Classification of the realizable elements of the Weyl groups $W_n$ of blowups of the projective plane, and of the automorphisms they determine

An element $w$ of the Weyl group $W_n$ — the Coxeter group generated by reflections in the roots of the Picard lattice $\mathbb{Z}^{1,n}$ of a rational surface obtained by blowing up the projective plane at $n$ points — is realizable if it is the induced action of an automorphism of some such surface. McMullen (2007) proved that any $w$ with spectral radius $\sigma(w) > 1$ and no periodic roots is realizable by an automorphism of positive entropy, and asked for a more complete classification of the realizable elements and of the automorphisms they determine. The element-wise classification of the realizable elements remains open. Kim proved an if-and-only-if realizability criterion for the family of marked blowups of a cuspidal cubic, showed that every unrealizable element in that family is one element of the group generated by the reflections in nodal roots away from a realizable element with the same spectral radius, exhibited a non-realizable element of $W_{14}$, proved that some essential element is not realizable on any anticanonical rational surface, and posed the sharpened question of whether every essential element of $\bigcup_{n \ge 10} W_n$ is realizable. This record states the residual classification problem: a necessary-and-sufficient criterion for realizability on arbitrary marked iterated blowups of $\mathbb{P}^2$, not necessarily anticanonical (covering elements with periodic roots and the parabolic case), the essential-element question, and the classification of realizations. The companion open questions from the same paper — virtual abelianness of the stabilizers $W_n^v$ and the exact range of $n$ for Siegel-disk realizations of the standard Coxeter element — remain open, and the plurianticanonical-section question of the same paper has a negative answer, due to Bedford and Kim (2009).

Background

Let $S$ be a rational surface obtained from the complex projective plane $\mathbb{P}^2$ by blowing up $n$ points, possibly infinitely near (that is, on exceptional curves of earlier blowups). Its Picard group $\operatorname{Pic}(S) \cong H^2(S, \mathbb{Z})$ is free abelian of rank $n+1$, with intersection form of signature $(1, n)$. A marking is an isometry $\phi$ from $H^2(S, \mathbb{Z})$ onto the Lorentz lattice $\mathbb{Z}^{1,n}$ with orthogonal basis $e_0, \dots, e_n$, where $e_0^2 = 1$ and $e_i^2 = -1$. The Weyl group $W_n$ is the Coxeter group generated by the reflections $s_\alpha(x) = x + (x \cdot \alpha)\,\alpha$ in the roots $\Phi_n$, the $W_n$-orbit of the simple roots $\alpha_0 = e_0 - e_1 - e_2 - e_3$ and $\alpha_i = e_i - e_{i+1}$ for $1 \le i \le n - 1$; every root has self-intersection $-2$. In particular $\Phi_n$ contains the transpositions $\pm(e_i - e_j)$ and the Cremona roots $\pm(e_0 - e_i - e_j - e_k)$ with $i, j, k \ge 1$ pairwise distinct; the reflections in these two families generate $W_n$, but for $n \ge 6$ they do not exhaust $\Phi_n$. For $3 \le n \le 8$, $W_n$ is the finite Weyl group attached to del Pezzo surfaces (smooth rational surfaces with ample anticanonical class; the corresponding root lattices are $A_1 \oplus A_2$, $A_4$, $D_5$, $E_6$, $E_7$, $E_8$), while for $n \ge 9$ it is infinite, and for $n \ge 10$ hyperbolic. A classical theorem of Nagata, in the form stated by McMullen, shows that $W_n$ is exactly the ambiguity of the marking: if both $(S, \phi)$ and $(S, \phi \circ w)$ are marked iterated blowups of $\mathbb{P}^2$, then $w \in W_n$.

An element $w \in W_n$ is realized (or realizable) if there exist a marked iterated blowup $(S, \phi)$ and a biholomorphic automorphism $F \in \operatorname{Aut}(S)$ such that $\phi \circ F^* = w \circ \phi$, where $F^*$ is the induced action on cohomology. Not every element is realizable: an automorphism must preserve the nef cone (the closed cone of divisor classes intersecting every effective curve non-negatively) and respect holomorphic invariants of the surface, whereas $W_n$ is a purely arithmetic object. By a theorem of Gromov and Yomdin, the topological entropy of a surface automorphism $F$ is $h(F) = \log \sigma(F^*)$, where $\sigma$ denotes spectral radius; this spectral radius is the dynamical degree $\delta(F)$. Hence a realization of $w$ has entropy $\log \sigma(w)$, so the classification of realizable elements contains in particular the question of which entropies occur. For $n \ge 10$ many elements have $\sigma(w) > 1$; the occurring spectral radii are constrained (they are algebraic units, and include Salem numbers — algebraic integers $> 1$ whose remaining conjugates lie in the closed unit disk with at least one on the unit circle), and every positive entropy of any compact complex surface automorphism is at least $\log \lambda_{\mathrm{Lehmer}}$, where $\lambda_{\mathrm{Lehmer}} \approx 1.176$ is Lehmer's number, the smallest known Salem number, which equals the spectral radius of the Coxeter element of $W_{10}$ (McMullen, Theorem 1.2).

A root is periodic for $w$ if its orbit under $w$ is finite. McMullen's Theorem 1.5 gives a sufficient condition for realizability: if $\sigma(w) > 1$ and $w$ has no periodic roots, then $w$ is realized by an automorphism of positive entropy, constructed by blowing up points of a cuspidal cubic (an irreducible plane cubic with a cusp; its strict transform is an anticanonical curve, i.e. a member of the linear system $|-K_S|$). He observed that the construction produces many distinct realizations, depending on the choice of eigenvector, and wrote: “It would be interesting to have a more complete classification of realizable elements in the Weyl group, and of the automorphisms they determine.” The same paper poses companion questions. For $v \in (\mathbb{C}^{1,n})^*$, where $(\mathbb{C}^{1,n})^* = \{v \in \mathbb{C}^{1,n} : 0 \notin v \cdot \Phi_n\}$, let $W_n^v = \{w \in W_n : [v] \text{ is an eigenvector for } w \text{ on } \mathbb{C}^{1,n}/\mathbb{C}k_n\}$ (with $k_n = 3e_0 - e_1 - \cdots - e_n$) be the stabilizer of the class $[v]$, which for suitable $v$ is the automorphism group of a marked blowup $(S^v, Y^v, \phi^v)$ with anticanonical curve $Y^v$: is $W_n^v$ always virtually abelian (does it contain an abelian subgroup of finite index)? A question posed in the source, similar to one formulated by Harbourne, asks whether some negative power of the canonical class $K_S$ of a minimal realization (a realization on a surface on which no $(-1)$-curve is invariant under the induced $\mathbb{Z}$-action — minimality in the sense of Manin, as used in the source) of an element of infinite order admits a holomorphic section. And Theorem 10.1 realizes the standard Coxeter element (the product of the simple reflections, each taken once) with a Siegel disk — a Fatou component on which the return map is conjugate to an irrational rotation — for all sufficiently large $n \not\equiv 2, 4 \pmod 6$, with explicit examples for $n = 11, 12$, leaving the exact range of $n$ undetermined.

The later literature settles the degree half of the classification: Uehara proved that every spectral radius $\lambda > 1$ of an element of $W_n$ occurs as the dynamical degree of a rational surface automorphism, using orbit data — a combinatorial datum (an integer, a bijection between the indeterminacy points of a tuple of birational maps, and the multiplicities of the blown-up points) prescribing the finite orbits of the base points and determining the blowups that resolve the map, a method introduced by Diller — and Kim re-proved the equality $\Lambda = \Delta$ between the Weyl spectrum $\Lambda$ (the set of spectral radii of elements of the groups $W_n$) and the set $\Delta$ of dynamical degrees of rational surface automorphisms by Coxeter-theoretic methods. Further families of positive-entropy examples were constructed by Déserti and Grivaux. The element-wise classification remains open. Kim calls $w \in W_n$ essential if it is not conjugate, after relabelling the basis vectors if necessary, into a subgroup $W_{n'}$ with $n' < n$; every element with $\sigma > 1$ is conjugate to an essential one with the same spectral radius (possibly in a smaller $W_{n'}$). She exhibited a non-essential element of $W_{14}$ — one conjugate into a subgroup $W_{10} \subset W_{14}$ — that is not realizable, the obstruction being detected by counting periodic points with the Lefschetz fixed-point formula; an element that is realized and non-essential would necessarily act on a non-minimal surface. She posed the sharpened question: “Is every essential element in $\cup_{n \ge 10} W_n$ realizable?” For the cuspidal family she proved: if $v$ is a leading eigenvector (an eigenvector for the spectral radius) of an essential $\omega$ with spectral radius $\lambda$, then $\omega$ is realized on the marked blowup $(S^v, Y^v, \phi^v)$ if and only if $\omega^{-1}(\alpha)$ is a positive root for every nodal root $\alpha$ (a root is nodal if it corresponds to an effective irreducible class on $(S^v, Y^v, \phi^v)$); and if not, some element $s$ of the group generated by the reflections in nodal roots makes $s \circ \omega$ realizable with the same spectral radius, findable in finitely many steps. She also proved that some essential element is not realizable on any anticanonical rational surface (a rational surface carrying an effective anticanonical divisor), so a complete criterion must go beyond anticanonical surfaces. On the companion questions: Bedford and Kim resolved the Harbourne question negatively, by constructing a minimal positive-entropy automorphism with no invariant curve (hence no plurianticanonical section); Uehara showed that every automorphism preserving a cuspidal anticanonical curve arises from orbit data, and that the automorphism groups of certain such surfaces are infinite cyclic; but the virtual abelianness of $W_n^v$ in general, and the Siegel-disk range of $n$, remain open.

Problem Statement

Classify the realizable elements of the Weyl groups $W_n$ and the automorphisms they determine, in the sense of the source: for every $n$ (the main open cases being $n \ge 10$, where $W_n$ is hyperbolic and elements of spectral radius greater than one first occur), determine exactly which elements $w \in W_n$ are realized by automorphisms $F$ of rational surfaces $S$ that are iterated blowups of $\mathbb{P}^2$ at $n$ points — that is, for which $w$ there exist a marked iterated blowup $(S, \phi)$ and $F \in \operatorname{Aut}(S)$ with $\phi \circ F^* = w \circ \phi$ — and describe the realizing pairs $(S, F)$. A complete answer should comprise: (a) a necessary-and-sufficient criterion for realizability on marked iterated blowups of $\mathbb{P}^2$ that are not necessarily anticanonical, expressed in terms of the Coxeter-theoretic data of $w$ (such as its action on the roots and its eigenvectors) rather than as a restatement of the definition of realizability, and extending McMullen's sufficient condition ($\sigma(w) > 1$ and no periodic roots) to elements with periodic roots and to the parabolic case ($\sigma(w) = 1$ with $w$ of infinite order) — this criterion settles, as its essential-element case, Kim's sharpened question of whether every essential element of $\bigcup_{n \ge 10} W_n$ is realizable, and, if the answer is negative, characterizes the realizable essential elements; and (b) for each realizable element, a classification of the automorphisms it determines. The degree-theoretic half — every spectral radius greater than one occurs as the dynamical degree of some rational surface automorphism — is already settled (Uehara; Kim) and is not required. None of the components alone exhausts the classification, which ranges over all elements of all $W_n$: results confined to a subfamily — a single anticanonical curve, bounded $n$, or elements without periodic roots alone — are progress but do not complete the problem. The companion questions from the same source (virtual abelianness of the stabilizers $W_n^v$; the exact range of $n$ for Siegel-disk realizations of the standard Coxeter element) are not part of this problem.

References

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