# The dynamical degrees of rational surface automorphisms

## 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

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 .

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 .

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

This theorem was shown by Uehara *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 .

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 *marked blowup* of McMullen .

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 .

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 .

To have a dynamical degree as stated by McMullen ,

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 It is shown in ’s.

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:

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.

This article is organized as follows: Section 2 discusses Coxeter groups and their essential elements. In Sections 3–5, we generalize McMullen’s definitions

## 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 in -orbits 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

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The group homomorphism is injective.

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.

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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

### 2.3. Essential elements

Let be a set of generators 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

The essential elements of and their properties were considered by Krammer

It is known

In this article, we will focus on essential elements.