In this talk, we consider the Weierstrass point on smooth algebraic curve and its weight. Firstly, I will introduce a lower bound of the (Weierstrass) weights of fixed points of automorphisms on an algebraic curve over the complex number field, which is given by Del Pozo in 2006. Then under a further (but reasonable) assumption, I will provide a classification for the curves on which this lower bound is attained. This is a joint work with Sakai Fumio and Kawasaki Masumi. Some other related topics will be mentioned in this talk.