An effective description of the Jelonek set (Q1602657)

Let \(f:X\to Y\) be a dominant polynomial map of affine varieties over an algebraically closed field \(K\). The map \(f\) is not finite at a point \(y\in Y\) iff there is no Zariski open neighbourhood \(U\) of \(y\) such that the restriction map \(f|_{f^{-1}(U)}:f^{-1}(U)\to U\) is finite. Let \(J_f\subset Y\) be the set of points at which \(f\) is not finite. The paper gives an effective method (in the language of the Gröbner basis) of the description of the ideal of the set \(J_f\). The first result of this type, in the case \(X=Y=\mathbb C^n\), was given by \textit{Z. Jelonek} [Ann. Pol. Math. 58, 259-266 (1999; Zbl 0806.14009) and Math. Ann. 315, 1-35 (1999; Zbl 0946.14039)].