Weierstrass points on cyclic covers

of the projective line

Author:
Christopher Towse

Journal:
Trans. Amer. Math. Soc. **348** (1996), 3355-3378

MSC (1991):
Primary 14H55, 11G30

DOI:
https://doi.org/10.1090/S0002-9947-96-01649-2

MathSciNet review:
1357406

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Abstract | References | Similar Articles | Additional Information

Abstract: We are interested in cyclic covers of the projective line which are totally ramified at all of their branch points. We begin with curves given by an equation of the form , where is a polynomial of degree . Under a mild hypothesis, it is easy to see that all of the branch points must be Weierstrass points. Our main problem is to find the total Weierstrass weight of these points, . We obtain a lower bound for , which we show is exact if and are relatively prime. As a fraction of the total Weierstrass weight of all points on the curve, we get the following particularly nice asymptotic formula (as well as an interesting exact formula):

where is the genus of the curve. In the case that (cyclic trigonal curves), we are able to show in most cases that for sufficiently large primes , the branch points and the non-branch Weierstrass points remain distinct modulo .

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Additional Information

**Christopher Towse**

Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1003

Email:
towse@math.lsa.umich.edu

DOI:
https://doi.org/10.1090/S0002-9947-96-01649-2

Keywords:
Weierstrass points,
branch points

Received by editor(s):
September 27, 1994

Received by editor(s) in revised form:
October 16, 1995

Article copyright:
© Copyright 1996
American Mathematical Society