Restrictions on Seshadri constants on surfaces

This talk will be divided in two parts. In the first I show that all positive rational numbers can be realized as Seshadri constants of ample line bundles on some surfaces. To the contrary, Seshadri constants of adjoint line bundles underlie strong numerical restrictions. In the second part I will show how geometric properties of a surface can be read of values of Seshadri constants of a fixed ample line bundle taken at very general points.