Let $S$ be a projective surface and $L$ an ample line bundle on $S$. The Severi variety $\mathcal V_{\mathcal L,\delta}$ parametrizes curves with $\delta$ nodes and no other singularities in the linear system $\mathcal L = \mathbb P(H^0(X,L))$. In these notes the author says that "general nonsense" of deformation theory gives the lower bound $h^0(\mathcal O_S(C)) - 1 - \delta$ for the dimension of $\mathcal V_{\mathcal L,\delta}$ at a curve $C$ (at least on a surface with $h^1(\mathcal O_S) = 0$.

This is a silly question, but does this imply that if we pick $\delta$ no larger than $h^0(\mathcal O_S(C)) - 2$ then we get nodal curves with $\delta$ nodes in $\mathcal L$? Or is this a case of "if the space isn't empty, then this is its dimension"?

In the latter case, are there any general conditions that let us know the space isn't empty? I'm particularily interested in the case where $K_S$ and $L$ are positive multiples of the same ample bundle and we try to find $\delta = h^0(S,L) - c$ nodes ($c$ some hopefully small constant).