The possible values of geometric genera of normal surface singularities

TL;DR

This paper demonstrates that for normal surface singularities with a fixed resolution graph, the possible geometric genera and certain cohomology dimensions form continuous intervals of integers, revealing a structured range of these invariants.

Contribution

It establishes that the sets of possible geometric genera and cohomology dimensions for fixed resolution graphs are intervals of integers, providing new insights into the structure of surface singularities.

Findings

Geometric genera form an interval of integers for fixed resolution graphs.

Cohomology dimensions $h^1$ form an interval of integers for fixed resolution and Chern class.

Results reveal a structured range of invariants for normal surface singularities.

Abstract

In this article we prove that the possible geometric genuses $p_g(\tX)$ corresponding to normal surface singularities $\tX$ with fixed negative definite resolution graph $\mathcal{T}$ form an interval of integers. Similarly let us have a resolution graph $\mathcal{T}$ and a fixed normal surface singularity $(X, 0)$ with resolution $\tX$ and resolution graph $\mathcal{T}$, furthermore consider a Chern class $l' \in L'$. We prove that the possible values of $h^1(\tX, \calL)$, where $\calL \in \pic^{l'}(\tX)$ form an interval of integers.