ZARISKI DENSITY OF POINTS WITH MAXIMAL ARITHMETIC DEGREE FOR SURFACES

The paper proves that for a smooth projective surface X/K with potentially dense rational points and any surjective self-morphism f with δ_f>1, the property (DR) holds (Definition 1.3 and Theorem 1.5), via reduction to minimal models and a case-by-case geometric analysis (K3/Enriques/abelian/elliptic surface cases) using Lemmas 3.2–3.3 and Proposition 4.5, together with elliptic-fibration inputs (Lemmas 3.5–3.6) . The model’s solution gives a different (height-theoretic) route: construct nef eigenclasses and canonical heights, use a dense set of points with α_f(x)=δ_f, and then enforce pairwise disjoint orbits by normalizing canonical heights. This approach is essentially correct but needs one auxiliary justification: that among a Zariski-dense set of points with α_f(x)=δ_f, the normalized canonical heights take infinitely many distinct values. This can be supplied using Northcott on curves where the eigen-divisor restricts to an ample class and the finiteness of f-periodic curves on surfaces where needed. With that small patch, both the paper’s argument and the model’s argument are correct, though they are substantially different in technique.