Rational maps of \(\mathbb{P}^{n}\) with prescribed fixed points and the smooth conic case (Q2878801)

In the paper under review, the authors classify rational maps \({\mathbb P}^n\dashrightarrow \mathbb P^n\) defined by forms of degree \(r+1\) in \(k[x_0, \ldots,x_n]\), where \(k\) is an algebraically closed field, whose set of fixed points contains a prescribed irreducible part defined by forms of degree \(r+1\). The proof of their main result (Theorem 2.1) has a gap in its part (2) and its part (3) must be restated in order to pay attention to that; in fact the authors have already written an erratum where an additional hypothesis was included in that statement [ibid. 14, No. 7, Article ID 1592001, 5 p. (2015; Zbl 1323.14011)]. Moreover, taking into account that additional hypothesis, Remark 2.3 (1) is okay but (2) need to be rewritten, as done in the erratum: indeed, their conclusion for plane Cremona maps fixing pointwise a curve does not work because, for example, there are a lot of non de Jonquières Cremona involutions in that situation (see [\textit{L. Bayle} and \textit{A. Beauville}, Asian J. Math. 4, No. 1, 11--17 (2000; Zbl 1055.14012)]).NEWLINENEWLINENext they consider the special case where \(r=1\) and the prescribed fixed part is a (maybe reducible or non reduced) hypersurface of degree 2, i.e. the one of quadratic rational maps fixing the points of a hyperquadric, and characterize when such a map become birational. In the particular case where \(n\) is 2, i.e., the one of quadratic rational maps of \({\mathbb P}^2\) fixing an (this time) irreducible conic, they classify these maps up to conjugate by a linear automorphism of the plane.