Complete spelling rules for the Monster tower over three-space

TL;DR

This paper establishes the complete set of spelling rules for the RVT codes in the Monster tower over three-space, providing a detailed combinatorial characterization of points in the tower.

Contribution

It derives the explicit spelling rules for RVT codes over three-dimensional bases, generalizing methods to arbitrary dimensions.

Findings

Determined which RVT words are realized in the three-space case.

Analyzed incidence relations between subtowers and Baby Monsters.

Presented a general method for identifying the birth level of Baby Monsters.

Abstract

The Monster tower, also known as the Semple tower, is a sequence of manifolds with distributions of interest to both differential and algebraic geometers. Each manifold is a projective bundle over the previous. Moreover, each level is a fiber compactified jet bundle equipped with an action of finite jets of the diffeomorphism group. There is a correspondence between points in the tower and curves in the base manifold. These points admit a stratification which can be encoded by a word called the RVT code. Here, we derive the spelling rules for these words in the case of a three dimensional base. That is, we determine precisely which words are realized by points in the tower. To this end, we study the incidence relations between certain subtowers, called Baby Monsters, and present a general method for determining the level at which each Baby Monster is born. Here, we focus on the case where the base manifold is three dimensional, but all the methods presented generalize to bases of arbitrary dimension.