Seshadri constant for a family of surfaces

Abstract

The aim of this note is to study local and global Seshadri constants for a family of smooth surfaces with prescribed polarization. We shall first observe that given $\alpha$ being smaller than the square root of the degree of polarization, the set of local Seshadri constants in the range $(0, \alpha]$ is finite. This in particular implies that the square root of the degree of polarization is the only possible accumulation point of the set of local Seshadri constants. Next we shall remark the Zariski closedness of the set of points whose local Seshadri constants are in any given interval $(0, a]$. As applications, we shall also add a few remarks on the lower semi-continuity of both local and global Seshadri constants with respect to parameters involved, and on the minimality and the maximality of their infimum and supremum.