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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)
##### Keywords:
algebraic actions; quotient map
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##### References:
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