Seshadri constants on hyperelliptic surfaces

We prove new results on single point Seshadri constants for ample line bundles on hyperelliptic surfaces. Given a hyperelliptic surface \(X\) and an ample line bundle \(L\) on \(X\), we show that the least Seshadri constant \(\varepsilon(L)\) of \(L\) is a rational number when \(X\) is not of type 6. We also prove new lower bounds for the Seshadri constant \(\varepsilon(L,1)\) of \(L\) at a very general point.