The determination of integral closures and geometric applications

Abstract

We express explicitly the integral closures of some ring extensions; this is done for all Bring-Jerrard extensions of any degree as well as for all general extensions of degree ≤5; so far such an explicit expression is known only for degree ≤3 extensions. As a geometric application, we present explicitly the structure sheaf of every Bring-Jerrard covering space in terms of coefficients of the equation defining the covering; in particular, we show that a degree-3 morphism π:Y→X is quasi-etale if and only if c1(π* OY) is trivial (details in Theorem 5.3). We also try to get a geometric Galoisness criterion for an arbitrary degree-n finite morphism; this is successfully done when n=3 and less satisfactorily done when n=5. © 2003 Elsevier Inc. All rights reserved.