Positivity on Blow-up of hyperelliptic surfaces

Let $X_r$ denote the blow-up of the hyperelliptic surface $X$ at $r$ very general points. In this paper, we first provide a criterion for the ampleness of a line bundle on $X_r$ and compare it with an existing result. We then study the multi-point Seshadri constants of ample line bundles on hyperelliptic surfaces $X$. Next, we compute single-point Seshadri constants on $X_r$ for specific ample line bundles on odd types. Furthermore, we show that the global Seshadri constants for certain ample line bundles on blow-up of hyperelliptic surfaces are rational.