On Higher Order Weierstrass Points of the Universal Curve*

Let X denote a compact Riemann surface of genus g. Let K denote the canonical line bundle on X and denote by K n the n-fold tensor power of K. Put dn=dimcH~ Kn)=(2n-1 ) (g -1 )+b ln . At each point P~X, there is a sequence of d, integers 1 = 71 (P) < 72(P) < " " < 7a~(P) < 2 n ( g 1) + 1, called the sequence of n-gaps at P. An integer 7 is an n-gap at P if and only if there exists O~H~ K ~) with a zero of order 7 1 at P. The point P is called an n-fold Weierstrass point if 7a~(P)>dn (cf. 1-14, 21]). The multiplicity, or weight, of an ndn fold Weierstrass point P is ~ (7i(P)-i). If we count these points with multi-