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In [1] (e.g. corollary 1.3), Tan gives an effective criterion that a linear system $|T+nH|$ is $k$-very ample. ------------------ P.S. $(-1)$-very ample $\Leftrightarrow$ $h^1(T+nH)=0$ $0$-very ample $\Leftrightarrow$ base points free $1$-very ample $\Leftrightarrow$ very ample ... ---------------- The criterion will tell you an effect lowerbound $n_k$ such that $|T+nH|$ is $k$-very ample whenever $n> n_k$.
