Algebraic group actions and the structure of affine algebraic varieties

1.线性化问题我们已经在与线性化问题密切相关的抵消问题上得到了一些结果：(1)存在一个无穷维仿射伪平面族，它不具有抵消性质。此外，Tom Dieck曲面被刻画为具有代数环面的非平凡作用的仿射伪平面。(2)带切片的三角导数的核是多项式环。2.广义Jacobian问题(1)我们研究了具有A^1-原函数的光滑仿射曲面的Jacobian问题，特别是仿射伪平面。通过与R.V.Gurjar和P.Russell的合作，我们在一定程度上弄清了仿射伪平面的结构，但不能解决仿射伪平面的广义Jacobian问题。(2)Miyanishi证明了当仿射伪平面的广义Jacobian问题满足广义Sard性质时，它有一个肯定的答案。

1.Linearization ProblemWe have had some results on the Cancellation Problem which is closely related to the Linearization Problem.(1)There exists an infinite dimensional family of affine pseudo-planes without cancellation property. Furthermore, tom Dieck surfaces are characterized as affine pseudo-planes with nontrivial actions of an algebraic torus.(2)The kernel of a triangular derivation with a slice is a polynomial ring.2.Generalized Jacobian Problem(1)We studied the Jacobian Problem for smooth affine surfaces with A^1-fibrations, especially, affine pseudo-planes. We succeeded in making clear of the structure of affine pseudo-planes to some extent with joint work with R.V.Gurjar and P.Russell, but could not resolve the Generalized Jacobian Problem for affine pseudo-planes.(2)Miyanishi showed that the Generalized Jacobian Problem for affine pseudo-planes has an affirmative answer if they satisfy the generalized Sard property.