2. deformations

Consider a singularity $(V,0)$ given by power series $f_1,\dots, f_k \in K\langle x_1, \dots, x_n\rangle$. The idea of deformation theory is to perturb the defining functions, that is to consider power series $F_1(t,x), \dots,$ $F_k(t,x)$ with $F_i(0,x) = f_i(x)$, where $t\in S$ may be considered as a small parameter of a parameter space $S$ (containing 0).

For $t\in S$ the power series $f_{i,t}(x) = F_i(t,x)$ define a singularity $V_t$, which is a perturbation of $V = V_0$ for $t\not= 0$ close to 0. It may be hoped that $V_t$ is simpler than $V_0$ but still contains enough information about $V_0$. 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 $F_i$ is flat if and only if any relation between the $f_i$, say

$$\sum r_i(x) f_i(x) = 0,$$

lifts to a relation

$$\sum R_i(t,x) F_i(t,x)=0,$$

with $R_i(x,0) = r_i(x)$. Equivalently, for any generator $(r_1, \dots, r_k)$ of syz$\,(f_1, \dots, f_k)$ there exists an element $(R_1, \dots, R_k) \in$   syz$\,(F_1, \dots, F_k)$ satisfying $R_i(0,x) = r_i(x)$. Hence, syzygies with respect to local orderings come into play.

There exists the notion of a semi-universal deformation of $(V,0)$ which contains essentially all information about all deformations of $(V,0)$.

For an isolated hypersurface singularity $f(x_1, \dots, x_n)$ the semi-universal deformation is given by

$$F(t,x) = f(x) + \sum^\tau_{j=1} t_j g_j(x),$$

where $1 =: g_1, g_2, \dots, g_\tau$ represent a $K$-basis of the Tjurina algebra

$$K\langle x \rangle /\langle f, \partial f/\partial x_1, \dots, \partial f/\partial x_n\rangle,$$

$\tau = \dim_K K\langle x \rangle/\langle f, \partial f/\partial x_1, \dots, \partial f/\partial x_n\rangle$ being the Tjurina number.

To compute $g_1,\ldots,g_\tau$ we only need to compute a standard basis of the ideal $\langle f, \tfrac{\partial f}{\partial x_1}, \dots, \tfrac{\partial f}{\partial x_n}\rangle$ with respect to a local ordering and then compute a basis of $K[x]$ 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 $C$ of degree 4, parametrised by $t \mapsto (t,t^2,t^3,t^4)$.

Homogeneous equations for the cone over $C$ are given by the $2 \times 2$-minors of the matrix:

$$m = \binom{x\;\;y\;\;z\;\;u}{y\;\;z\;\;u\;\;v} \in$$   Mat$$\,_{2\times 4}(K[x,y,z,u,v]).$$

SINGULAR commands for computing the semi-universal deformation:

LIB "deform.lib";
ring r         = 0,(x,y,z,u,v),ds;
matrix m[2][4] = 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 $Js =\langle BD, AD-D^2,-CD\rangle \subset K[A,B,C,D]$ defines the required base space which consists of a 3-dimensional component $(D=0)$ and a transversal 1-dimensional component $(B=C=A-D=0)$. 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 (Fs and Js as above)

$$K[[A,B,C,D]]/J_s \longrightarrow K[[A,B,C,D,x,y,z,u,v]]/J_s + F_s.$$

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:

1. Computation of standard bases, normal forms and resolutions for local orderings;

2. computation of Ext groups (cf. 4.1) for computing infinitesimal deformations and obstructions;

3. computation of Massey products for determining obstructions to lift, recursively, infinitesimal deformations of a given order to higher order;

4. one of the main difficulties in point 3 is the necessity to compute a completely reduced normal form with respect to a local ordering. In general, such a normal form exists only as formal power series. In the present situation, however, the reduction has to be carried out only for a subset of the variables in a fixed degree and, hence, the complete reduction is finite.