The following is proved: Given a family of projective reduced curves
X
→
T
X \to T
(
T
T
irreducible), if
X
t
{X_t}
(the general curve) is integral and
X
0
{X_0}
is a special curve (having irreducible components
X
1
,
…
,
X
r
{X_1}, \ldots ,{X_r}
), then
∑
i
=
1
r
g
i
(
X
i
)
⩽
g
(
X
t
)
\sum \nolimits _{i = 1}^r {{g_i}({X_i}) \leqslant g({X_t})}
, where
g
(
Z
)
=
g(Z) =
geometric genus of
Z
Z
. Conversely, if
A
A
is a reduced plane projective curve, of degree
n
n
with irreducible components
X
1
,
…
,
X
r
{X_1}, \ldots ,{X_r}
, and
g
g
satisfies
∑
i
=
1
r
g
i
(
X
i
)
⩽
g
⩽
1
2
(
n
−
1
)
(
n
−
2
)
\sum \nolimits _{i = 1}^r {{g_i}({X_i}) \leqslant g \leqslant \frac {1} {2}(n - 1)(n - 2)}
, then a family of plane curves
X
→
T
X \to T
(with
T
T
integral) exists, where for some
t
0
∈
T
,
X
t
0
=
Z
{t_0} \in T,{X_{{t_0}}} = Z
and for
t
t
generic,
X
t
{X_t}
is integral and has only nodes as singularities. Results of this type appear in an old paper by G. Albanese, but the exposition is rather obscure.