On Castelnuovo’s inequality for algebraic curves. I
HTML articles powered by AMS MathViewer
- by Robert D. M. Accola PDF
- Trans. Amer. Math. Soc. 251 (1979), 357-373 Request permission
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
-
G. Castelnuovo, Ricerche di geometria sulle curve algebriche, Atti Accad. Sci. Torino 24 (1889) (Memorie Scelte, Zanichelli, Bologna, 1937, p. 19).
—, 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).
—, Sulla linearita delta involuzioni piu volte infinite appartenente ad una curva algebrica, Atti Accad. Sci. Torino 28 (1893) (Memorie Scelte, p. 115).
- Julian Lowell Coolidge, A treatise on algebraic plane curves, Dover Publications, Inc., New York, 1959. MR 0120551
- Robert J. Walker, Algebraic Curves, Princeton Mathematical Series, vol. 13, Princeton University Press, Princeton, N. J., 1950. MR 0033083
Additional Information
- © Copyright 1979 American Mathematical Society
- 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