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Salem numbers and Enriques surfaces. (English) Zbl 1404.14043

Given an automorphism \(g\) of a smooth projective algebraic surface \(X\) over an algebraically closed field \(\mathbb K\), one calls dynamical degree \(\lambda(g)\) of \(g\) the eigenvalue of \(g^*\) on \(\text{Num}(X)\) having maximal absolute value. Hyperbolic automorphism are such that \(\lambda(g)>1\).
The purpose of the paper is to show hyperbolic automorphisms with small dynamical degree on an Enriques surface and look for the smallest Salem number which can be obtained as dynamical degree of such an automorphism. It is known that the smallest known Salem number, the Lehmer number, can be obtained as dynamical degree of an automorphism of a rational surface or a \(K3\) surface, but not an Enriques surface (result in [K. Oguiso, Adv. Stud. Pure Math. 60, 331–360 (2010; Zbl 1215.14039)]). This motivates the question of the author in the current paper.
Methods to find Salem numbers include the study of elliptic pencils and projective involutions on the quartic surface \(H(C)\) defined by the determinant of the Hessian matrix of the polynomial defining the nonsingular cubic surface \(C\subset\mathbb P^3\). The method is improved considering Eckardt points on \(C\) and also with the introduction of Coble surface of Hessian type.
However, as the author says, some of the obtained Salem numbers do not seem optimal, that is why at the end of the article we are left with some further questions on the smallest Salem number in a given degree realized on an Enriques or Coble surface, or the dimension of an equi-hyperbolic family of projective surfaces.

MSC:

14J28 \(K3\) surfaces and Enriques surfaces
14J50 Automorphisms of surfaces and higher-dimensional varieties
14J26 Rational and ruled surfaces
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)

Citations:

Zbl 1215.14039
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References:

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