The dynamical degrees of rational surface automorphisms

By Kyounghee Kim

Abstract

The induced action on the Picard group of a rational surface automorphism with positive entropy can be identified with an element of the Coxeter group associated to diagram. It follows that the set of dynamical degrees of rational surface automorphisms is a subset of the spectral radii of elements in the Coxeter group. This article concerns the realizability of an element of the Coxeter group as an automorphism on a rational surface with an irreducible reduced anticanonical curve. For any unrealizable element, we explicitly construct a realizable element with the same spectral radius. Hence, we show that the set of dynamical degrees and the set of spectral radii of the Coxeter group are, in fact, identical. This has been shown by Uehara [Ann. Inst. Fourier (Grenoble) 66 (2016), pp. 377–432] by explicitly constructing a rational surface automorphism. This construction depends on a decomposition of an element of the Coxeter group. Our proof is conceptual and provides a simple description of elements of the Coxeter group, which are realized by automorphisms on anticanonical rational surfaces.

1. Introduction

A compact complex surface admitting an automorphism is rare. However, it is shown Reference 2Reference 3Reference 17 that there are infinite families of rational surfaces admitting automorphisms. Several methods for constructing rational surfaces and studying the dynamics of automorphisms on them have been considered, and the dynamical properties of such automorphisms have been studied in several recent papers (for example, Reference 4Reference 5Reference 6Reference 15).

Suppose is a rational surface obtained by blowing up a set of (possibly infinitely near) points in . The Picard group is generated by the class of a strict transform of a generic line in and the classes of total transforms of exceptional curves over points in . There is a natural isomorphism between with a Minkowski inner product and a Picard group . By Nagata Reference 18Reference 19, if is an automorphism with infinite order, then there is an element of the Weyl group generated by a set of reflections acting on such that the induced action is equivalent to via the isomorphism , in other words, the following diagram commutes.

\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} \mathbb{Z}^{1,n} \arrow[r, "\omega"] \arrow[d, "\phi",labels=left ] & \mathbb{Z}^{1,n} \arrow[d, "\phi" ] \\ Pic(S) \arrow[r, "F_*"] & Pic(S) \end{tikzcd}

In this case, we say realizes . The generators of this Weyl group are reflections through a set of vectors , such that

This Weyl group is isomorphic to the Coxeter group associated to the Coxeter diagram . (See Reference 12.)

If is a rational surface automorphism, the dynamical degree is given by

Let be the set of dynamical degrees of rational surface automorphisms and be the set of spectral radii of Weyl group elements:

where denotes the dynamical degree and denotes the spectral radius. According to Diller and Favre Reference 8, the dynamical degree of an automorphism is given by the spectral radius of . Thus, we have . In this article, we show .

Theorem A.

For any , there is an element such that for a rational surface automorphism with . Thus we have

This theorem was shown by Uehara Reference 22 by constructing an automorphism with a given dynamical degree as a composition of quadratic birational maps. The main idea in Reference 22 is identifying possible orbit data (See Section 7.3 for the definition.) for quadratic birational maps. The brief sketch of Uehara’s construction is the following: Let be a reflection through , and be a reflection through for with . Also, let be a span of reflections through . With these notation, we see that for any , we have for some . Thus for with we have

where

and is a set of three distinct integers in . By inspecting the above decomposition, for each , one can obtain the smallest nonnegative integers and such that

This allows to obtain orbit data (See Section 7.3 for the definition.) for a quadratic birational map whose critical images correspond to for each . By assuming all fix a given cusp cubic , we have and for some . Diller in Reference 7 showed that the multiplier and the parameters for three critical images uniquely determine (up to a linear conjugacy) a birational map properly fixing a cubic with arithmetic genus . By repeatedly applying Diller’s construction, Uehara obtained a system of equations of to get an explicit formula for the desired birational map .

Uehara’s construction required expressing as a composition of generators of . However, this is a difficult task. Furthermore, the construction depends on the orbit data, which is not uniquely determined by .

In this article, we generalize McMullen’s construction in Reference 17 by allowing infinitely near points in the base locus with the assumption of the existence of an anticanonical curve. The isomorphism together with the base points is called the marked blowup of . McMullen Reference 17 constructs a rational surface automorphism from a marked blowup with an anticanonical curve provided that all base points are distinct points in . A brief sketch of McMullen’s construction is as follows:

Suppose . If and give two markings for , each marking gives a projection to . Each marking would have a different base locus, and these two markings are associated with a birational map , which does not necessarily lift to automorphism.

The most well-known example would be the case with , and is the Cremona involution , the reflection through the vector . Let be a marked blowups with where is the class of the strict transform of generic line in and is the class of the exceptional curve over a point . Let be the corresponding blowup along three distinct points . The marked blowup with the projection satisfies and thus the class , is the class of exceptional curves over a point .

\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd}[column sep=small] & S \arrow["\pi",dl,labels=above left] \arrow[dr,"\pi' "] \\ \p^2 \arrow[rr, "f", dashed] & & \p^2 \end{tikzcd}

Since , we have where is a conic passing through while . Also, the exceptional curve over satisfies , and where is the line joining . The class of the strict transform of the line joining , is given by . Thus we see the birational map is quadratic and satisfies

In addition, if there is an automorphism on sending one base locus to another, by composition of an automorphism and a birational map , we can obtain an automorphism on by lifting .

\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd}[column sep=small] & S \arrow["\pi'",dl,labels=above left] \arrow[rr, "F"] && S \arrow["\pi",dl,labels=above left] \arrow[dr,"\pi' "] \\ \p^2 \arrow[rr, "g"] & &\p^2 \arrow[rr, "f", dashed] & & \p^2 \end{tikzcd}

To have a dynamical degree , as stated by McMullen Reference 16, we need to blow up at least points in . Thus, if two sets of points with are in general position, an automorphism on sending to is special, or in fact, might not exist.

One can overcome this difficulty by imposing the condition of the existence of an anticanonical curve . If a point in the base locus has no infinitely near points in the base locus, then is in the strict transform of . Suppose are in the base locus satisfying a point is an infinitely near point of of order and no other points are infinitely near to , then there is a successive blowups of a single point with . It follows that each strict transform of in contains . In either case, the set of base points in belongs to the projection . If is a rational curve with arithmetic genus , it is easy to tell whether there is such an automorphism from to .

We say is essential if is not conjugate into a proper subgroup of generated by ’s. It is shown in Reference 13 that there is a nonessential element such that is not realizable by a rational surface automorphism. This is shown by counting periodic points using the Lefschetz fixed point formula. In fact, this is conjugate to the Coxeter element , and it is known Reference 2Reference 17 that is realized by a rational surface automorphism. In this article, we let denote the set of essential elements in and focus on realizing essential elements in .

Let with the complex Minkowski form. For each we defined a marked blowup (See Section 6 for the explicit construction.) by blowing up points on a cuspidal cubic. Let us denote as the set of positive roots and as the set of negative roots. Also, let be a set of nodal roots in and let be the span of all reflections through nodal roots:

Theorem B.

Let and let be an essential element in with the spectral radius . Suppose is a leading eigenvector for . Then we have:

(1)

is realized by an automorphism on if and only if for all .

(2)

If is not realized by an automorphism on , there is such that is realized by a rational surface automorphism on and we have .

Also, via an example (see Theorems 8.4 and 8.5), we show that there is an essential element such that cannot be realized by any automorphisms on anticanonical rational surfaces. Then, using Theorem B, we find a realizable with the same spectral radius. The method in Theorem B can be done in finitely many steps due to Corollary 3.6.

Theorem C.

There is an essential element such that cannot be realized by an automorphism on an anticanonical rational surface.

This article is organized as follows: Section 2 discusses Coxeter groups and their essential elements. In Sections 35, we generalize McMullen’s definitions Reference 17 of marked blowups and marked cubics by allowing infinitely near points as base points. Also, we show that the set of nodal roots is finite, provided a unique anticanonical curve exists. In Section 6, we prove Theorem A and Theorem B. A couple of basic properties of quadratic birational maps and their orbit data are discussed in Section 7. In this section, we also briefly explain Diller’s construction in Reference 7 and a few more cases omitted in Reference 7. In Section 8, using a specific example, we prove Theorem C and show how to use Theorem B.

2. Coxeter group

Let be a lattice of signature with the basis such that

Using this basis as coordinates, the Minkowski inner product is given by

The canonical vector is given by . The subgroup orthogonal to is generated by a set of vectors

The generating vectors satisfy

For each , we can define a reflection on :

We see that , , and for . The group generated by those reflections fixes and preserves the Minkowski product, and we have

where

This group is called the Coxeter group. Let be a inner product space over with the basis . Then each gives the unique reflection in preserving . The root system is the -orbits in of generating vectors ’s.

2.1. Root system and simple roots

Each vector in is called a root, and each generating vector is called a simple root. Each is a linear combination with either for all or for all . We say is positive if with for all and is negative if for all . Let denote the set of all positive roots and for the set of negative roots.

Let us list a couple of useful facts about root systems. (See Reference 12 for details.)

The group homomorphism is injective.

.

For all , .

2.2. Length function

For each , is a word in and the length is given by the minimum possible length of a word in for . Note that can have several different words of length . One useful fact of the length of is its relation to the number of positive roots sent to negative roots Reference 12, that is, for , where is the set of positive roots sent by to negative roots.

2.3. Essential elements

Let be a set of generators of .

Definition 2.1.

We say is essential if there is no subgroup generated by a proper subset such that

Let us denote the set of essential elements of .

Thus if is nonessential, then can be identified as an element of a smaller Coxeter group generated by . If , then is finite and its Coxeter graph is given by for some Reference 12, Chapter 2. If , we may assume that . Then we can treat as an element of . If is generated by , then the spectral radius of . Thus, from the definition, we see that:

Lemma 2.2.

If with the spectral radius , then there is such that is conjugate to an essential element and .

The essential elements of and their properties were considered by Krammer Reference 14 and Paris Reference 20.

Proposition 2.3 Reference 14 .

Suppose a Coxeter group . Then

every essential element is of infinite order,

is essential if and only if is essential, and

if is essential then is a finite index subgroup of the centralizer of in .

It is known Reference 16Reference 20 that the Coxeter element is essential, and the product of in any other is conjugate to . In fact, the Coxeter element has the smallest spectral radius among all essential elements in .

Theorem 2.4 Reference 16, Theorem 1.2 .

Let . Then the spectral radius .

In this article, we will focus on essential elements.

Lemma 2.5.

If , then for each , there is such that .

Proof.

Suppose there is such that for all . Then