Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On Castelnuovo's inequality for algebraic curves. I


Author: Robert D. M. Accola
Journal: Trans. Amer. Math. Soc. 251 (1979), 357-373
MSC: Primary 14H35; Secondary 14C10, 30F10
MathSciNet review: 531984
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {W_p}$ be a Riemann surface of genus p admitting a simple linear series $ g_n^r$ where $ n\, =\, m(r - 1)\, +\, q,\,\, q\, =\, 2,\,3,...,\,r - 1$, or r. Castelnuovo's inequality states that (1) $ 2p\, \leqslant\, 2f(r,n,1)\, =\, m(m - 1)(r - 1)\, +\, 2m(q - 1)$. By further work of Castelnuovo, equality in (1) and $ q\, <\, r$ implies that $ {W_p}$ admits a plane model of degree $ n\, -\, r\, +\, 2$ with $ r\, -\, 2$ m-fold singularities and one $ (n\, -\, r\, +\, 1\, -\, m)$-fold singularity. Formula (1) generalizes as follows. Suppose $ {W_p}$ admits s simple linear series $ g_n^r$ where $ n\, =\, m(rs - 1)\, +\, q$ and $ q = - (s - 1)r + 2,\, - (s - 1)r + 3,\ldots,r - 1$, or r. For q consider the cases $ v = 0,1,\ldots,s - 1$ as follows: case $ v\, =\, 0:2 \leqslant q \leqslant r$, case $ v > 0:2 \leqslant q + vr \leqslant r + 1$. Then (2) $ 2p \,\leqslant\, 2f(r,\,n,\,s)\, =\, {m^2}(r{s^2}\, -\, s) \,+\, ms(2q \,-\, 1\, -\, r)\, -\, v (v \, -\, 1)r\, -\, 2v (q \,-\, 1)$. Examples show that (2) is sharp. Finally, if $ n\, = \,m'r\, + \,q'$, $ q'\, = \,1,\,2,\, \ldots ,\,r\, - \,1$, or r and $ {W_p}$ admits $ m'\, + \,1$ simple $ g_n^{r}$'s then (3) $ 2p\, \leqslant \,2f\,(r\, + \,1,\,n\, + \,1,\,1)\, = \,m'\,(m'\, - \,1)r\, + \,2m'\,q'\,$. Since $ f(r,\,n,\,2)\, < \,f(r,\,n,\,1)$ we obtain as a corollary: if $ p\, = \,f(r,\,n,\,1)$ then $ {W_p}$ admits at most one simple $ g_n^r$.


References [Enhancements On Off] (What's this?)

  • [1] G. Castelnuovo, Ricerche di geometria sulle curve algebriche, Atti Accad. Sci. Torino 24 (1889) (Memorie Scelte, Zanichelli, Bologna, 1937, p. 19).
  • [2] -, Sui multiple di una serie lineare di gruppe di punti appartenente ad una curva algebrica, Rend. Circ. Mat. Palermo 7 (1893), 89-110 (Memorie Scelte, p. 95).
  • [3] -, Sulla linearita delta involuzioni piu volte infinite appartenente ad una curva algebrica, Atti Accad. Sci. Torino 28 (1893) (Memorie Scelte, p. 115).
  • [4] Julian Lowell Coolidge, A treatise on algebraic plane curves, Dover Publications, Inc., New York, 1959. MR 0120551 (22 #11302)
  • [5] Robert J. Walker, Algebraic Curves, Princeton Mathematical Series, vol. 13, Princeton University Press, Princeton, N. J., 1950. MR 0033083 (11,387e)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 14H35, 14C10, 30F10

Retrieve articles in all journals with MSC: 14H35, 14C10, 30F10


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0531984-2
PII: S 0002-9947(1979)0531984-2
Keywords: Riemann surface, algebraic curve, linear series
Article copyright: © Copyright 1979 American Mathematical Society