The Severi variety V_{n,d} of a smooth projective surface S is defined as the
subvariety of the linear system |O_S(n)|, which parametrizes curves with d
nodes. We show that, for a general surface S of degree k in P^3 and for all
n>k-1, d=0,...,dim(|O_S(n)|), there exists one component of V_{n,d} which is
reduced, of the expected dimension dim(|O_S(n)|)-d. Components of the expected
dimension are the easiest to handle, trying to settle an enumerative geometry
for singular curves on surfaces. On the other hand, we also construct examples
of reducible Severi varieties, on general surfaces of degree k>7 in P^3.Comment: AMSTeX, AMSppt style, 14 page
