The kernel of the Kodaira-Spencer map of the versal \(\mu\)-constant deformation of an irreducible plane curve singularity with \({\mathbb{C}}^ r\)-action (Q911668)

Let \(f(x,y)=x^ a+y^ b\), a and b relatively prime, and \(F(x,y,t)=f(x,y)+ \sum^{r}_{i=1}t_ im_{\mu-r+i} \), \(t=(t_ 1,...,t_ r)\), where \(\{m_ 1,...,m_{\mu}\}\) is a monomial basis of \({\mathbb{C}}[x,y]/(\partial f/\partial x,\partial f/\partial y)\) ordered by degree \((\deg(x^{\alpha}y^{\beta})= \alpha b+\beta a)\) and where \(m_{\mu -r+1},...,m_{\mu}\) have degree greater than \(\deg(f)=ab\). If \(X=\{F=0\}\subset {\mathbb{C}}^ 2\times {\mathbb{C}}^ r\) then the projection \(\pi: X\to {\mathbb{C}}^ r\) defines a family of plane curves which have an isolated singularity at 0 of fixed topological type and \(\pi\) is versal with respect to this property. The authors are interested in constructing coarse moduli spaces for plane curve singularities with the same topological type as \(\{f=0\}\). For this purpose one has to collapse the analytically trivial subfamilies which are given as the integral manifolds of the kernel L of the Kodaira-Spencer map \(Der_{{\mathbb{C}}}{\mathbb{C}}[t]\to {\mathbb{C}}[x,y,t]/(F,\partial F/\partial x,\partial F/\partial y)\), \(\partial \mapsto\) class of \(\partial F.\) The authors present a fast algorithm for computing the Lie-algebra L for given a and b. This would be the first step in constructing a stratification of \({\mathbb{C}}^ r\) into locally closed invariant subvarieties on which the action of L admits a geometric quotient giving the desired coarse moduli space.