blowing up general k points on the plane Del Pezzo surfaces are obtained by blowing up $1  \leq k \leq 8$  points on general position in $\mathbb{P}^2$. What does it happen when the number of points is larger than nine? In this sense, Beauville's book in surfaces presents the topic in the context of linear system of cubics: Nine points in the plane determine a cubic curve, and del Pezzo surfaces $S_{9-k}$, with $k\leq 6$, are embedded into $\mathbb{P}^{9-k}$ by the linear system of cubics through the $k $ points. Is there a nice interpretation of the surfaces obtained by linear system of plane curves of degree $d$?
I suppose this is well known but I cannot find a reference. Thanks!!
 A: 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   
A: The canonical divisor of the blow up $\pi: X\to \mathbb P^2$ at $k$ ordinary points is 
$$
K_X = -3\pi^*L +\sum_{i=1}^k E_i,
$$
where $L\subset \mathbb P^2$ is a hyperplane and $E_i$ is an exceptional curve of the first kind. Choosing the representatives right and an easy computation shows that 
$$
K_X^2 = 9 - k,
$$
so $-K_X$ could possibly be ample only if $0\leq k\leq 8$.
As Artie points out, embedding into $\mathbb P^{9-k}$ only works for $0\leq k\leq 6$. It is easy to see that this cannot be true for $k>6$ as then $9-k\leq 2$ and there is no way you can embed a surface different from $\mathbb P^2$ into $\mathbb P^2$, $\mathbb P^1$, or $\mathbb P^0$.
So, the interesting question is what you get if $k=7$ or $8$.
It is relatively easy to see that $-K_X$ is not very ample (unlike in the $0\leq k\leq 6$ case): By looking at the short exact sequences,
$$
0\to \pi^*\omega_{\mathbb P^2}^{-1}(-\sum_{i=1}^r E_i) \to \pi^*\omega_{\mathbb P^2}^{-1}(-\sum_{i=1}^{r-1} E_i)\to \mathscr O_{E_r}\to 0
$$
one can see easily that 
$$
\dim H^0(X, \omega_X^{-1}) = 10-k.
$$
In fact the $\mathbb P^{9-k}$ above is just the projectivization of this linear space. 

Remark Since the OP is talking about embedding a blow-up of $\mathbb P^2$ at $k$ points into $\mathbb P^{9-k}$, I assume they mean the classical definition of Del Pezzos, although the fact why $k$ can't be bigger than $8$ works for any definition, in particular for the now commonly used one asking only for that $-K_X$ is ample.

