Surjectivity of quotient maps for algebraic $(\mathbb{C},+)$-actions and polynomial maps with contractible fibres

Details Surjectivity of quotient maps for algebraic $(\mathbb{C},+)$-actions and polynomial maps with contractible fibres Surjectivity of quotient maps for algebraic $(\mathbb{C},+)$-actions and polynomial maps with contractible fibres

2006-02-10

arxiv.org/abs/math/0602227v1

Transformation Groups, Vol. 7, No. 1, 2002, pp. 3-14

Mathematics

Algebraic Geometry

12 pages

Scientific paper

In this paper, we establish two results concerning algebraic $(\mathbb{C},+)$-actions on $\mathbb{C}^n$. First let $\phi$ be an algebraic $(\mathbb{C},+)$-action on $\mathbb{C}^3$. By a result of Miyanishi, 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 $\phi$ 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 components of their connected fibres are contractible. For such maps, we prove the existence of an algebraic $(\mathbb{C},+)$-action $\phi$ on $\mathbb{C}^n$ for which $F$ is invariant. Moreover we give some conditions so that $F^*(\mathbb{C}[t_1,...,t_{n-1}])$ is the ring of invariants of $\phi$.

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