# Making Hironaka's theorem explicit for hypersurfaces

Given a smooth hypersurface $H$ in $\mathbb{C}^n$, a theorem of Hironaka promises that one can find a strict normal crossings compactification $\bar{H}$ inside of a projective variety $X$. For me, this result is the definition of a "black box fact". What are some techniques for computing such a compactification in practice?

More practically, I am interested in obtaining a strict normal crossings compactification of the Koras-Russell cubic which is the hypersurface in $\mathbb{C}^4$ cut out by the equation:

$$x + x^2y + z^2 + t^3 = 0$$

I should confess that I haven't really tried in this example. I could just compactify the hypersurface in say $\mathbb{C}P^n$ (or some weighted projective space) and then start blowing things up and if I am lucky enough eventually arrive at something smooth. But I'm wondering how professional algebraic geometers go about doing these things in a more systematic way.

In the case of this cubic, I suggest to keep the $x$ fibration (which is natural because of the automorphisms group) and to see that the remainining coordinates give something which looks like a del Pezzo surface of degree $1$. Such a surface has equation $z^2+t^3=F6(w, y)$ in a weighted projective space. Hence, going to $\mathbb{P}^1\times \mathbb{P}(1,1,2,3)$ seems to be natural. I didnt check if it is already smoothor if you need to blow up more but I guess that the del Pezzo fibration obtained has to be checked.