Let
W
p
{W_p}
be a Riemann surface of genus p admitting a simple linear series
g
n
r
g_n^r
where
n
=
m
(
r
−
1
)
+
q
,
q
=
2
,
3
,
.
.
.
,
r
−
1
n\, =\, m(r - 1)\, +\, q,\,\, q\, =\, 2,\,3,...,\,r - 1
, or r. Castelnuovo’s inequality states that (1)
2
p
⩽
2
f
(
r
,
n
,
1
)
=
m
(
m
−
1
)
(
r
−
1
)
+
2
m
(
q
−
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
q\, >\, r
implies that
W
p
{W_p}
admits a plane model of degree
n
−
r
+
2
n\, -\, r\, +\, 2
with
r
−
2
r\, -\, 2
m-fold singularities and one
(
n
−
r
+
1
−
m
)
(n\, -\, r\, +\, 1\, -\, m)
-fold singularity. Formula (1) generalizes as follows. Suppose
W
p
{W_p}
admits s simple linear series
g
n
r
g_n^r
where
n
=
m
(
r
s
−
1
)
+
q
n\, =\, m(rs - 1)\, +\, q
and
q
=
−
(
s
−
1
)
r
+
2
,
−
(
s
−
1
)
r
+
3
,
…
,
r
−
1
q = - (s - 1)r + 2,\, - (s - 1)r + 3,\ldots ,r - 1
, or r. For q consider the cases
v
=
0
,
1
,
…
,
s
−
1
v = 0,1,\ldots ,s - 1
as follows: case
v
=
0
:
2
⩽
q
⩽
r
v\, =\, 0:2 \leqslant q \leqslant r
, case
v
>
0
:
2
⩽
q
+
v
r
⩽
r
+
1
v > 0:2 \leqslant q + vr \leqslant r + 1
. Then (2)
2
p
⩽
2
f
(
r
,
n
,
s
)
=
m
2
(
r
s
2
−
s
)
+
m
s
(
2
q
−
1
−
r
)
−
v
(
v
−
1
)
r
−
2
v
(
q
−
1
)
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
′
n\, = \,m’r\, + \,q’
,
q
′
=
1
,
2
,
…
,
r
−
1
q’\, = \,1,\,2,\, \ldots ,\,r\, - \,1
, or r and
W
p
{W_p}
admits
m
′
+
1
m’\, + \,1
simple
g
n
r
g_n^{r}
’s then (3)
2
p
⩽
2
f
(
r
+
1
,
n
+
1
,
1
)
=
m
′
(
m
′
−
1
)
r
+
2
m
′
q
′
2p\, \leqslant \,2f\,(r\, + \,1,\,n\, + \,1,\,1)\, = \,m’\,(m’\, - \,1)r\, + \,2m’\,q’\,
. Since
f
(
r
,
n
,
2
)
>
f
(
r
,
n
,
1
)
f(r,\,n,\,2)\, > \,f(r,\,n,\,1)
we obtain as a corollary: if
p
=
f
(
r
,
n
,
1
)
p\, = \,f(r,\,n,\,1)
then
W
p
{W_p}
admits at most one simple
g
n
r
g_n^r
.