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Is there some nice compactification of the affine space with a del Pezzo surface of degree $\le 4$ (for example with a cubic surface) ?

More precisely, I would like a projective algebraic variety $X$ and an open subset $U\subset X$ isomorphic to $\mathbb{A}^3$, such that $X\setminus U$ contains a del Pezzo surface of degree $\le 4$, and not too many other components (say $0$, $1$ or $2$). The variety $X$ being smooth would be perfect, but terminal singularities are OK.

Of course, it is easy to do this with $\mathbb{P}^2$, and there are some constructions with quadrics or del Pezzo surfaces. We can also blow-up some points and get del Pezzo surfaces, but this create some artificial planes.

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  • $\begingroup$ Just a remark: if $S$ and $X$ are smooth and $X\smallsetminus S\cong \mathbb{A}^3$, the Gysin exact sequence gives you $b_2(X)=1$ and $b_2(S)=b_4(X)$, hence $b_2(S)=1$ by Poincaré duality. So if $S$ is a del Pezzo surface, $S\cong \mathbb{P}^2$. $\endgroup$
    – abx
    Commented Apr 2, 2014 at 13:03
  • $\begingroup$ Thanks, nice remark. I was also thinking that smooth was too much. $\endgroup$
    – Jérémy Blanc
    Commented Apr 2, 2014 at 13:04
  • $\begingroup$ Do you have some similar formulas when the singularities are mild? $\endgroup$
    – Jérémy Blanc
    Commented Apr 2, 2014 at 13:50
  • $\begingroup$ If $X\smallsetminus S$ is a normal crossing divisor, the Gysin exact sequence is replaced by the Leray spectral sequence for $X\smallsetminus S\hookrightarrow X$. But then the cohomology of the intersection of the components will appear, so it might be tricky (I didn't check). $\endgroup$
    – abx
    Commented Apr 2, 2014 at 14:28

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