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, delt a, the Severi variety V-n,delta(0) (S) is the subvariety of the linear syst em \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) h as at least one component that is reduced, of the expected dimension dim(\O -S(n)\) - delta. We also construct examples of reducible Severi varieties o n general surfaces of degree d greater than or equal to 8.