On the weight of higher order Weierstrass points

Abstract

Introduction. Let C be a complete nonsingular curve of genus g^2 over an algebraicallyclosedfieldk of characteristiczero and D adivisoron C with dim|Z)| §0. Then we may definethe notion of Z>Weierstrass points(see e.g.[3]). Let P be a point on C and /=dim|J9|+l. If vis a positiveinteger such that dim L(D~{v-I)P)>dim L{D-vP), we callthisinteger v a Z)-gap at P. There are exactly /￡>-gapsand the sequence of />gaps i>i(P),-･-,vi(P)at P, vl{P)<- <vi(P),is calledthe Z)-gapsequence at P. The multiplicityof the Wronskian of D at a point P can be computed as ￡(vi(P)―*)･This integer is called the Dweight at P and denoted by wD{P). When wD(P) is positive,we callthe point / a /)-Weierstrass point. Itis well known that for the canonical divisorK,