A dynamical characterisation of smooth cubic affine surfaces of Markov type

Using valuative techniques, we show that a smooth affine surface with a non-elementary automorphism group and completable by a cycle of rational curves is either the algebraic torus or a smooth cubic affine surface of Markov type. Furthermore we show that smooth cubic affine surfaces of Markov type do no admit dominant endomorphisms which are not automorphisms.

💡 Research Summary

The paper investigates smooth affine surfaces X₀ over an algebraically closed field K that admit a non‑elementary automorphism group and can be completed by a cycle of rational curves. A non‑elementary group is defined as one containing two loxodromic automorphisms with no common iterates; loxodromic means the dynamical degree λ(f) exceeds 1 (equivalently, positive topological entropy when K = ℂ). Building on previous work (Abb23) and Gizatullin’s classification, the author first recalls that any smooth affine surface with a loxodromic automorphism must be rational and its completion Δ is either a tree (zigzag) or a cycle of rational curves.

The core of the paper focuses on the cycle case. Using valuation theory, the author studies the space of valuations centered at infinity of X₀. Inside this space there exists an Aut(X₀)-invariant circle, homeomorphic to the inverse limit of the dual graphs of cyclic completions. When Aut(X₀) is non‑elementary, its action on this circle has large orbits, which forces strong constraints on the intersection form of divisors supported on the boundary Δ. By invoking Goodman’s theorem (1969) that the boundary of any affine surface is the support of an ample effective Cartier divisor, the author shows that the intersection matrix must be of Minkowski type (one positive eigenvalue, the rest negative). This can happen only when Δ consists of exactly three rational curves forming a triangle.

Consequently, X₀ must be the complement of a triangle of lines Δ in a smooth cubic surface S ⊂ ℙ³. Such surfaces are known as “cubic affine surfaces of Markov type” because their coordinate rings satisfy the classical Markov equation x² + y² + z² = 3xyz. The paper proves that these surfaces, together with the algebraic torus Gₘ², are the only smooth affine surfaces with a non‑elementary automorphism group that admit a cyclic completion. The distinction between the two cases is whether X₀ possesses non‑constant invertible regular functions; this occurs precisely for the torus.

In the second major part, the author studies dominant endomorphisms of Markov‑type cubic affine surfaces. Assuming characteristic zero (in particular K = ℂ), the paper shows that any dominant endomorphism f : X₀ → X₀ must be an automorphism. The argument proceeds by first proving that the quasi‑Albanese variety QAlb(X₀) is trivial, which implies that X₀ has no non‑constant invertible regular functions and that the divisor class group injects into the Néron–Severi space with a non‑degenerate intersection form. Using the valuation circle again, one shows that f must be proper and unramified; thus f is a covering map of the complex manifold X₀(ℂ). Since X₀(ℂ) is simply connected (a fact known from the description of Markov surfaces as complements of a triangle in a smooth cubic), any covering must be a homeomorphism, hence an automorphism. The paper contrasts this rigidity with the singular Cayley cubic, which does admit non‑automorphic dominant endomorphisms descending from monomial maps on Gₘ², illustrating that smoothness is essential.

Throughout, the author employs several sophisticated tools: the Picard‑Manin spaces of Cartier and Weil classes (c‑NS and w‑NS), the action of birational maps on these spaces, log Kodaira dimension κ(X₀) (showing κ = 0 precisely for the cyclic case and κ = –∞ for the tree case), and the quasi‑Albanese variety. Proposition 3.6 characterises the torus via the non‑triviality of QAlb, while Proposition 3.3 links QAlb = 0 to the equality K_X₀̂ = K̂ and vanishing of the Albanese of any projective compactification.

The main results are summarized as:

• Theorem A: A smooth affine surface with a non‑elementary automorphism group and a cyclic completion is either Gₘ² or a smooth cubic affine surface of Markov type.
• Theorem B: For a smooth Markov‑type cubic affine surface over a field of characteristic zero, every dominant endomorphism is an automorphism.

These theorems complete the dynamical classification of smooth affine surfaces with large automorphism groups, extending the Gizatullin‑type dichotomy from trees to cycles and revealing a striking rigidity phenomenon for Markov surfaces. The work bridges valuation theory, birational geometry, and complex dynamics, and provides a clear answer to the long‑standing question of which smooth affine surfaces admit rich dynamical automorphism groups.