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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

- Robert D. M. Accola,
*On generalized Weierstrass points on Riemann surfaces*, Modular functions in analysis and number theory, Lecture Notes Math. Statist., vol. 5, Univ. Pittsburgh, Pittsburgh, PA, 1983, pp. 1–19. MR**732958** - E. Ballico and L. Gatto,
*Weierstrass points on singular curves*. - Nadia Chiarli,
*A Hurwitz type formula for singular curves*, C. R. Math. Rep. Acad. Sci. Canada**6**(1984), no. 2, 67–72. MR**740598** - William Fulton,
*Intersection theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR**732620** - Arnaldo García,
*On Weierstrass points on Artin-Schreier extensions of $k(x)$*, Math. Nachr.**144**(1989), 233–239. MR**1037171**, DOI https://doi.org/10.1002/mana.19891440116 - Arnaldo García and R. F. Lax,
*Weierstrass weight of Gorenstein singularities with one or two branches*, Manuscripta Math.**81**(1993), no. 3-4, 361–378. MR**1248761**, DOI https://doi.org/10.1007/BF02567864 - Arnaldo García and R. F. Lax,
*Weierstrass points on Gorenstein curves in arbitrary characteristic*, Comm. Algebra**22**(1994), no. 12, 4841–4854. MR**1285713**, DOI https://doi.org/10.1080/00927879408825108 - A. Garcia and R.F. Lax,
*On canonical ideals, intersection numbers, and Weierstrass points on Gorenstein curves*, J. Alg. (to appear). - R. F. Lax,
*Weierstrass points on rational nodal curves*, Glasgow Math. J.**29**(1987), no. 1, 131–140. MR**876157**, DOI https://doi.org/10.1017/S0017089500006741 - R. F. Lax and Carl Widland,
*Weierstrass points on rational nodal curves of genus $3$*, Canad. Math. Bull.**30**(1987), no. 3, 286–294. MR**906350**, DOI https://doi.org/10.4153/CMB-1987-041-7 - Joseph Lewittes,
*Automorphisms of compact Riemann surfaces*, Amer. J. Math.**85**(1963), 734–752. MR**160893**, DOI https://doi.org/10.2307/2373117 - Anna Oneto and Elsa Zatini,
*Finite morphisms of Gorenstein curves*, Commutative algebra (Trento, 1981) Lecture Notes in Pure and Appl. Math., vol. 84, Dekker, New York, 1983, pp. 197–210. MR**686945** - Keiichi Watanabe,
*Certain invariant subrings are Gorenstein. I, II*, Osaka Math. J.**11**(1974), 1–8; ibid. 11 (1974), 379–388. MR**354646** - C. Widland,
*Weierstrass points on Gorenstein curves*, Louisiana State University, 1984. - Carl Widland and Robert Lax,
*Weierstrass points on Gorenstein curves*, Pacific J. Math.**142**(1990), no. 1, 197–208. MR**1038736**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
14H55

Retrieve articles in all journals with MSC (1991): 14H55

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

Keywords:
Weierstrass point,
rational nodal curve

Received by editor(s):
September 14, 1994

Communicated by:
Eric Friedlander

Article copyright:
© Copyright 1996
American Mathematical Society