A Coarse Stratification of the Monster Tower

Abstract

Abstract. The monster tower is a tower of spaces over a specified base; each space in the tower is a parameter space for curvilinear data up to a specified order. We describe and analyze a natural stratification of these spaces.The monster tower, also known in the algebro-geometric literature as the Semple tower, is a tower of smooth spaces (varieties) over a specified smooth base M. Each space M(k) in the tower, called a monster space, is a parameter space for curvilinear data up to order k on M. We will describe a coarse stratification of each monster space, with each stratum corresponding to a code word created out of a certain alphabet according to rules that we will specify. These strata parametrize curvilinear data "of the same type." The monster space can be regarded as an especially nice compactification of the parameter space for curvilinear data of nonsingular curves on M (as explained in Section 2), with the added points representing the data of singular curves; in our stratification the nonsingular data points will form a single open dense stratum.Versions of this coarse stratification have been observed by virtually everyone who has studied the monster construction. Here we develop the theory in full generality, beginning with a base space of arbitrary dimension and at all levels. A finer stratification would result from a thorough analysis of the orbits under the action of a suitable group acting on the base (whose action can be lifted to the tower) or, working locally, of the pseudogroup of local diffeomorphisms at a selected point. Results such as those in Section 5.7 of [12] show that one can expect there to be infinitely many strata, i.e., that there are moduli. We seem to be very far, however, from a full understanding of where and why moduli occur.

Complete spelling rules for the Monster tower over three-space

“…Finally, current work with algebraic geometers [4] extends and generalizes the results of this paper to the case of an n-dimensional base. An interesting open question here concerns the existence of moduli in orbits of the action of the diffeomorphism group of the base space.…”

“…We assume rules (1) -(3) have been proved. Here we will provide details for rule (4); the remaining proofs are very similar. To this end, let ω be an RVT code of length k, ending with the letter T 2 .…”

“…And we discovered in [7] that algebraic geometers have long studied these objects under different names. We have begun pursuing these connections [6] and working with algebraic geometers to consolidate understanding and improve existing terminology and techniques [4]. Here, we study the RVT code for the tower, which is invariant under the action of the diffeomorphism group.…”

Cartan Prolongation of a Family of Curves Acquiring a Node

“…We begin by briefly explaining the construction of the monster tower, closely following the account in Section 1 of our recent paper with Castro and Shanbrom [5]. That paper also gives a brief history of the construction, which was discovered independently in two separate strands of the mathematical literature.…”

“…Removing all the lower-dimensional loci from an intersection locus gives a code word stratum, an open dense subset of the intersection locus. In this manner we have stratified S(k), as in [5]. In that paper we also gave equations defining each intersection locus in each chart; all of them are linear equations in the local coordinates.…”