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 P-3 of degree d greater than or equal to 5 has geometric genus g >1 +de gC(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 P-3 and on a general projectively Cohen-Macaulay surface in P-4.