Remarks on Seshadri constants

Abstract

Given a smooth complex projective variety $X$ and an ample line bundle $L$ on $X$. Fix a point $x\in X$. We consider the question, are there conditions which guarantee the maxima of the Seshadri constant of $L$ at $x$, i.e $\eps(L,x)=\root n \of {L^n}$? We give a partial answer for surfaces and find examples where the answer to our question is negative. If $(X,\Theta)$ is a general principal polarized abelian surface, then $\eps(\Theta,x)={4/3}<\sqrt{2}=\sqrt{\Theta^2}$ for all $x\in X$.