Some results of regularity for Severi varieties of projective surfaces

For a linear system |C| on a smooth projective surface S, whose general member is a smooth, irreducible curve, the Severi variety V_{|C|, d} is the locally closed subscheme of |C| which parametrizes curves with only d nodes as singularities. In this paper we give numerical conditions on the class of divisors and upper bounds on d, ensuring that the corresponding Severi variety is smooth of codimension d. Our result generalizes what is proven in previous papers in the literature. We also consider examples of smooth Severi varieties on surfaces of general type in P^3 which contain a line.