Geodesic
Effective construction of a~nonsingular in codimension one algebraic variety over a~zero-characteristic ground field
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XIX, Tome 387 (2011), pp. 167-188

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Let $k$ be a field of zero-characteristic finitely generated over a primitive subfield. Let $f$ be a polynomial of degree at most $d$ in $n$ variables with coefficients from $k$ and irreducible over an algebraic closure $\overline k$. Then we construct a nonsingular in codimension one algebraic variety $V$ and a finite birational isomorphism $V\to\mathcal Z(f)$ where $\mathcal Z(f)$ is the hypersurface of all common zeroes of the polynomial $f$ in the affine space. The working time of the algorithm for constructing $V$ is polynomial in the size of the input. 
@article{ZNSL_2011_387_a7,
     author = {A. L. Chistov},
     title = {Effective construction of a~nonsingular in codimension one algebraic variety over a~zero-characteristic ground field},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {167--188},
     publisher = {mathdoc},
     volume = {387},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_387_a7/}
}
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A. L. Chistov. Effective construction of a~nonsingular in codimension one algebraic variety over a~zero-characteristic ground field. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XIX, Tome 387 (2011), pp. 167-188. http://geodesic.mathdoc.fr/item/ZNSL_2011_387_a7/