Focal loci of families and the genus of curves on surfaces

In this article we apply the classical method of focal loci of families to give a lower bound for the genus of curves lying on general surfaces. First we translate and reprove Xu's result that any curve $C$ on a general surface in $\mathbb{P}^{3}$ of degree $d \geq 5$ has geometric genus $g > 1 + \hbox {deg} C (d - 5) / 2$. Then we prove a similar lower bound for the curves lying on a general surface in a given component of the Noether-Lefschetz locus in $\mathbb{P}^{3}$ and on a general projectively Cohen-Macaulay surface in $\mathbb{P}^{4}$.