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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
MathSciNet review: 1357406
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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 $y^{n}=f(x)$, where $f$ is a polynomial of degree $d$. 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, $BW$. We obtain a lower bound for $BW$, which we show is exact if $n$ and $d$ 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):

\begin{equation*}\lim _{d\to \infty }\frac {BW}{g^{3}-g}=\frac {n+1}{3(n-1)^{2}}, \end{equation*}

where $g$ is the genus of the curve. In the case that $n=3$ (cyclic trigonal curves), we are able to show in most cases that for sufficiently large primes $p$, the branch points and the non-branch Weierstrass points remain distinct modulo $p$.


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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: http://dx.doi.org/10.1090/S0002-9947-96-01649-2
PII: S 0002-9947(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