Monomialization of strongly prepared morphisms from nonsingular \(n\)-folds to surfaces (Q1881120)

Let \(\varphi: X\to Y\) be a dominant morphism of nonsingular varieties over a field of characteristic zero. \(\varphi\) is called monomial if for every point \(x\in X\) there exists an étale neighbourhood \(U\) of \(x\), uniformizing parameters \((u_1,\ldots,u_n)\) on \(U\), regular parameters \(v_1,\ldots,v_m\) of the local ring \(\mathcal O_{Y,\varphi(x)}\) and a matrix \(| | a_{ij}| | \) of nonnegative integers of rank \(m\) such that \(v_i=u_1^{a_{i1}}\cdots u_n^{a_{in}}\), \(\forall i=1,\ldots,m\). A natural question is: given the dominant morphism \(\varphi: X\to Y\), do there exist two morphisms \(\sigma: X_1\to X\) and \(\tau: Y_1\to Y\) which are products of blowups of nonsingular centers, and a monomial morphism \(\varphi: X_1\to Y_1\) such that \(\tau\circ\varphi_1=\varphi\circ\sigma\)? The answer is ``yes'' if \(Y\) is a curve, or if \(Y\) is a surface and \(\dim(X)\leq 3\). The main result of the paper under review is the following: Theorem. Suppose that \(\varphi: X\to Y\) is a strongly prepared morphism of nonsingular varieties, with \(\dim(X)=n\geq 2\) and \(Y\) a surface. Then there exists a finite sequence of quadratic transforms \(\tau: Y_1\to Y\) and monoidal transforms \(\sigma: X_1\to X\) of nonsingular two-codimensional centers such that the induced morphism \(\varphi_1: X_1\to Y_1\) is monomial. Applications of this result are also given.