Some Properties of Finite Morphisms on Double Points

Abstract

For a finite morphism f : X → Y of smooth varieties such that f maps X birationally onto X′=f(X), the local equations of f are obtained at the double points which are not triple. If $\cal C$ is the conductor of X over X′, and $D = Sing(X') ⊂ X'$, $Δ ⊂ X$ are the subschemes defined by $\cal C$, then D and Δ are shown to be complete intersections at these points, provided that $\cal C$ has “the expected” codimension. This leads one to determine the depth of local rings of X′ at these double points. On the other hand, when $\cal C$ is reduced in X, it is proved that X′ is weakly normal at these points, and some global results are given. For the case of affine spaces, the local equations of X′ at these points are computed.