Spatial curve singularities and the monster/semple tower

Abstract

The Monster tower ([MZ01], [MZ10]), known as the Semple Tower in Algebraic Geometry ([Sem54], [Ber10]), is a tower of fibrations canonically constructed over an initial smooth n-dimensional base manifold. Each consecutive fiber is a projective n — 1 space. Each level of the tower is endowed with a rank n distribution, that is, a subbundle of its tangent bundle. The pseudogroup of diffeomorphisms of the base acts on each level so as to preserve the fibration and the distribution. The main problem is to classify orbits (equivalence classes) relative to this action. Analytic curves in the base can be prolonged (= Nash blown-up) to curves in the tower which are integral for the distribution. Prolongation yields a dictionary between singularity classes of curves in the base n-space and orbits in the tower. This dictionary yielded a rather complete solution to the classification problem for n = 2 ([MZ10]). A key part of this solution was the construction of the ‘RVT’ classes, a discrete set of equivalence classes built from verifying conditions of transversality or tangency to the fiber at each level ([MZ10]). Here we define analogous ‘RC’ classes for n > 2 indexed by words in the two letters, R (for regular, or transverse) and C (for critical, or tangent). There are 2^k−1 such classes of length k and they exhaust the tower at level k. The codimension of such a class is the number of C’s in its word. We attack the classification problem by codimension, rather than level. The codimension 0 class is open and dense and its structure is well known. We prove that any point of any codimension 1 class is realized by a curve having a classical A _2k singularity (k depending on the type of class). Following ([MZ10]) we define what it means for a singularity class in the tower to be “tower simple”. The codimension 0 and 1 classes are tower simple, and tower simple implies simple in the usual sense of singularity. Our main result is a classification of the codimension 2 tower simple classes in any dimension n. A key step in the classification asserts that any point of any codimension 2 singularity is realized by a curve of multiplicity 3 or 4. A central tool used in the classification are the listings of curve singularities due to Arnol’d ([Arn99], Bruce-Gaffney ([BG82]), and Gibson-Hobbs ([GH93]).

We also classify the first occurring truly spatial singularities as subclasses of the codimension 2 classes. (A point or a singularity class is “spatial” if there is no curve which realizes it and which can be made to lie in some smooth surface.) As a step in the classification theorem we establish the existence of a canonical arrangement of hyperplanes at each point, lying in the distribution n-plane at that point. This arrangement leads to a coding scheme finer than the RC coding. Using the arrangement coding we establish the lower bound of 29 for the number of distinct orbits in the case n = 3 and level 4. Finally, Mormul ([Mor04], [Mor09]) has defined a different coding scheme for singularity classes in the tower and in an appendix we establish some relations between our coding and his.