AK-invariant, some conjectures, examples and counterexamples (Q2724127)

This paper surveys what is known about the AK-invariant of an algebraic variety, and which important open questions remain. The author first defined this invariant to distinguish certain 3-folds from affine space \({\mathbb C}^3\), and for this reason most other authors refer to it as the ML-invariant. Specifically, for an algebraic variety \(X\), \(\text{ML}(X)\) is the intersection of all fixed rings of algebraic \(G_a\)-actions on \(X\), or equivalently the intersection of all kernels of locally nilpotent derivations of the coordinate ring \(k[X]\). This ring is also referred to as the ring of absolute constants of \(X\). For example, Makar-Limanov showed that for the Russell cubic 3-folds \(X\), \(\text{ML}(X)\neq {\mathbb C}\), whereas \(\text{ML}({\mathbb C}^3)={\mathbb C}\). Some of the most important questions about \(\text{ML}(X)\) are connected to the cancellation problem. These are embodied in the 3 conjectures found in the paper, for example:NEWLINENEWLINENEWLINEConjecture 1: \(\text{ML}(A)= \text{ML}(A[x])\) when \(A\) is a UFD. NEWLINENEWLINENEWLINEAny reader interested in this important new idea will find this paper to be a well-written, informal introduction to the subject.