Rational surface automorphisms preserving cuspidal anticanonical curves

Automorphism groups of rational surfaces

“…Now assume that X admits a cuspidal anticanonical curve Y , and that there is an automorphism F ∈ Aut(X, Y ) such that F is not birationally conjugate to a linear map on P 2 , and the determinant δ(F ) is not a root of unity. Then a result mentioned in [15] says that there is a birational morphism π : X → P 2 , which is decomposed as (2)…”

“…P Again let F ∈ Aut(X, Y ) be an automorphism such that F is not birationally conjugate to a linear map on P 2 , and the determinant δ(F ) is not a root of unity, and let w be the element of W N realized by F . Then it turns out (see [15]) that δ(F ) is an eigenvalue of w and that there is an eigenvector ξ of w with eigenvalue δ(F ) such that composition (2) is expressed as π = π ξ : X → P 2 . Moreover, the characteristic polynomial of w is expressed as χ w (t) := det(tI − w) = R w (t)S w (t), where R w (t) is a product of cyclotomic polynomials and S w (t) is an irreducible non-cyclotomic polynomial, which is known to be the so-called Salem polynomial (see e.g.…”

“…which is independent of the choice of coordinates on Y * , is called the determinant of F . Remark 1.1 A rational surface X admitting a cuspidal anticanonical curve Y may be obtained from a blowup π : X → P 2 of points lying on (the smooth locus of) a cuspidal cubic curve in P 2 (see [15], Proposition 2.1). Note that a cuspidal cubic curve is characterized as a reduced cubic curve C with the properties that (1) it is irreducible and (2) it admits an automorphism whose determinant is not a root of unity, which is a consequence of the classification of Pic 0 (C) (see [10]).…”

“…The condition (2) is essential in our study of automorphisms on rational surfaces, and it seems to be difficult to characterize automorphisms whose determinants are roots of unity in a simple way (see e.g. [4,14,15]). Moreover, the condition (1) enables us to construct more automorphisms preserving anticanonical curves.…”

Entropy of Real Rational Surface Automorphisms

“…One might hope that the holomorphic version of the Lefschetz fixed point formula [GH,Chapter 3.4] f (p)=p 1 det(id − Df (p)) = 1 would be useful here and for proving Lemma 5.2 below in a more conceptual fashion. That is, the multipliers at the two fixed points on C(R) are known (see [Ueh2,Lemma 5.2]), and the product of the multipliers at each of the other two fixed points must equal δ. This leaves us to determine only one multiplier at each of the latter.…”