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 multipoint Seshadri constant e(K-X, x(1), x(2), ..., x(r)) when it lies between 0 and 1/r, where KX is the canonical line bundle on X. In the second part, we assume X of the form C x 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.