On automorphisms of flexible varieties. (English) Zbl 1485.14116
The well-known Abhyankar-Moh Theorem on embedding of the line in the plane asserts that any polynomial injective mapping of \(\mathbb{C}\) into \(\mathbb{C }^{2}\) can be extended to a polynomial automorphism of \(\mathbb{C}^{2}\) (we treat the domain \(\mathbb{C}\) as embedded in \(\mathbb{C}^{2}\), \(\mathbb{ C\cong C\times }\{0\}\subset \mathbb{C}^{2})\) onto \(\mathbb{C}^{2}.\) The authors consider a generalization (in many aspects) of this extension problem: when can an isomorphism \(f:Y_{1}\rightarrow Y_{2}\) of subschemes of a variety \(X\) (over algebraically closed field \(k\) of characteristic zero) be extended to an automorphism of the whole \(X\) ? The main results concern the case \(Y_{1},Y_{2}\) are curves (one-dimensional suschemes of \(X)\) and \(X\) a flexible variety (normal quasi-affine variety for which algebraic automorphisms of \(X\) generated by one-parameter unipotent subgroups of \( \operatorname{Aut}(X)\) act transitively on the smooth part of \(X).\) They give several criteria under which such extensions exist.
Reviewer: Tadeusz Krasiński (Łódź)
MSC:
| 14R99 | Affine geometry |
| 32M17 | Automorphism groups of \(\mathbb{C}^n\) and affine manifolds |
| 14J50 | Automorphisms of surfaces and higher-dimensional varieties |
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