Smoothness of the images of members of a linear system under an endomorphism of the affine plane
Endomorphisms of the complex plane are studied. Let \(f,g\in \mathbb C [x,y]\) and let \(\varphi=(f,g)\) be an endomorphism of \( \mathbb C^2.\) For \(b\in \mathbb C\) let \(C_b=\{x=b\}.\) The question whether the image \(\varphi (C_b)\) is smooth for a general \(b\) is connected to the Jacobian conjecture. For a pair \(f,g\) and a point \(a\in \mathbb C \) the authors define a special polynomial \(H_a\) and prove that the smoothness of \(\varphi (C_b)\) for general \(b\) is equivalent to the non-vanishing of \(H_a\) for almost all \(a\in \mathbb C.\) Then they introduce a hypersurface \(V\in \mathbb C ^3\) such that its coordinate ring is \(\mathbb C [x,f,g].\) The normalization \(\tilde V\) of \(V\) appears to be a \(\mathbb C ^1\)-fibration with only cyclic singularities. The inclusion \(\mathbb C [x,f,g]\subset \mathbb C [x,y] \) induces a morphism \(\pi:V\to \mathbb C^2.\) It is provend that: - if \(\pi\) is finite, then \(\tilde V\) is isomorphic to \(\mathbb C^2/G,\) where \(G\) is a cyclic group; - if \(\mathbb C (x,f,g)= \mathbb C (x,y) ,\) then \(\tilde V\) is isomorphic to \(\mathbb C^2.\)
scientific article; zbMATH DE number 5527492
scientific article