Rational surface automorphisms preserving cuspidal anticanonical curves (Q289866)

Let \(\pi : X \rightarrow \mathbb{P}^2\) be the blow up of \(\mathbb{P}^2\) at finitely many points and let \(F : X \rightarrow X\) be an automorphism of \(X\) preserving a curve \(C\in |-K_X|\) with at most cusp singularities. The main result of this paper is that if \(\pi F \pi^{-1}\) is not an automorphism of \(\mathbb{P}^2\), there exists a birational map \(\phi : \mathbb{P}^2 \dasharrow \mathbb{P}^2\) fixing \(\tilde{C}=\pi_{\ast}(C)\) which lifts to an automorphism \(\bar{\phi}\) of \(X\) such that \(F\) is birationally equivalent to \(\bar{\phi}\). The automorphism \(\bar{\phi}\) is constructed by a method called realization of orbit data that is described in the paper and which first appeared in [\textit{T. Uehara}, Ann. Inst. Fourier 66, No. 1, 377--432 (2016; Zbl 1360.14042)].