Effective construction of a nonsingular in codimension one algebraic variety over a zero-characteristic ground field

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.