Some results of regularity for Severi varieties of projective surfaces

For a linear system $|C|$ on a smooth projective surface $S$, whose general element is a smooth, irreducible curve, the Severi variety $V_{|C|, \delta}$ is the locally closed subscheme of $|C|$ which parametrizes irreducible curves with only $\delta$ nodes as singularities. In this paper we give numerical conditions on the class of divisors and upper-bounds on $\delta$ ensuring that the corresponding Severi variety is everywhere smooth of codimension $\delta$ in $|C|$ (regular, for short). In particular, we focus on surfaces of general type, since for such surfaces less is known than what is proven for other cases. Our result generalizes some results of Chiantini-Sernesi (1997) and of Greuel-Lossen-Shustin (1997 - in the case of nodes) as it is shown by some examples of Severi varieties on blown-up surfaces or surfaces in $\P^3$ which are elements of a component of the Noether-Lefschetz locus. We also consider examples of regular Severi varieties on surfaces in $\P^3$ of general type which contain a line.