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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
DOI: https://doi.org/10.1090/S0002-9947-1979-0531984-2
MathSciNet review: 531984
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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).
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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0531984-2
Keywords: Riemann surface, algebraic curve, linear series
Article copyright: © Copyright 1979 American Mathematical Society

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