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VersalDeformations :: versalDeformation

versalDeformation -- computes a power series representation of a versal deformation

Description

Here we provide an overview of our approach to solving deformation problems. For details on using the command versalDeformation, please see the documentation links below. For simplicity, we restrict to the case of the versal deformation of an isolated singularity,

First we fix some notation. Let S be a polynomial ring over some field k, and let I be an ideal of S defining a scheme X=Spec S/I with isolated singularities. Consider a free resolution of S/I:

...→Sl →Sm →S →S/I→0

with differentials R0:Sl→Sm and F0:Sm→S. Let φiHom(Sm/ Im R0,S) for i=1,...,n represent a basis of T1(S/I)≅Hom(Sm/ Im R0,S)/ Jac F0. We introduce deformation parameters t1,...,tn with the ring T=S[t1,...,tn] and consider the map F1: Tm→T defined as F1=F0+∑tiφi. Let a be the ideal generated by t1,...,tn. It follows that there is a map R1: Tl→Tm with R1= R0 mod a satisfying the first order deformation equation F1R1= 0 mod a2.

Our goal is to lift this equation to higher order, that is, for each i>0, to find Fi: Tm→T with Fi=Fi-1 mod ai and Ri: Tl→Tm with Ri= Ri-1 mod ai satisfying FiRi= 0 mod ai+1. In general, there are obstructions to doing this, governed by the d-dimensional k vector space T2(S/I). Thus, we instead aim to solve

(FiRi)tr+Ci-2Gi-2= 0

mod ai+1. Here, Gi-2: k[t]→k[t]d and Ci-2: Td→Tl are congruent modulo ai to Gi-3 and Ci-3, respectively. Furthermore, we require that Gi and Ci vanish for i<0, and C0 is of the form V D, where V∈Hom(Sd,Sl) gives representatives of a basis for T2(S/I) and D∈Hom(Sd,Sd) is a diagonal matrix. The Gi now give equations for the miniversal base space of X.

Our implementation solves the above equation step by step. Given a solution (Fi,Ri,Gi-2,Ci-2) modulo ai+1, the package uses Macaulay2’s built in matrix quotients to first solve for Fi+1 and Gi-1 (by working over the ring T/I+ Im (Gi-2)tr +ai+2) and then solve for Ri+1 and Ci-1. For the actual computation, we avoid working over quotient rings involving high powers of a by representing the (Fi,Ri,Gi-2,Ci-2) as lists of matrices which keep track of the orders of the tj involved.

Ways to use versalDeformation :