THE ( g - 1 ) - SUPPORT COVER OF THE CANONICAL LOCUS

Abstract

One of Max Noether's achievements [I, 21 was a characterization of nonhyperelliptic compact Riemann surfaces S by the property that the vector space of holomorphic q-differentials, q an integer > 1, is spanned by q-fold products of holomorphic differentials chosen from a basis i?,, . . . ,-9, of the vector space of holomorphic differentials on S. Noether's proof depended on showing that for a non-hyperelliptic surface there always exists a positive divisor D of degree g 2 on S, such that for any s E S there is exactly one holomorphic differential -9 (up to multiplication by a constant) whose divisor, (S), contains D + s in its support. An alternate proof of Noether's theorem based on the following result has been suggested: S is non-hyperelliptic if and only if there exists on S a holomorphic differential -9 whose divisor (8) = s, + . * + s2,-* has the property that s, # s,, i # j, and if 8 is a non-zero holomorphic differential on S and i,, . . . , i g _ , are integers with 1 S i , < i2 < < is_, S 2g 2 then

Cite this paper

@inproceedings{Farkas2012THEG, title={THE ( g - 1 ) - SUPPORT COVER OF THE CANONICAL LOCUS}, author={H. Farkas and Michael D. Fried}, year={2012} }