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 $E_n, n\ge 10$ 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 $S$ 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 $S$ is a rational surface obtained by blowing up a set $P$ of $n$ (possibly infinitely near) points in $\mathbf{P}^2$. The Picard group $Pic(S)$ is generated by the class $\mathbf{e}_0$ of a strict transform of a generic line in $\mathbf{P}^2$ and the classes $\mathbf{e}_i$ of total transforms of exceptional curves over points in $P$. There is a natural isomorphism $\phi$ between $\mathbb{Z}^{1,n}$ with a Minkowski inner product and a Picard group $Pic(S)$. By Nagata Reference 18Reference 19, if $F:S \to S$ is an automorphism with infinite order, then there is an element $\omega$ of the Weyl group $W_n$ generated by a set of reflections acting on $\mathbb{Z}^{1,n}$ such that the induced action $F_* : Pic(S) \to Pic(S)$ is equivalent to $\omega$ via the isomorphism $\phi$, in other words, the following diagram commutes.
In this case, we say $F$realizes$\omega \in W_n$. The generators of this Weyl group $W_n$ are reflections through a set of vectors $\alpha _i \in \mathbb{Z}^{1.n}$,$i=0,\dots , n-1$ such that
Let $\Delta$ be the set of dynamical degrees of rational surface automorphisms and $\Lambda$ be the set of spectral radii of Weyl group elements:
$$\begin{gather*} \Delta = \{ \delta (F) : F \text{ is a rational surface automorphism.}\} \quad \text{and} \\ \Lambda = \{ \lambda (\omega ) : \omega \in W_n \text{ for some } n\ge 1 \} \end{gather*}$$
where $\delta$ denotes the dynamical degree and $\lambda$ denotes the spectral radius. According to Diller and Favre Reference 8, the dynamical degree $\delta (F)$ of an automorphism $F$ is given by the spectral radius of $F_*$. Thus, we have $\Delta \subset \Lambda$. In this article, we show $\Lambda = \Delta$.
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 $\kappa$ be a reflection through $\alpha _0$, and $\kappa _I$ be a reflection through $e_0 - e_i-e_j-e_k$ for $I \subset \{ 1, \dots , n\}$ with $|I|=3$. Also, let $\Sigma$ be a span of reflections through $\alpha _i, i\ge 1$. With these notation, we see that for any $r \in \Sigma$, we have $r^{-1} \kappa \, r = \kappa _I$ for some $I \subset \{ 1, 2, \dots , n\}$. Thus for $\omega = r_0\, \kappa \, r_1\, \kappa \, r_2\, \cdots \, \kappa \, r_m \in W_n$ with $r_i \in \Sigma , i = 1, \dots , m$ we have
$$\begin{equation*} \omega = r \kappa _{I_m} \cdots \kappa _{I_1}, \end{equation*}$$
and $I_j=\{ j_1, j_2, j_3\}$ is a set of three distinct integers in $\{1,2, \dots , n\}$. By inspecting the above decomposition, for each $j_i$, one can obtain the smallest nonnegative integers $t$ and $s_k$ such that
This allows to obtain orbit data (See Section 7.3 for the definition.) for a quadratic birational map $f_j$ whose critical images correspond to $e_{j_1}, e_{j_2}, e_{j_3}$ for each $j=1, \dots , m$. By assuming all $f_j$ fix a given cusp cubic $C = \{ \gamma (t) \}$, we have $e_{j_i} = \gamma (p_{j,i})$ and $f_j|_{C} \,\gamma (t) = \gamma (d_j t + c_j)$ for some $p_{j,j}, d_j, c_j \in \mathbb{C}$. Diller in Reference 7 showed that the multiplier $d_j$ and the parameters for three critical images uniquely determine (up to a linear conjugacy) a birational map properly fixing a cubic with arithmetic genus $=1$. By repeatedly applying Diller’s construction, Uehara obtained a system of equations of $p_{j, i}, d_j, c_j$ to get an explicit formula for the desired birational map $f = f_m \circ \cdots \circ f_1$.
Uehara’s construction required expressing $\omega \in W_n$ as a composition of generators of $W_n$. However, this is a difficult task. Furthermore, the construction depends on the orbit data, which is not uniquely determined by $\omega \in W_n$.
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 $\phi :\mathbb{Z}^{1,n} \to Pic(S)$ together with the base points is called the marked blowup of $S$. 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 $\mathbf{P}^2$. A brief sketch of McMullen’s construction is as follows:
Suppose $\omega \in W_n$. If $\phi$ and $\phi \circ \omega$ give two markings for $S$, each marking gives a projection to $\mathbf{P}^2$. Each marking would have a different base locus, and these two markings are associated with a birational map $f$, which does not necessarily lift to automorphism.
The most well-known example would be the case with $n=3$, and $\omega$ is the Cremona involution $\kappa$, the reflection through the vector $e_0-e_1-e_2-e_3$. Let $(S,\phi )$ be a marked blowups with $\phi (e_i) = \mathbf{e}_i$ where $\mathbf{e}_0$ is the class of the strict transform $\tilde{H}$ of generic line $H$ in $\mathbf{P}^2$ and $\mathbf{e}_i$ is the class of the exceptional curve $\mathcal{E}_i$ over a point $p_i \in \mathbf{P}^2$. Let $\pi :S \to \mathbf{P}^2$ be the corresponding blowup along three distinct points $\{p_1,p_2,p_3\} \in \mathbf{P}^2$. The marked blowup $(S, \phi \circ \kappa )$ with the projection $\pi ':S \to \mathbf{P}^2$ satisfies $\phi ( \kappa e_i) = \mathbf{e}'_i$ and thus the class $\mathbf{e}'_i = \mathbf{e}_0 - \mathbf{e}_1-\mathbf{e}_2-\mathbf{e}_3 + \mathbf{e}_i$,$i=1,2,3$ is the class of exceptional curves over a point $q_i \in \mathbf{P}^2$.
Since $\omega \mathit{e}_0 = 2 \mathit{e}_0 - \mathit{e}_1-\mathit{e}_2-\mathit{e}_3$, we have $\pi ' \tilde{H} = Q$ where $Q$ is a conic passing through $p_1,p_2,p_3$ while $\pi \tilde{H} = H$. Also, the exceptional curve $\mathcal{E}_i$ over $p_i$ satisfies $\pi (\mathcal{E}_i) = p_i$, and $\pi '(\mathcal{E}_i) = L'_{j,k}$ where $L'_{i,j}$ is the line joining $q_i,q_j$. The class of the strict transform of the line joining $p_i,p_j$,$L_{i,j}$ is given by $[L_{i,j} ]= \mathbf{e}_0 - \mathbf{e}_i-\mathbf{e}_j = \mathbf{e}'_k$. Thus we see the birational map $f$ is quadratic and satisfies
In addition, if there is an automorphism $g$ on $\mathbf{P}^2$ sending one base locus to another, by composition $f\circ g$ of an automorphism $g$ and a birational map $f$, we can obtain an automorphism on $S$ by lifting $f \circ g$.
To have a dynamical degree $>1$, as stated by McMullen Reference 16, we need to blow up at least $10$ points in $\mathbf{P}^2$. Thus, if two sets of points $P, P'$ with $|P|=|P'|\ge 10$ are in general position, an automorphism on $\mathbf{P}^2$ sending $P$ to $P'$ is special, or in fact, might not exist.
One can overcome this difficulty by imposing the condition of the existence of an anticanonical curve $Y$. If a point $p$ in the base locus has no infinitely near points in the base locus, then $p$ is in the strict transform of $Y$. Suppose $p^{(i)}, i=1, \dots , k$ are in the base locus satisfying a point $p^{(i)}$ is an infinitely near point of $p^{(i-1)}$ of order $1$ and no other points are infinitely near to $p^{(1)}$, then there is a successive blowups $\pi _i : S_i = \mathrm{B}\ell _{p^{(i)}} \to S_{i-1}$ of a single point $p^{(i)}$ with $S_0 = \mathbf{P}^2$. It follows that each strict transform of $Y$ in $S_i$ contains $p^{(i)}$. In either case, the set of base points in $\mathbf{P}^2$ belongs to the projection $X=\pi (Y)$. If $X$ is a rational curve with arithmetic genus $1$, it is easy to tell whether there is such an automorphism from $P$ to $P'$.
We say $\omega \in W_n, n\ge 10$ is essential if $\omega$ is not conjugate into a proper subgroup of $W_n$ generated by $\alpha _i$’s. It is shown in Reference 13 that there is a nonessential element $\omega \in W_{14}$ such that $\omega$ is not realizable by a rational surface automorphism. This is shown by counting periodic points using the Lefschetz fixed point formula. In fact, this $\omega$ is conjugate to the Coxeter element $\omega _{10} \coloneq s_0 \cdots s_{10} \in W_{10}$, and it is known Reference 2Reference 17 that $\omega _{10}$ is realized by a rational surface automorphism. In this article, we let $W_n^{ess}$ denote the set of essential elements in $W_n$ and focus on realizing essential elements in $\cup _{n\ge 10} W_n^{ess}$.
Let $\mathbb{C}^{1,n} = \mathbb{Z}^{1,n} \otimes \mathbb{C}$ with the complex Minkowski form. For each $v \in \mathbb{C}^{1,n}$ we defined a marked blowup$(S^v, Y^v, \phi ^v) = \mathrm{B}\ell (X,\rho ^v)$ (See Section 6 for the explicit construction.) by blowing up points on a cuspidal cubic. Let us denote $\Phi ^+_n$ as the set of positive roots and $\Phi ^-_n = - \Phi ^+_n$ as the set of negative roots. Also, let $\Phi ^+_v$ be a set of nodal roots in $(S^v, Y^v, \phi ^v)$ and let $\Sigma _v$ 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 $\omega \in W_{n\ge 10}$ such that $\omega$ cannot be realized by any automorphisms on anticanonical rational surfaces. Then, using Theorem B, we find a realizable $\omega '$ 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 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 $\mathbb{Z}^{1,n}$ be a lattice of signature $(1,n)$ with the basis $( \mathit{e}_0,\dots , \mathit{e}_n)$ such that
The canonical vector is given by $\kappa _n = (-3,1,1,\dots ,1)\in \mathbb{Z}^{1,n}$. The subgroup $L_n = \kappa _n^\perp \subset \mathbb{Z}^{1,n}$ orthogonal to $\kappa _n$ is generated by a set of vectors
We see that $s_i(\kappa _n) = \kappa _n$,$s_i(\alpha _i) = - \alpha _i$, and $s_i(\alpha _j) = \alpha _j$ for $j \ne i$. The group $W_n$ generated by those reflections $s_0, \dots , s_{n-1}$ fixes $\kappa _n$ and preserves the Minkowski product, and we have
This group $W_n \subset O(L_n)$ is called the Coxeter group. Let $V_n$ be a inner product space over $\mathbb{R}$ with the basis $\{\alpha _0, \alpha _1, \dots , \alpha _{n-1}\}$. Then each $s_i \in W_n$ gives the unique reflection in $O(V_n)$ preserving $L_n$. The root system $\Phi _n = \cup _i W_n \alpha _i$ is the $W_n$-orbits in $V_n$ of generating vectors $\alpha _i$’s.
2.1. Root system and simple roots
Each vector in $\alpha \in \Phi _n$ is called a root, and each generating vector $\alpha _i$ is called a simple root. Each $\alpha \in \Phi _n$ is a linear combination $\sum _i c_i \alpha _i$ with either $c_i \ge 0$ for all $i$ or $c_i \le 0$ for all $i$. We say $\alpha \in \Phi _n$ is positive if $\alpha = \sum _i c_i \alpha _i$ with $c_i \ge 0$ for all $i=0, \dots , n-1$ and $\alpha$ is negative if $c_i \le 0$ for all $i$. Let $\Phi _n^+$ denote the set of all positive roots and $\Phi _n^-$ 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 $W_n \to GL(\mathbb{Z}^{1,n})$ is injective.
•
$\Phi _n = \Phi _n^+ \cup \Phi _n^-$.
•
For all $\alpha _i$,$s_i ( \Phi _n^+ \setminus \{ \alpha _i\}) = \Phi _n^+ \setminus \{ \alpha _i\}$.
2.2. Length function
For each $\omega \in W_n$,$\omega$ is a word in $s_i, i=0, \dots , n-1$ and the length $\ell (\omega )$ is given by the minimum possible length of a word in $s_i$ for $\omega$. Note that $\omega \in W_n$ can have several different words of length $\ell (\omega )$. One useful fact of the length of $\omega$ is its relation to the number of positive roots sent to negative roots Reference 12, that is, for $\omega \in W_n$,$\ell (\omega ) = | \Pi (\omega )|$ where $\Pi (\omega )$ is the set of positive roots sent by $\omega$ to negative roots.
Let $S = \{ s_0, s_1, \dots , s_{n-1} \}$ be a set of generators of $W_n$.
Thus if $\omega \in W_n$ is nonessential, then $\omega$ can be identified as an element of a smaller Coxeter group generated by $S'$. If $S' \subset \{ s_1, s_2,\dots , s_{n-1} \}$, then $W '$ is finite and its Coxeter graph is given by $A_m$ for some $m\le n$Reference 12, Chapter 2. If $s_0 \in S'$, we may assume that $S' \subset \{ s_0, s_1, \dots , s_{n-2} \}$. Then we can treat $\omega$ as an element of $W_{n-1}$. If $\omega \in W_n$ is generated by $s_1, \dots , s_n$, then the spectral radius $\lambda (\omega )$ of $\omega =1$. Thus, from the definition, we see that:
The essential elements of $W_n, n\ge 10$ and their properties were considered by Krammer Reference 14 and Paris Reference 20.
It is known Reference 16Reference 20 that the Coxeter element $\omega = s_0 s_1 \cdots s_{n-1} \in W_n$ is essential, and the product of $s_0, \dots , s_{n-1}$ in any other is conjugate to $\omega$. In fact, the Coxeter element has the smallest spectral radius among all essential elements in $W_n$.
In this article, we will focus on essential elements.