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Rational nodal curves with no smooth Weierstrass points


Authors: Arnaldo Garcia and R. F. Lax
Journal: Proc. Amer. Math. Soc. 124 (1996), 407-413
MSC (1991): Primary 14H55
DOI: https://doi.org/10.1090/S0002-9939-96-03298-4
MathSciNet review: 1322924
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Abstract: Let $X$ denote the rational curve with $n+1$ nodes obtained from the Riemann sphere by identifying 0 with $\infty $ and $\zeta ^j$ with $-\zeta ^j$ for $j=0,1,\dots ,n-1$, where $\zeta $ is a primitive $(2n)$th root of unity. We show that if $n$ is even, then $X$ has no smooth Weierstrass points, while if $n$ is odd, then $X$ has $2n$ smooth Weierstrass points.


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  • 1 R.D.M. Accola, On generalized Weierstrass points on Riemann surfaces, Modular functions in analysis and number theory, Lecture Notes Math. Stat., University of Pittsburgh, Pittsburgh, PA, 1978, pp. (1--19), MR 85f:14016.
  • 2 E. Ballico and L. Gatto, Weierstrass points on singular curves.
  • 3 N. Chiarli, A Hurwitz type formula for singular curves, C. R. Math. Rep. Acad. Sci. Canada 6 (1984), 67--72, MR 85e:14037.
  • 4 W. Fulton, Intersection Theory, Springer, Berlin-Heidelberg-New York, 1984, MR 85k:14004.
  • 5 A. Garcia, On Weierstrass points on Artin-Schreier extensions of $k(x)$, Math. Nachr. 144 (1989), 233--239, MR 91f:14021.
  • 6 A. Garcia and R.F. Lax, Weierstrass weight of Gorenstein singularities with one or two branches, Manuscripta Math. 81 (1993), 361--378, MR 94j:14032.
  • 7 A. Garcia and R.F. Lax, Weierstrass points on Gorenstein curves in arbitrary characteristic, Comm. Algebra 22 (1994), 4841--4854, MR 95f:14063.
  • 8 A. Garcia and R.F. Lax, On canonical ideals, intersection numbers, and Weierstrass points on Gorenstein curves, J. Alg. (to appear).
  • 9 R. F. Lax, Weierstrass points on rational nodal curves, Glasgow Math. J. 29 (1987), 131--140, MR 88a:14016.
  • 10 R.F. Lax and C. Widland, Weierstrass points on rational nodal curves of genus 3, Canad. Math. Bull. 30 (1987), 286--294, MR 88h:14039.
  • 11 J. Lewittes, Automorphisms of compact Riemann surfaces, Amer. J. Math. 85 (1963), 732--752, MR 28:4102.
  • 12 A. Oneto And E. Zatini, Finite morphisms of Gorenstein curves, Commutative algebra (Trento, 1981), Lecture Notes in Pure and Appl. Math. No. 84, Dekker, New York, 1983, pp. (197--210), MR 84g:14051.
  • 13 K. Watanabe, Certain invariant subrings are Gorenstein II, Osaka J. Math. 11 (1974), 379--388, MR 50:7124.
  • 14 C. Widland, Weierstrass points on Gorenstein curves, Louisiana State University, 1984.
  • 15 C. Widland and R.F. Lax, Weierstrass points on Gorenstein curves, Pac. J. Math. 142 (1990), 197--208, MR 91b:14037.

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Additional Information

Arnaldo Garcia
Affiliation: IMPA, Estrada Dona Castorina 110, 22.460 Rio de Janeiro, Brasil
Email: garcia@impa.br

R. F. Lax
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
Email: lax@math.lsu.edu

DOI: https://doi.org/10.1090/S0002-9939-96-03298-4
Keywords: Weierstrass point, rational nodal curve
Received by editor(s): September 14, 1994
Communicated by: Eric Friedlander
Article copyright: © Copyright 1996 American Mathematical Society

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