Salem Numbers and Enriques Surfaces

ABSTRACT

It is known that the dynamical degree of an automorphism g of an algebraic surface S is lower semi-continuous when (S, g) varies in an algebraic family. In this article, we report on computer experiments confirming this behavior with the aim to realize small Salem numbers as the dynamical degrees of automorphisms of Enriques surfaces or rational Coble surfaces.

Keywords:

• Salem numbers
• Enriques surfaces
• dynamics

2000 AMS Subject Classification:

• 14J28
• 14J50
• 14J26
• 37F45

Acknowledgments

This article owes much to the conversations with Paul Reschke whose joint work with Bar Roytman makes a similar experiment with automorphisms of K3 surfaces isomorphic to a hypersurface in the product . I am also grateful to Keiji Oguiso and Xun Yu for useful remarks on my talk on this topic during the “Conference on K3 surfaces and related topics” held in Seoul, Korea, November 16–20, 2015. I thank the referee whose numerous critical comments have greatly improved the exposition of the article.

Notes

1 Over , the group is isomorphic to the subgroup of modulo torsion generated by algebraic cycles.

2 We call such curves ( − 2)-curves because they are characterized by the property that their self-intersection is equal to −2.

3 The terminology is due to S. Mukai.

4 In [Dolgachev and Keum 02], p. 3034, the second case was overlooked.

5 According to [Shimada], 100 hours of computer computations using random choice of automorphisms suggests that 4.33064… is the minimal Salem number realized by an automorphism of a general Enriques surface of Hessian type.

6 In this case, the Clebsch diagonal surface must be given by the equation s₁ = s₃ = 0, where s_k are elementary symmetric polynomials in 5 variables.

7 The Hessian surfaces and were studied in detail in [Dardanelli and van Geemen 07], in particular, the authors compute the Picard lattices of their minimal resolutions.

8 It is proved in [Mukai and Ohashi 15], Theorem 1, that this group is the whole group of automorphisms of the surface.

9 This remark is due to V. Nikulin.