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

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, irreducible over an algebraic closure [`(k)] bar{k} . Then we construct an algebraic variety V nonsingular in codimension one and a finite birational isomorphism V → Z(f), where Z(f) is the hypersurface of all common zeros of the polynomial f in the affine space. The running time of the algorithm for constructing V is polynomial in the size of the input. Bibliography: 8 titles.