Some results on Seshadri constants on surfaces of general type

We prove two new results for Seshadri constants on surfaces of general type. Let X be a surface of general type. In the first part, inspired by Bauer and Szemberg (Manuscripta Math 126(2):167–175, 2008 ), we list the possible values for the multi-point Seshadri constant ε ( K X , x 1 , x 2 , … , x r ) when it lies between 0 and 1/ r , where K X is the canonical line bundle on X . In the second part, we assume X of the form C × C , where C is a general smooth curve of genus g ⩾ 2 . Given such X and an ample line bundle L on X with some conditions on it, we show that the global Seshadri constant of L is a rational number.