A vanishing theorem for finitely supported ideals in regular local rings
A cohomological vanishing property is proved for finitely supported ideals in an arbitrary d-dimensional regular local ring. (Such vanishing implies some refined Briancon-Skoda-type results, not otherwise known in mixed char- acteristic.) It follows that the adjoint e I of a finitely supported ideal I has order ord e I = sup(ordI +1 −d, 0), and that taking adjoints of finitely supported ideals commutes with taking strict transforms at infinitely near points. In particular, e I is also finitely supported.
Configurations of infinitely near points
We present a survey of some aspects and new results on configurations, i.e. disjoint unions of constellations of infinitely near points, local and global theory, with some applications and results on generalized Enriques diagrams, singular foliations, and linear systems defined by clusters.
Abstract dilatations and infinitely near points
In the following pages there is developed an abstract theory of infinitely near points for two-dimensional regular local rings which have infinite residue fields. The scope of the theory may be indicated by the fact that it contains generalizations of Noether's formula for intersection multiplicities (Theorem 8), the theory of proximate points (§ 8) and the conditions for a set of integers to be curve multiplicities at a given finite sequence of consecutive points (§ 10). To this extent our account resembles that given by van der Waerden ((6), Chap. 9), but the starting point and (with the exception of the concluding section) the type of reasoning employed are very different. There is also a marked contrast with the methods used by Zariski (7) in an account of related topics.