Consider a singularity given by power series . The idea of deformation theory is to perturb the defining functions, that is to consider power series with , where may be considered as a small parameter of a parameter space (containing 0).
For the power series define a singularity , which is a perturbation of for close to 0. It may be hoped that is simpler than but still contains enough information about . For this hope to be fulfilled, it is, however, necessary to restrict the possible perturbations of the equations to flat perturbations, which are called deformations.
Grothendieck's criterion of flatness states that the perturbation given by the is flat if and only if any relation between the , say
There exists the notion of a semi-universal deformation of which contains essentially all information about all deformations of .
For an isolated hypersurface singularity the semi-universal deformation is given by
To compute we only need to compute a standard basis of the ideal with respect to a local ordering and then compute a basis of modulo the leading monomials of the standard basis. For complete intersections we have similar formulas.
For non-hypersurface singularities, the semi-universal deformation is much more complicated and up to now no finite algorithm is known in general. However, there exists an algorithm to compute this deformation up to arbitrary high order cf. Ll,Ma1, which is implemented in SINGULAR.
As an example we calculate the base space of the semi-universal deformation of the normal surface singularity, being the cone over the rational normal curve of degree 4, parametrised by .
Homogeneous equations for the cone over are given by the -minors of the matrix:
SINGULAR commands for computing the semi-universal deformation:
LIB "deform.lib"; ring r = 0,(x,y,z,u,v),ds; matrix m = x,y,z,u,y,z,u,v; ideal f = minor(m,2); versal(f); setring Px; Fs; ==> Fs[1,1]=-u2+zv+Bu+Dv ==> Fs[1,2]=-zu+yv-Au+Du ==> Fs[1,3]=-yu+xv+Cu+Dz ==> Fs[1,4]=z2-yu+Az+By ==> Fs[1,5]=yz-xu+Bx-Cz ==> Fs[1,6]=-y2+xz+Ax+Cy Js; ==> Js[1,1]=BD ==> Js[1,2]=AD-D2 ==> Js[1,3]=-CD
The ideal defines the required base space which consists of a 3-dimensional component and a transversal 1-dimensional component . This was the first example, found by Pinkham, of a base space of a normal surface having several components of different dimensions.
The full versal deformation is given by the map (
Although, in general, the equations for the versal deformation are formal power series, in many cases of interest (as in the example above) the algorithm terminates and the resulting ideals are polynomial.
Ingredients for the semi-universal deformation algorithm: