Affine pseudo-planes with torus actions
The paper under review is devoted to the study of affine pseudo-planes \(X\), that are smooth affine surfaces defined over \(\mathbb{C}\) which are endowed with an \(\mathbb{A}^1\)-filtration such that every fiber is irreducible and only one fiber is a multiple fiber. Recall that a smooth affine surface \(X\) is called an \(\text{ML}_i\)-surface if the Makar-Limanov invariant \(\text{ML}(X)\) has dimension \(i\) (\(i=0,1,2\)) over \(\mathbb{C}\). It is known that every affine pseudo-plane which is an \(\text{ML}_0\)-surface admits a hyperbolic \(\mathbb{G}_m\)-action. The authors prove that if there is a hyperbolic \(\mathbb{G}_m\)-action on \(X\), where \(X\) is an \({\text{ML}}_1\) affine pseudo-plane, then the universal covering \(\tilde X\) is isomorphic to an affine hypersurface \(x^ry=z^d-1\) in the affine 3-space \(\mathbb{A}^3\) and \(X\) is the quotient of \(\tilde X\) be the cyclic group of order \(d\) via the action \((x,y,z)\to (\xi x, \xi^{-r}y, \xi^az)\), where \(r\geq 2, d\geq 2, 0<a<d\) and \(\text{gcd}(a,d)=1\). Also it is shown that any \(\mathbb{Q}\)-homology plane with the logarithmic Kodaira dimension equal to \(-\infty\) and a nontrivial \(\mathbb{G}_m\)-action is an affine pseudo-plane. In the last section, the automorphism group of an \(\text{ML}_1\) affine pseudo-plane with a nontrivial \(\mathbb{G}_m\)-action is described.
scientific article; zbMATH DE number 5118041