A note on k-very ampleness of a bundle on a blown up plane (Q1403520)

Let \(\pi : X \to \mathbb P^2\) be the blowing-up at \(r\) general points \(P_1,\dots ,P_r\) and \(E_i:= \pi ^{-1}(P_i)\) the exceptional divisors. Fix integers \(k\geq 2\) and \(d \geq 2k+3\). Set \(L:= \pi ^\ast (\mathcal {O}_P(d))(-E_1- \cdots -E_r))\). Since \(L\cdot E_i = 1\), \(L\) is not \(k\)-ample. Take an ``admissible '' \(0\)-cycle on \(X\), i.e. assume \(\sum _{i=1}^{r} l(Z\cap E_i) \leq 2\). Here the author gives a sharp criterion for the \(k\)-very ampleness of \(L| Z\).