Abelian differentials with double zeros

Author:
Hershel M. Farkas

Journal:
Proc. Amer. Math. Soc. **28** (1971), 155-162

MSC:
Primary 30.45

MathSciNet review:
0274739

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Abstract: In this note we give a simple proof of the following theorem: The locus of points in the Torelli space of compact Riemann surfaces of genus whose underlying surfaces do not permit a basis for the abelian differentials of first kind each of whose elements is a differential with double zeros, has positive codimension in the Torelli space.

**[F1]**Hershel M. Farkas,*Special divisors and analytic subloci of Teichmueller space*, Amer. J. Math.**88**(1966), 881–901. MR**0213546****[F2]**. -,*On the Schottky relation and its generalization to arbitrary genus*, Ann. of Math. (to appear).**[L]**Joseph Lewittes,*Riemann surfaces and the theta function*, Acta Math.**111**(1964), 37–61. MR**0156964****[R]**H. E. Rauch,*Functional independence of theta constants*, Bull. Amer. Math. Soc.**74**(1968), 633–638. MR**0226000**, 10.1090/S0002-9904-1968-11969-X**[SG]**Giovanni Sansone and Johan Gerretsen,*Lectures on the theory of functions of a complex variable. I. Holomorphic functions*, P. Noordhoff, Groningen, 1960. MR**0113988**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1971-0274739-7

Keywords:
Abelian differentials,
theta functions,
Riemann surfaces moduli

Article copyright:
© Copyright 1971
American Mathematical Society