Characterization of negative line bundles whose Grauert blow-down are quadratic transforms

We show that the Grauert blow-down of a holomorphic negative line bundle $L$ over a compact complex space is a quadratic transform if and only if $k_0{L}^{\ast }$ is very ample and $(k_0+1)L^{\ast }$ is globally generated, where $k_0$ is the initial order of $L^{\ast }$, namely, the minimal integer such that $k_0^{\ast }$ has nontrivial holomorphic section.