Suppose I have a smooth projective surface $S\subset P^n$ embedded by a very ample linear system $|L|$. Consider now the generalized Severi varieties that parametrize curves oon $S$ with a given number of singularities of prescribed type. E.g.: one nodes, two nodes, one cusp, one node and one cusp, one tacnod, one triple point and one node... etc.

My question is: is the dimension of such varieties invariant when the surface undergoes blow-up and blow-down transformations?

I imagine that possibly one needs some hypothesis of irreducibilty on the curves.

Suppose I have a smooth projective surface $S\subset P^n$ embedded by a very ample linear system $|L|$. Consider now the generalized Severi varieties that parametrize curves on $S$ belonging to $|L|$, with a given number of singularities of prescribed type. E.g.: one nodes, two nodes, one cusp, one node and one cusp, one tacnode, one triple point and one node... etc.

My question is: is the dimension of such varieties invariant when the surface undergoes blow-up and blow-down transformations? For example: let $\sigma:\tilde{S}\to S$ be a blow up and $V(L)$ a certain Severi variety associated to $L$, are the dimensions of $V(L)$ and $V(\sigma^*L)$ the same?

Well-posing the question in the case of blow-downs seems a little trickier, but ideas are welcome.

I imagine that possibly one needs some hypothesis of irreducibilty on the curves.