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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.

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By now there are more tractable proofs of resolution of singularities than Hironaka's, so it no longer has to be a black box. A relatively elementary approach is described in Kollár's Lectures on Resolution of Singularities. There are many references there to algorithmic resolutions. In particular, in the case of hypersurfaces there is even a Maple program to do it. Check out Bodnár-Schicho's paper Automated Resolution of Singularities for Hypersurfaces and the references in there. (I'm not trying to be comprehensive in this answer to provide all the references, because there are lots of them. But probably most references that are at least 6-8 years old are referenced in the above two sources.)

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The question is natural and I would like to say that in practice, it depends a lot of the given hypersurface. The way you suggest seems to be a good idea in general, but the blow ups can sometimes be long to compute. I dont know a good algorithm in a software and do it always by "hand". Note that there are in general choices in the blowups. For example you have a cuspidal cubic curve at infinity and blowing up the pt and then the curve seems natural but if you have reducible sets of singularities there is no canonical choice.

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.

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