\[quadricResol\]), and we have the following locally free resolution $$0\to G_0^{\oplus e_{m,0}}\to\dots\to \bigoplus_{j=0}^{m-l}G_{j}^{\oplus e_{l,j}}\to\dots\to
\bigoplus_{j=0}^{m}G_{j}^{\oplus e_{0,j}}\to \mathcal{E}(d_{\min})
\to 0.$$ By tensoring $\mathcal{O}(-d_{\min})$, we get a locally free resolution of $\mathcal{E}$ in term of a full strong exceptional sequence $(\mathcal{O}(-d_{\min}),\mathcal{S}(-d_{\min}),\mathcal{O}(1-d_{\min}), \dots,\mathcal{O}(n-d_{\min}))$. Moreover the fact that $\mathcal{E}$ is nef imposes several constraints on $e_{l,j}$’s and $d_{\min}$; some easy constraints are that $e_{l,j}=0$ if $l+j>d_{\min}+c_1(\mathcal{E})+1$ (Propositions \[easyConst\] and \[firstConstraint\] (2) (a)) and that $d_{\min}\geqq 0$ if $c_1(\mathcal{E})<r$ (Proposition \[firstConstraint\] (2) (d)). Therefore if $c_1(\mathcal{E})$ is small then $d_{\min}$ has very few possible values and most $e_{l,j}$’s vanish. Note that the above resolution contains superfluous direct summands, so that we have to remove redundant direct summands. If $c_1(\mathcal{E})=1$, other constraints among $e_{l,j}$’s and a more detailed analysis of the resolution enable us to do so and we get the desired resolution as described in the classification due to Peternell-Szurek-Wiśniewski [@pswnef].

In the subsequent papers [@Nefofc1=3OnPN] and [@Nefofc1=(21)OnQ2], following our framework, we classify nef vector bundles on a projective space with the first Chern class three and the second Chern class less than eight, and nef vector bundles on a smooth quadric surface with the first Chern class $(2,1)$.

Notation and conventions {#convention}
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In 