vector bundle on X with $$\det (\mathcal {E})\cong \mathcal {L}$$det(E)≅L and no trivial factor. Here we prove that $$r\le h^0(\mathcal {L})-1$$r≤h0(L)-1 and prove the existence of some $$\mathcal {E}$$E for all r with $$n\le r \le h^0(\mathcal {L})-1$$n≤r≤h0(L)-1. In the case $$n=3$$n=3 we show that if $$h^0(\mathcal {L}) -r$$h0(L)-r is very small (in terms of the t-connectedness of $$(X,\mathcal {L})$$(X,L)), then $$\mathcal {E}$$E is obtained by a standard procedure from $$\mathcal {L}$$L. If $$h^1(\mathcal {O}_X)>0$$h1(OX)>0 we prove the upper bound for r substituting the integer $$h^0(\mathcal {L})-2$$h0(L)-2 with an integer $$\rho (\mathcal {L})$$ρ(L) with $$h^0(\mathcal {L})-2+ \max \{0,2h^1(\mathcal {O}_X)-h^1(\mathcal {L})\}\le \rho (\mathcal {L})\le h^0(\mathcal {L})-2+2h^1(\mathcal {O}_X)$$h0(L)-2+max{0,2h1(OX)-h1(L)}≤ρ(L)≤h0(L)-2+2h1(OX).