Cyclic multiple planes, branched covers of \(S{^n}\) and a result of D. L. Goldsmith (Q2850554)

The authors use the result that for any complex algebraic variety \(X\) and a closed subvariety \(Y\) of \(X\) there is a triangulation of \(X\) for which \(Y\) is a subcomplex. Two generalizations of Zariski's result on irregularity of cyclic multiple planes are offered. In both the conclusion is that the first Betti number of a surface \(S\) is \(0\). One result assumes that \(f(x,y)\in\mathbb{C}[x,y]\) is reduced, \(n=p^l\) for some prime \(p\) and \(l\geq 1\) an integer, where \(f=0\) is connected and \(S=\{ z^n-f=0\}\). In the second \(f(x,y)\) is irreducible, \(n=p^l\) and \(S\) is a resoltution of singularities of \(\{ z^n-f=0\}\). These results are valid in higher dimensions e.g., for \(x_{m+1}^n-f(x_1,\ldots,x_m)=0\), \(m\geq 2\). Finally, the authors show how to derive a know theoretic result from a theorem of Goldsmith (which they prove by correcting the original proof which contains an error).NEWLINENEWLINEFor the entire collection see [Zbl 1266.14004].