When classifying objects in algebraic geometry, one usually fixes discrete invariants, such as the genus of a projective curve, and then one would like to have a distinct view on the set of objects with fixed invariants with respect to some equivalence relation. For small invariants it is sometimes possible to enumerate the equivalence classes and to provide normal forms. For bigger invariants this usually fails and a way to describe the objects is to construct a classifying space such that each point of this space corresponds to a unique equivalence class. In algebraic geometry this classifying space should again be an algebraic variety, together with certain functorial properties. These ideas lead to the notion of a fine, respectively coarse, moduli space ([MuF], [Ne]).
Classically, moduli spaces have been constructed for global algebraic objects such as projective varieties, or for vector bundles on a fixed projective variety. During the past years there has also been some progress in constructing moduli spaces for singularities (cf. [GHP]) and for Cohen-Macaulay modules on a fixed local ring of a curve singularity ([GP3], see also [Gr1] for a survey). Indeed, the methods of proof are constructible and can be transferred to algorithms and finally to programmes.
In the following, we describe an algorithm to compute a moduli space for isolated hypersurface singularities, following [GHP]. The algorithm has been developed and implemented in SINGULAR by T. Bayer ([Ba]).
, , be a
a semiquasihomogeneous power series, i.e.,
Two power series are called right equivalent, , if there exists a holomorphic coordinate change such that , or, equivalently, , where is the algebra automorphism corresponding to .
In a series of papers, V.I. Arnold classified all isolated hypersurface singularities w.r.t. right equivalence up to modality 2, by giving normal forms [AGV].
Here we should like to present an algorithm to compute a moduli space for semiquasihomogeneous power series with fixed principal part w.r.t. right equivalence. In [GHP], also a moduli space for contact equivalence was constructed, but that construction is more involved and not treated here.
To start with we need an algebraic variety which parametrises all semiquasihomogeneous power series (up to right equivalence) and then to identify equivalent objects. Indeed, equivalent objects belong to the same orbit of an algebraic group action and the aim is to compute explicitly the group, the action of the group and, finally, the quotient space.
Giving , we compute the set of exponents so that is a monomial basis of the Milnor algebra . This requires a standard basis computation for a local ordering (cf. [GP1]). Then we select and set with .
Usually there exist such that . However, we have the following fact (proved in [GHP] by using the Gauß-Manin connection): let , and assume . Then , that is for .
In the theorem of Arnold,
, which implies that is unique. Moreover,
is a normal subgroup of
, and the quotient
The following steps are needed for computing the moduli space:
SINGULAR example for computing the moduli space (we omit intermediate commands):
> LIB "qhmoduli.lib"; > ring R = 0, (x,y,z), ls; // define a local ring > poly f = x2y + x2z + y5 - z5; // principal partStep 0: compute a basis for the semi-universal unfolding
Step 1, 2 and 3: compute the equations of the stabilizer of , compute the induced action on , linearise the action with equivariant embedding
This shows already that moduli spaces have a complicated structure, even for relatively small examples.