Seshadri constants and the spaces of curves with prescribed singularities and positions

We will give optimal bounds for Seshadri constants of an ample line bundle at multiple points on a complex projective surface X . We also present a solution to the long-studied classical problem on the existence of curves on X with given topological singularities at r arbitrary points p 1 ,..., p r . Namely, we obtain a universal lower bound on the degree of curves for the existence. It is independent of the position of the singular points.