On the irreducibility of Severi varieties on K3 surfaces

Let $(S,L)$ be a polarized $K3$ surface of genus $p⩾11$ such that $\mathrm{P}\mathrm{i}\mathrm{c}(S)=\mathbf{Z}[L]$, and $\delta$ a non-negative integer. We prove that if $p⩾4\delta -3$, then the Severi variety of $\delta$-nodal curves in $\mathrm{\mid }L\mathrm{\mid }$ is irreducible.