  In \cite{R} the author investigated invariants of relatively generic
structures on surface singularities generalising results of \cite{NNA1} and
\cite{NNA2} about generic analytic structures and generic line bundles to the
case of the relative setup, where we fix a given analytic type or line bundle
on a smaller subgraph or more generally on a smaller cycle and we choose a
relatively generic line bundle or analytic type on the large cycle and managed
to compute some of it's invariants, like geometric genus or $h^1$ of natural
line bundles.
  In \cite{NNAD} the authors investigated the images of Abel maps $c^{l'}(Z) :
\eca^{l'}(Z) \to \pic^{l'}(Z)$, where $l' \in - S'(|Z|)$, especially the
dimensions of the images of these maps and gave two algorithms to compute these
invariants from cohomology numbers of cycles and from periodic constants of
singularities we get from $\tX$ by blowing it up at generic points
sequentially. Furthemore in \cite{NNAD} the authors gave explicit combinatorial
formulas in the case of generic singularities.
  In this paper we want to generalise the theorems from \cite{NNAD} to the
relatively generic case. In this case we fix a subsingularity $\tX_1$ for a
subgraph $\mathcal{T}_1 \subset \mathcal{T}$ and a relatively generic
singularity $\tX$ corresponding to $\tX_1$. Furthermore we fix a line bundle
$\calL$ on $\tX_1$ and a Chern class $l' \in - S'$, such that $c^1(\calL) =
R(l')$. Our main goal in the article is to compute $\dim(c^{l'}(\eca^{ l',
\calL}(Z)))$ from invariants of the subsingularity $\tX_1$ and to conclude a
few corollaries.

Algebraic Geometry;Algebraic Topology