The base components of the dualizing sheaf of a curve on a surface

TL;DR

This paper investigates the structure of the fixed part of the dualizing sheaf for a 1-connected curve on a smooth surface, revealing that the fixed part consists of smooth rational curves with genus less than one.

Contribution

It establishes that the divisorial fixed part of the dualizing sheaf, if non-empty, has arithmetic genus less than one and consists of smooth rational components.

Findings

Fixed part has arithmetic genus < 1

Components are smooth rational curves

Provides structural insight into dualizing sheaf on curves

Abstract

This note studies the structure of the divisorial fixed part of the dualizing sheaf of a 1-connected curve D on a smooth surface S. It is shown that if the divisorial fixed part F of the dualizing sheaf is non empty then it has arithmetic genus<1 and each component of F is a smooth rational curve.