Chapter 16 - The desingularized Kummer

Summary

Introduction. In one formulation of the classical theory of the jacobian of a curve C of genus 2, there occurs a nonsingular surface S intermediate between the jacobian J(C) and the Kummer K. This surface is birationally equivalent canonically both to K and to K*, and is a minimum desingularization of each of them. The configuration links the abelian variety self-duality of the jacobian with a protective space duality for K. We shall give an exposition of this in Chapter 17.

In our context, when the ground field is not algebraically closed, things cannot be quite like that. The Kummer is not projectively self-dual and we cannot expect the dual of the jacobian, when appropriately defined, to be an abelian variety: it is, at best, a principal homogeneous space. Nevertheless, there is a nonsingular surface S, which is canonically a minimum desingularization of both K and K*. It arises naturally from a further study of the algorithms of Chapter 6, and is a natural half-way house in the duplication process.

In this chapter, we define and study S. At the end, we find an unramified natural covering, which we would expect to be the natural arena for a generalized abelian group duality. This will be a subject of later investigation.

The results described in this chapter were noticed only in the course of preparing these prolegomena. We have thus had to leave a number of loose ends.