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Surjectivity of quotient maps for algebraic \((\mathbb{C},+)\)-actions and polynomial maps with contractible fibers. (English) Zbl 1047.14031
Summary: We establish two results concerning algebraic \((\mathbb{C}, +)\)-actions on \(\mathbb{C}^n\). First, let \(\varphi\) be an algebraic \((\mathbb{C}, +)\)-action on \(\mathbb{C}^3\). By a result of M. Miyanishi [Normal affine subalgebras of a polynomial ring, in: Algebraic and topological theories. Kinokuniya, Tokyo 37–51 (1985)], its ring of invariants is isomorphic to \(\mathbb{C}[t_1,t_2]\). If \(f_1,f_2\) generate this ring, the quotient map of \(\varphi\) is the map \(F:\mathbb{C}^3\to\mathbb{C}^2\), \(x\mapsto(f_1(x), f_2 (x))\). By using some topological arguments we prove that \(F\) is always surjective. Second, we are interested in dominant polynomial maps \(F: \mathbb{C}^n\to\mathbb{C}^{n-1}\) whose connected componeuts of their generic fibers are contractible. For such maps, we prove the existence of an algebraic \((\mathbb{C},+)\)-action \(\varphi\) on \(\mathbb{C}^n\) for which \(F\) is invariant. Moreover we give some conditions so that \(F^*(\mathbb{C}[t_1, \dots,t_{n-1}])\) is the ring of invariants of \(\varphi\).

MSC:
14L30 Group actions on varieties or schemes (quotients)
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