Arithmetic families of smooth surfaces and equisingularity of embedded schemes

This article deals with the foundations of a theory of equisingularity for families of zero-dimensional sheaves of ideals on smooth algebraic surfaces, in the arithmetic context, i.e., where one works with schemes defined over Dedekind rings. Here, different equisingularity conditions are analyzed and compared, based on one of the following requirements: 1) each member of the the family has the same desingularization tree, 2) the family admits a simultaneous desingularization, 3) a naturally associated family of curves is equisingular. Similar conditions had been investigated, in the context of Complex Local Analytic Geometry, by J. J. Risler. Received: 17 November 1997 / Revised version: 19 April 1999     

The continuum as a formal space

A constructive definition of the continuum based on formal topology is given and its basic properties studied. A natural notion of Cauchy sequence is introduced and Cauchy completeness is proved. Other results include elementary proofs of the Baire and Cantor theorems. From a classical standpoint, formal reals are seen to be equivalent to the usual reals. Lastly, the relation of real numbers as a formal space to other approaches to constructive real numbers is determined. Received: 11 November 1996     

Embedded constant mean curvature surfaces with special symmetry

We give sharp, necessary conditions on complete embedded CMC surfaces with three ends and an extra reflection symmetry. The respective submoduli space is a two-dimensional variety in the moduli space of general CMC surfaces. Fundamental domains of our CMC surfaces are characterized by associated great circle polygons in the three-sphere. Received: 23 January 1998 / Revised version: 23 October 1998     

Triades et familles de courbes gauches

Sans résumé Received: 3 April 1998 / in final form: 16 December 1998