Relative systoles in hyperelliptic translation surfaces (Q6596245)

A (compact) translation surface \(S = (X, \omega)\) is a pair consisting of a compact Riemann surface together with a holomorphic \(1\)-form \(\omega\), which determines a singular flat metric. When \(X\) has genus \(g\), \(\omega\) has zeros whose orders add up to \(2g-2\) which correspond to points with cone angle \(2\pi(k+1)\) for a zero of order \(k\). A \textit{saddle connection} is a geodesic trajectory connecting two zeros with no zeros in the interior, and the length of the shortest saddle connection is called the \textit{relative systole} of \(S\), and denoted \(Sys(S)\). The collection of translation surfaces with fixed orders of zeros is known as a stratum, and a natural question is to understand the \textit{maximum} of \(Sys(S)\) on a stratum, that is, the \textit{longest shortest saddle connection}. For the torus, this given by the hexagonal torus, and related constructions of covers often yield the answer for higher-genus surfaces. The paper under review studies when local maxima which are not global maxima are possible on connected components of strata, and shows that this is possible if and only if connected component is not hyperelliptic.