Moduli spaces and invariants

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]).

Let $\mathbf{w} = (w_1, \dots, w_n) \in \mathbf{Z}^n$, $w_i > 0$, be a weight vector and $f \in \mathbf{C}\{x_1, \dots, x_n\}$ a semiquasihomogeneous power series, i.e.,

$$f = f_0 + \sum_{\langle \mathbf{w},\alpha\rangle > d} c_\alp... ... \sum_{\langle \mathbf{w},\alpha\rangle = d} c_\alpha x^\alpha$$

such that the quasi-homogeneous (or weighted homogeneous) principal part $f_0$ has an isolated singularity at the origin. We denote the class of all such power series (resp. singularities) by $\mathcal{C}_{f_0}$.

Two power series $f,g$ are called right equivalent, $f \stackrel{r}{\sim} g$, if there exists a holomorphic coordinate change $\phi^{\char93 }: (\mathbf{C}^n,0) \rightarrow (\mathbf{C}^n,0)$ such that $f = g \circ \phi^{\char93 }$, or, equivalently, $f = \phi(g)$, where $\phi \in Aut(\mathbf{C}\{x_1, \dots, x_n\})$ is the algebra automorphism corresponding to $\phi^{\char93 }$.

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 $f_0$, we compute the set of exponents $B \subset \mathbf{N}^n$ so that $\{x^\alpha \mid \alpha \in B\}$ is a monomial basis of the Milnor algebra $M_{f_0} = \mathbf{C}\{x_1, \dots, x_n\}/\langle f_{0,x_1}, \dots, f_{0,x_n}\rangle$. This requires a standard basis computation for a local ordering (cf. [GP1]). Then we select $B_- = \{\alpha \in B \vert \langle \mathbf{w},\alpha\rangle > d\}$ and set $T_- = \mathbf{C}^k$ with $k = \vert B_-\vert$.

The polynomial

$$F_t(x) = f_0(x) + \sum_{\alpha \in B_-} t_\alpha x^\alpha$$

is the miniversal $\mu$-constant unfolding of $f_0$. By a theorem of Arnold ([AGV]), for any $f \in \mathcal{C}_{f_0}$ there exists a $t \in T_-$ such that $f \stackrel{r}{\sim} F_t$. The next step in the algorithm is to compute, for a given $f \in \mathcal{C}_{f_0}$, a coordinate change $\phi$ and a $t \in T_-$ such that $\phi(f) = F_t$. The computation follows Arnold's proof, constructing $\phi$ degree by degree until the maximal weighted degree $\langle \mathbf{w}, \alpha \rangle$, $\alpha \in B_-$.

Usually there exist $t \neq t' \in T_-$ such that $F_t \stackrel{r}{\sim} F_{t'}$. However, we have the following fact (proved in [GHP] by using the Gauß-Manin connection): let $f,g \in \mathcal{C}_{f_0}$, $\varphi \in Aut \mathbf{C}\{x\}$ and assume $\varphi(f) = g$. Then $ord_\mathbf{w}(\varphi) \ge 0$, that is $ord_\mathbf{w}\bigl(\varphi(x_i) - x_i\bigr) \ge w_i$ for $i = 1, \dots, n$.

In the theorem of Arnold, $ord_\mathbf{w}(\phi) > 0$, which implies that $t \in T_-$ is unique. Moreover, $Aut_{> 0}\mathbf{C}\{x\}=\{\varphi \in Aut\mathbf{C}\{x\} \mid ord_\mathbf{w}(\varphi) > 0\}$ is a normal subgroup of $Aut_{\geq 0} \mathbf{C}\{x\}=\{\varphi \in Aut\mathbf{C}\{x\} \mid ord_\mathbf{w}(\varphi) \ge 0\}$, and the quotient

$$G^{\mathbf{w}}= Aut_{\ge 0} \mathbf{C}\{x\}/Aut_{> 0} \mathbf{C}\{x\}$$

acts algebraically on $T_-$. Let $G_{f_{0}}^{\mathbf{w}} \subset G^{\mathbf{w}}$ denote the subgroup which fixes $f_0$ and denote by $E_{f_0} \subset Aut(T_-)$ the image of $G_{f_{0}}^{\mathbf{w}}$. Then $E_{f_0}$ is a finite group acting algebraically on $T_-$ and the geometric quotient $T_-/E_{f_0}$ is the desired coarse moduli space for unfoldings in $\mathcal{C}_{f_0}$ modulo right equivalence (cf. [GHP]).

The following steps are needed for computing the moduli space:

0.

Compute miniversal $\mu$-constant unfolding,

1.

compute $G_{f_{0}}^{\mathbf{w}}$,

2.

compute the action of $G_{f_{0}}^{\mathbf{w}}$ on $T_-$ using Arnold's theorem,

3.

compute $E_{f_0}$ and linearise to get $E'_{f_0}$ acting linearly on some $\mathbf{C}^\ell$, $\ell \ge k$, and compute an equivariant embedding $i : T_- \hookrightarrow \mathbf{C}^\ell$,

4.

determine generating invariant polynomials for $E'_{f_0}$,

5.

determine the relations between the invariants to get the equations for $i(T_-)/E'_{f_0} \cong T_-/E_{f_0}$, which is the desired 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 part

Step 0: compute a basis for the semi-universal unfolding

> ideal B = UpperMonomials(f); B;
B[1]=y3z3, B[2]=x2y3, B[3]=x2y2  

Hence, $F = f + t_1y^3z^3 + t_2x^2y^3 + t_3x^2y^2$ is the miniversal $\mu$-constant unfolding. The dimension of the moduli space is 3.

Step 1, 2 and 3: compute the equations of the stabilizer of $f$, compute the induced action on $T_-=\mathbf{C}^3$, linearise the action with equivariant embedding $T_- \hookrightarrow \mathbf{C}^4$

> list stab = StabEqn(f);  ....  // commands omitted
> actionid;           //linearised action of on   
actionid[1]=s(1)*t(1), actionid[2]=-s(3)*t(2)+s(3)*t(4)+s(5)*t(2)
actionid[3]=s(4)*t(3), actionid[4]=s(5)*t(4)

Step 4: compute generators for the invariant ring of $E'_f$

> def T = InvariantRing(groupid, actionid); setring T;
> invars;            //there are 21 invariants of degree 3 to 10
invars[1]=t(1)*t(4)^2, invars[2]=t(2)*t(3)*t(4)-t(3)*t(4)^2, ...

Step 5: compute equations of the moduli space

> def R4 = ImageVariety(V, invars);  //V is the ideal of
> setring R4; imageid;       //simplified equation of moduli space
imageid[1]=Y(5)^2-Y(4)*Y(6), imageid[2]=Y(3)*Y(5)-Y(2)*Y(6), ...
imageid[55]=9*Y(1)^5+2816*Y(2)^2*Y(6)*Y(9)+296*Y(6)*Y(7)^2
  -152*Y(2)*Y(6)*Y(8)-960*Y(6)*Y(7)*Y(9)-9*Y(6)*Y(10)

Hence, the moduli space for $x^2 y + x^2 z + y^5 - z^5$ is a 3-dimensional affine subvariety of $\mathbf{C}^{10}$ defined by 55 equations of degrees between 2 and 5.

This shows already that moduli spaces have a complicated structure, even for relatively small examples.

Problems:

1. Extend the algorithms to construct moduli spaces for singularities with respect to contact equivalence. This will contain completely new parts since we need not only handle finite groups but unipotent groups.

2. Moduli spaces for torsion free modules on curve singularities have been constructed in [GP3] with constructive proofs. Again unipotent group actions come into play. It would be desirable to develop and implement algorithms and test conjectures related to the structure of these moduli spaces.