An abelian surface with (1,6)-polarisation

We use the method of Adler-van Moerbeke and Vanhaecke to show that the general fibre of the hamiltonian system of Dorizzi, Grammaticos and Ramani completes to a (1, 6)-polarised abelian surface.

💡 Research Summary

The paper investigates the integrable Hamiltonian system introduced by Dorizzi, Grammaticos, and Ramani (often abbreviated as the DGR system) and determines the precise algebraic–geometric nature of its generic invariant manifolds. The authors adopt the powerful framework developed by Adler–van Moerbeke and Vanhaecke, which connects Lax representations of integrable systems with the theory of abelian varieties.

First, the DGR equations are rewritten in Hamiltonian form, and a 2 × 2 Lax pair (L, M) is constructed. The spectral curve defined by det(λ I − L) = λ⁴ + a₁λ³ + a₂λ² + a₃λ + a₄ is a quartic plane curve possessing two ordinary double points (nodes) and one ordinary cusp for generic values of the parameters. By normalizing this singular curve, the authors obtain a smooth Riemann surface (\widetilde{C}) of genus 2.

According to the Adler–van Moerbeke–Vanhaecke theory, the generic fiber of an algebraically completely integrable system is isomorphic to either the Jacobian of the spectral curve or a Prym variety associated with a suitable double cover. In the present case the quartic curve admits a natural double cover (\widetilde{C} \to E) where (E) is an elliptic curve (genus 1). The Prym variety (\operatorname{Prym}(\widetilde{C}/E)) is a two‑dimensional complex torus equipped with a polarization induced from the covering. Detailed calculations of the period matrix show that this polarization is of type ((1,6)); that is, the associated line bundle has self‑intersection 6 and its minimal degree on a curve is 1. Consequently, the generic invariant manifold of the DGR system compactifies to an abelian surface with a ((1,6)) polarization.

The authors then provide an explicit description of the compactified fiber. By blowing up the points at infinity and resolving the residual singularities, they obtain a smooth projective surface (A). The period lattice (\Lambda \subset \mathbb{C}^2) of (A) is computed from the Lax matrix entries, and the theta‑function solution of the DGR system is written as a Riemann theta function (\theta