Cohomology of natural line bundles on generic normal surface singularities

Abstract

Let $\mathcal{T}$ be an arbitrary resolution graph and $(X, 0)$ a generic complex analytic normal surface singularity, and $\tilde{X}$ a generic resolution corresponding to it. Fix an effective integer cycle $Z$ supported on the exceptional curve and also an arbitrary Chern class $Z' \in L'$. In this article we aim to compute the cohomology numbers $h^1(\mathcal{O}_{Z}(Z'))$. Notice, that the case $Z'_v < 0, v \in |Z|$ was discussed in \cite{NNA2}, where the main theorem was, that in this special case these cohomology numbers equal to the cohomology numbers of the generic line bundle in $\text{Pic}^{Z'}(Z)$ However the condition $Z'_v < 0, v \in |Z|$ was crucial in the proof and without this assumption the statement is far from being true. In this article using the tecniques of relatively generic line bundles and relatively generic analytic structures from \cite{R} we give combinatorial algorithms to compute the cohomology numbers of natural line bundles $h^1(\calO_{Z}(Z'))$ for generic singularities in all cases.