Delta invariant of curves on rational surfaces I. The analytic approach

TL;DR

This paper provides a new analytic formula for the delta invariant of reduced curve germs on rational surface singularities, linking topological, analytic, and algebraic perspectives.

Contribution

It introduces a concrete expression for the delta invariant based on the embedded topological type and relates it to adjoint ideals and the Riemann–Roch correction term.

Findings

Delta invariant expressed via embedded topological type

Identification with an analytic invariant from adjoint ideals

Connection to the local correction term in Riemann–Roch formula

Abstract

We prove that if (C,0) is a reduced curve germ on a rational surface singularity (X,0) then its delta invariant can be recovered by a concrete expression associated with the embedded topological type of the pair (X,C). Furthermore, we also identify it with another (a priori) embedded analytic invariant, which is motivated by the theory of adjoint ideals. Finally, we connect our formulae with the local correction term at singular points of the global Riemann--Roch formula, valid for projective normal surfaces, introduced by Blache.