# Is the factorial cDV-singularity $T_1^2 + T_2^3 + T_3^4T_4$ any quotient of any affine space by any group?

the reason for my question is the following: the two-dimensional canonical singularities are the ADE-singularities, which all are quotients of either affine space or another ADE-singularity by finite abelian Groups and even more, quotients of affine 2-space by finite solvable Groups but one - namely $E_8$, which is factorial and a quotient of $A_1$ by the perfect Group $A_5$.

Now a generalization of the ADE-singularities in Dimension 3 are the compound du Val singularities, which are canonical and are formally equivalent to $f(x,y,z)+tg(x,y,z,t)$, where f is the polynomial of an ADE-singularity.

Now some of these are again quotients of affine 3-space, while a lot more than the one before ($E_8$) are factorial. For example the one in the heading. Which means in particular, that they cannot be a Quotient of affine space by a solvable Group.

As you may see now, it would be very nice, if these singularities were also some quotients by any Groups. Any idea? I am quite sure that it can't be a finite Group and affine 3-space.

Thank you and looking forward to your suggestions, Lukas

• For the record, at time of writing this question has attracted a vote to close as "unclear what you're asking". I strongly disagree with this reason given – Yemon Choi May 8 '17 at 11:17

Let $$B=k[x,y,z,u]$$ be a polynomial ring in four variables over a field $$k$$ of characteristic $$0$$, and let $$X$$ be 4-dimensional affine space over $$k$$. A locally nilpotent derivation $$D$$ of $$B$$ induces an algebraic action of the additive group $$G=(k,+)$$ on $$X$$ via the exponential mapping $${\rm exp} (tD)$$, $$t \in k$$. The ring of invariants of the group action is equal to the kernel of the derivation, which is a UFD of transcendence degree $$3$$ over $$k$$ (It is not known whether the kernel must be affine).
Consider the triangular derivation $$D: u \to z \to y \to x^n$$ and $$x \to 0$$, where $$n$$ is a positive integer. The kernel of $$D$$ is of the form $$k[x,f,g,h]$$, where $$x^{2n}u+f^3+g^2=0$$. The corresponding algebraic variety $$X/G$$ is therefore of the type you are looking for.