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Let $X_k$ be $\mathbb{CP}^2$ blown up at $k$-points (where $k$ is from $0$ to $8$). I think it is known that $X_k$ can be embedded in $\mathbb{CP}^n$ for some $n$.

$\textbf{Question:}$ Can $X_k$ be described as the zero set of $n-2$ homogeneous polynomials in $\mathbb{CP}^n$, for some $n$?

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  • $\begingroup$ Of course, $X_6$ is a cubic in $\mathbb P^3$. If you allow weighted projective spaces, then $X_7$ is a degree $4$ hypersurface in $\mathbb P(1,1,1,2)$. I can't think of other constructions off the top of my head. $\endgroup$
    – Lev Borisov
    May 20, 2015 at 11:08
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    $\begingroup$ "I think": yes, any projective variety can be embedded in some $\mathbf P^n$. "Can $X_k...$ ?" Assume wlog that all the polynomials have degree $\geq 2$. Use adjunction to calculate the anticanonical bundle of a complete intersection of $n-2$ things in $\mathbf P^n$. Observe that the answer is positive only for $n=3,d=2,3$ or $n=4,(d_1,d_2)=(2,2).$ $\endgroup$
    – user5117
    May 20, 2015 at 11:08
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    $\begingroup$ To add to Artie Prendergast-Smith's comment: this means that $X_k$ is a complete intersection iff $k = 0,5,$ or $6$ (the case $n=3,d=2$ he mentioned does correspond to a del Pezzo, but not to a blow-up of $\mathbb{P}^2$). $\endgroup$
    – R.P.
    May 20, 2015 at 11:20
  • $\begingroup$ @René: good point (about quadrics)! $\endgroup$
    – user5117
    May 20, 2015 at 11:23
  • $\begingroup$ @Rene: If I understand your (and Smith's) remark correctly, the answer to my question is no in general; in particular if $k=1,2,3,4$, $7$ or $8$? The answer is only yes if $k=0,5$ or $6$. $\endgroup$
    – Ritwik
    May 20, 2015 at 14:14

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