Invariants of relatively generic surface singularities II. Images of Abel maps

Abstract

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.