On the Severi varieties of surfaces in P^3

For a smooth surface S in P-3 of degree d and for positive integers n, delta, the Severi variety V-n,delta(0) (S) is the subvariety of the linear system O-S(n) which parametrizes curves with delta nodes. We show that for S general, n greater than or equal to d and for all delta with 0 less than or equal to delta less than or equal to dim(O-S(n)), then V-n,delta(0)(S) has at least one component that is reduced, of the expected dimension dim(O-S(n)) - delta. We also construct examples of reducible Severi varieties on general surfaces of degree d greater than or equal to 8.