I am referring to line 1-2 of page 472 in Griffiths Harris.
M is a compact complex manifold of dimension 2 that may be embedded in projective space. L is an arbitrary line bundle on M. How can one choose a divisor D on M sufficiently positive so that both the linear series |D| and |L+D| contain smooth, irreducible divisors? Thanks.
To ensure that $|D|$ and $|L+D|$ both contain smooth irreducible divisors, it's enough to arrange that they are both very ample linear systems. (Then Bertini's theorem says that the general member of each is smooth and irreducible.)
To ensure they are both very ample, use the following fact: for any line bundle $L$ on $M$, we can write $L=A-B$, with $A$ and $B$ both very ample. Then taking $D=B$ we get $L+D=A$, so both $D$ and $L+D$ are very ample, as required.