On specializations of curves. I

Abstract:

The following is proved: Given a family of projective reduced curves $X \to T$ ($T$ irreducible), if ${X_t}$ (the general curve) is integral and ${X_0}$ is a special curve (having irreducible components ${X_1}, \ldots ,{X_r}$), then $\sum \nolimits _{i = 1}^r {{g_i}({X_i}) \leqslant g({X_t})}$, where $g(Z) =$ geometric genus of $Z$. Conversely, if $A$ is a reduced plane projective curve, of degree $n$ with irreducible components ${X_1}, \ldots ,{X_r}$, and $g$ satisfies $\sum \nolimits _{i = 1}^r {{g_i}({X_i}) \leqslant g \leqslant \frac {1} {2}(n - 1)(n - 2)}$, then a family of plane curves $X \to T$ (with $T$ integral) exists, where for some ${t_0} \in T,{X_{{t_0}}} = Z$ and for $t$ generic, ${X_t}$ 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.