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>4 has geometric genus g > 1 + 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 P^3 and on a general projectively Cohen-Macaulay surface in P^4.