When a chain of blowups defines an automorphism of \({\mathbb{C}}^2\) (Q1585632)

The paper gives a new proof of the following Vitushkin theorem: Let \(\sigma_1 :V\to \mathbb C \mathbb P^2=\mathbb C^2\cup L\) be a composition of blow-ups at points in the line \(L\) at infinity. \(\sigma_1 \) is said to define an automorphism of \(\mathbb C^2\) if and only if there exists a similar composition of blow-ups \(\sigma_2:V\to \mathbb C \mathbb P^2 \) at points in the line \(L\) at infinity with the same exceptional divisor as \(\sigma_1 \) such that proper transformation of \(L\) by \(\sigma_2 \) is the last glued curve \(E'\) by \(\sigma_1 \). Then, in fact, \(\sigma_2\sigma_1^{-1}|\mathbb C^2 \) is an automorphism of \(\mathbb C^2 \). Then the Vitushkin theorem says: \(\sigma_1 \) defines an automorphism of \(\mathbb C^2 \) if and only if there exists a homology class \(S\in H_2(V,\mathbb Z)\) such that \(S^2=1\), \(S\cdot K_V=-3\), \(S\cdot E'=1\) and \(S\cdot E_i=0\) for the remaining glued curves \(E_i\).