A minor correction: the blowup of $\mathbb{P}^{2}$ at 9 points cannot be del Pezzo, since its anticanonical class has self-intersection equal to 0.

Much of the recent interest in the blowup $X_{k}$ of $\mathbb{P}^{2}$ at $k \geq 9$ points in general position centers around the ample cone of $X_{k},$ rather than a specific embedding of $X_{k}$ in projective space.

Let $H$ be the pullback of the hyperplane class via the blowup ${\pi}: X_{k} \rightarrow \mathbb{P}^{2},$ and let $E=\sum_{i=1}^{k}E_{i}$ be the sum of the $k$ exceptional divisors on $X_{k}.$ The anticanonical class $3H-E$ fails to be ample for $k \geq 9,$ but we can instead ask the following: for which positive integers $d,r$ is the divisor $dH-rE$ ample?

Since $H$ spans a boundary ray of the ample cone of $X_{k},$ we know that $H-tE$ is ample for $0 < t << 1,$ e.g. that $dH-rE$ is ample for $d >> r.$ So what we are really interested in is
\begin{equation}
t_{k}:=\sup \{ t > 0 : H-tE \hskip5pt {\rm ample} \}
\end{equation}
An upper bound for $t_{k}$ may be obtained from the positive value of $t$ for which $(H-tE)^{2}=0,$ i.e. ${1}/{\sqrt{k}}.$

$\textbf{Nagata's conjecture:} \hskip10pt t_{k}=1/{\sqrt{k}}.$

This statement holds when $k=m^{2}$ is a perfect square which is at least 9; this can be seen by looking at the ample cone of the blowup of $\mathbb{P}^{2}$ at a general complete intersection of two degree-$m$ plane curves and noting that the ample cone of a surface can only shrink upon specialization.

There is a large body of work on Nagata's conjecture and its generalizations. A nice overview can be found in "Remarks on the Nagata Conjecture" by B. Strycharz-Szemberg and T. Szemberg, available at

www.uni-due.de/~mat903/preprints/nagata1.pdf