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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 \(\varphi\).

14L30 Group actions on varieties or schemes (quotients)
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[1] [Ch] ?. ?. ?????,??????????? ????????????? ?????????, ?????, ?., 1985. English transl.: E. M. Chirka,Complex Analytic Sets, Kluwer Academic Publishers, 1989.
[2] [Da] D. Daigle,On some properties of locally nilpotent derivations, J. Pure and Appl. Algebra114 (1997), 221-230. · Zbl 0885.13003
[3] [De] J. K. Deveney, Ga-actions on ?3 and ?7, Commun. in Algebra22(15) (1994), 6295-6302. · Zbl 0867.13002
[4] [D-F] J. K. Deveney, D. R. Finston,On locally trivial G a-actions, Transformation Groups2 (1997), No. 2, 137-145. · Zbl 0913.14012
[5] [Kr] H. Kraft,Challenging problems on affine n-space, SĂ©minaire Bourbaki802, 1994-95.
[6] [Miy] M. Miyanishi,Normal affine subalgebras of a polynomial ring, in:Algebraic and Topological Theories. To The Memory of Dr. Takehito Miyata, Kinokuniya, Tokyo (1985), 37-51.
[7] [Mum] D. Mumford,Algebraic Geometry I: Complex Projective Varieties, Grundl. der math. Wissensch., Vol. 221, Springer-Verlag, Berlin, Heidelberg, New York, 1976. Russian transl.: ?. ???????, ?????????????? ?????????. ??????????? ??????????? ????????????, ???, ?., 1979. · Zbl 0356.14002
[8] [Wi] J. Winkelmann, On free holomorphic ?-actions on ? n and homogenous Stein manifolds, Math. Ann.286 (1990), No. 1-3, 593-612. · Zbl 0708.32004
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