A study of H. Martens’ Theorem on chains of cycles - Coppens, Marc (MATHEMATICA SCANDINAVICA, 2025)
Let C be a smooth curve of genus g and let d,r be integers with 1 ≤ r ≤ g-2 and 2r≤ d ≤ g-2+r. H. Martens’ Theorem states that (W^r_d(C))=d-2r implies C is hyperelliptic. It is known that for a metric graph Γ of genus g such statement using (W^r_d(Γ )) does not hold. However replacing (W^r_d(Γ )) by the so-called Brill-Noether rank w^r_d(Γ ) it was stated as a conjecture. Using a similar definition in the case of curves one has (W^r_d(C))=w^r_d(C). Let Γ be a chain of cycles of genus g and let r,d be integers with 1 ≤ r ≤ g-2 and 2r≤ d ≤ g-3+r. If w^r_d(Γ )=d-2r then we prove Γ is hyperelliptic. In case g ≥ 2r+3 then we prove there exist non-hyperelliptic chains of cycles satisfying w^r_g-2+r(Γ )=g-2-r, contradicting the conjecture. We give a complete description of all counterexamples within the set of chains of cycles to the statement of H. Martens’ Theorem. Those counterexamples also give rise to chains of cycles such that w^r_g-2+r(Γ ) eq w^1_g-r(Γ ). This shows that the Riemann-Roch duality does not hold for the Brill-Noether ranks of metric graphs.