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.