The dynamical degrees of rational surface automorphisms (Q6554728)

This work provides an alternative proof of a theorem by \textit{T. Uehara} [Ann. Inst. Fourier 66, No. 1, 377--432 (2016; Zbl 1360.14042)] by generalizing a construction of \textit{C. T. McMullen} [Publ. Math., Inst. Hautes Étud. Sci. 95, 151--183 (2002; Zbl 1148.20305)].\N\NTo elaborate on the context, let \(S\) be a rational surface obtained by blowing up a set \(P\) of \(n\) (possibly infinitely near) points in the projective plane \(\mathbf{P}^2\). 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 (of signature \((1,n)\)) and a Picard group \(Pic(S)\).\N\NBy \textit{M. Nagata} [Mem. Coll. Sci., Univ. Kyoto, Ser. A 32, 351--370 (1960; Zbl 0100.16703); Mem. Coll. Sci., Univ. Kyoto, Ser. A 33, 271--293 (1960; Zbl 0100.16801)], if \(F\colon S\to S\) 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 \(F_\ast\colon Pic(S)\to Pic(S)\) is equivalent to \(\omega\) via the isomorphism \(\phi\). Such \(F\) is said to \textit{realize} \(\omega\in W_n\).\N\NAs elements of \(W_n\) can be represented as \((1+n)\times(1+n)\) matrices with integer entries, the spectral radius \(\lambda(\omega)\) of an element \(\omega\in W_n\) is well-defined. For a rational surface automorphism \(F\colon S\to S\), its dynamical degree \(\delta(F)\) is given by the limit of \((\text{algebraic degree of }F^n)^{1/n}\) as \(n\to\infty\). The sets \(\Lambda\) of spectral radii of \(\omega\in W_n\) for some \(n\geq 1\) and \(\Delta\) of dynamical degrees of rational surface automorphisms are the same. This is the theorem of \textit{T. Uehara} mentioned earlier, but also Theorem A of this work: for each \(\lambda\in\Lambda\), it is realized as the spectral radius of a realizable Weyl group element \(\omega\in W_n\), i.e., there exists a rational surface automorphism \(F\) with \(F_\ast=\omega\) and \(\delta(F)=\lambda(\omega)=\lambda\).\N\NTo have a dynamical degree \(>1\), as stated by McMullen, we need to blow up at least 10 points in \(\mathbf{P}^2\). Theorem B of this work focuses on (1) when an essential element \(\omega\in W_n\) (i.e., elements that cannot be conjugated into a proper subgroup of \(W_n\) generated by reflections) is realizable and (2) if not realizable, how it can be modified to become realizable while maintaining the spectral radius. This result comes after generalizing definitions of \textit{C. T. McMullen} [Publ. Math., Inst. Hautes Étud. Sci. 105, 49--89 (2007; Zbl 1143.37033)] concerning marked blowups and marked cubics by allowing infinitely near points as base points (see Sections 3--5 of this work).\N\NRealizability in Theorem B depends on the specific form of the marked blowup surfaces. However, in Theorem C, it is shown that there is an essential element \(\omega\in\bigcup_{n=10}^\infty W_n\) such that \(\omega\) cannot be realized by an automorphism on an anticanonical rational surface. The theorem is, in fact, a consequence of Theorem B (see Section 8 of this work).