Hello,


I usually see the following theorem without seeing any reference or explanation


Theorem: Let X be a blowup of $P^2$ at generic r points. If $r\leq 8$ then there are only finitely many (-1) curves on $X$ , while if $r\geq 9$ there are infinitely many of them. 


This is a little mysterious to me. I imagine the proof would be finding a family of curves in $P^2$ containing these points which are centers of the blowup, so that the multiplicities of those curves at these points are large enough to make the strict transform of these curves (-1) curves. However, I don't know what family to be chosen. Can someone give a proof or a reference containing a proof of this fact? 



Reference needed for negative curves on blowup of the projective plane at generic points
It is a result of Nagata but it is easy to explain (at least to se the difference between $r\le 8$ and $r\ge 9$):


The Picard group of the blow-up is generated by $L$, the pull-back of a general line, and $E\_1,...,E\_r$ the exceptional curves. Any curve disctinct from the $E\_i$'s is linearly equivalent to $C=dL-\sum a\_i E\_i$ for some integers $d,a\_1,\dots,a\_r$ with $d>0$, which is the degree of the image of the curve in $\mathbb{P}^2$. The integer $a\_i=E\_i\cdot C$ is the multiplicity of this curve at $a\_i$ and is then non-negative.


If $C$ is a $(-1)$-curve, we have $C^2=-1$ and $C\cdot C+C\cdot K\_S=-2$ (adjunction formula) so 


$\sum a\_i=3d-1$


$\sum (a\_i)^2=d^2+1$


By Cauchy-Schwartz we have $(\sum a\_i)^2\le r\sum (a\_i^2)$, which yields $9d^2-6d+1\le rd^2+r$. For $r\le 8$, we find thus only finitely many possiblities for $d$ ($d\le 6$), hence only finitely many $(-1)$-curves and for $r\ge 9$ we get infinitely many solutions.


It remains to see that the set of infinitely many numeric solutions gives infinitely many curves in the case $r\ge 9$. By a dimension count, we find that $|C|$ is not empty for each numeric solution. The fact that the points are in general position "should" imply that $|C|$ counts in fact an irreducible element. (This last part is the only one where something has to be checked; it seems natural but is not so easy, see the Nagata and Harbourne-Hirschowitz conjecture).


