Minimal Models

Abstract

One of the purposes of this chapter is to explain the geometric meaning of the notion of residue introduced in Section 10.2.1.

This chapter is quite independent of the rest of the book and can be used as an introduction to the theory developed in [42] and [46].

The main point to be understood is that the nonlinear coordinates ui used in Section 10.2.1, represent local coordinates around a point at infinity of a suitable geometric model of a completion of the variety $\mathcal{A}_X$. In fact, we are thinking of models, proper over the space $U\supset\mathcal{A}_X$, in which the complement of $\mathcal{A}_X$ is a divisor with normal crossings. In this respect the local computation done in Section 10.2.1, corresponds to a model in which all the subspaces of the arrangement have been blown up, but there is a subtler model that gives rise to a more intricate combinatorics but possibly to more efficient computational algorithms, due to its minimality.

In order to be able to define our models, we need to introduce a number of notions of a combinatorial nature.