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# When does  the blow-up of $CP^2$ at N points  embed in $CP^4$?

Write $X_N$ for this blow up. Place the N points in 'general position' as needed. Then $X_6$ embeds in $CP^2$ as a smooth cubic surface. (See, eg, Griffiths and Harris.) But there is no other $N$ (except $N=0$) for which $X_N$ embeds in $CP^3$. (Proof: The topology of the blow-up disagrees with that of a smooth surface of degree $d$ in $CP^3$. (Gompf-Stipsisz p. 21.) On the other hand, $X_N$ embeds in $CP^5$ simply because any smooth algebraic surface $X$ so embeds. (Harris, `Algebraic Geometry, a first course', p. 193.)

Embarrassingly, I don't even know the answer for $N=1$ where $X_1$ is the 1st Hirzebruch surface! (I'm betting it does embed.)

Motivation: This question began in an attempt to better understand the 27 lines on the cubic and my initial surprise at how the construction described in GH of $X_6$ yielded a smooth surface in $CP^3$, and how all such surfaces arise through that construction by varying the 6 points. I am hoping answers might help me understand the moduli of blow-ups as I move the N points about the plane, and orient me as a novice to algebraic surfaces.

ag gt
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