Weights of Weierstrass points in double coverings of curves of genus one or two

Abstract

L e t X be a p r o j e c t i v e n o n s i n g u l a r c u r v e o f g e n u s g , d e f i n e d o v e r an a l g e b r a i c a l l y c l o s e d f i e l d o f c h a r a c t e r i s t i c z e r o ( e . g . X i s a R i e m a n n S u r f a c e ) . We w i l l a l w a y s a s s u m e t h a t X is non-hyperelliptic. In [6], Kate gives a sufficient condition for X to be elliptic-hyperelliptic (i.e. a double covering of an elliptic curve). His main result is: if g = 8 or g m II and there exists a point P on X with weight (denoted by w(P)) equal to (g2-5g+10)/2, then X is elliptic-I~]pere~ipt~. Here we prove the following two theorems: