On Castelnuovo’s inequality for algebraic curves. I

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 $g_n^r$.