Residue Balancing on Singular Curves

TL;DR

This paper develops a comprehensive theory of residue maps on singular algebraic curves, linking local residue conditions to global differentials and their deformation properties, with applications to moduli spaces.

Contribution

It introduces the scheme--theoretic residue span theorem and a refined residue balancing principle applicable to arbitrary singularities, advancing the understanding of dualizing sheaves on singular curves.

Findings

Residue functionals at nodes span the dual of canonical sheaves when nodes are sufficiently numerous.

Global residue conditions are equivalent to local balancing conditions at singularities.

A refined balancing principle accounts for higher-order and conductor-level residue constraints.

Abstract

This paper investigates residue maps and their spanning properties for singular algebraic curves, with particular emphasis on three interconnected themes: the \emph{scheme--theoretic residue span}, the \emph{residue--balancing principle}, and \emph{residue balancing in the presence of arbitrary singularities}. Starting from the theory of dualizing sheaves on nodal curves, we reinterpret canonical and higher--order differentials as meromorphic objects on the normalization whose local principal parts are constrained by explicit residue conditions. A key result is the scheme--theoretic residue span theorem, which asserts that % for nodal curves of geometric genus $g$ with $\delta$ nodes, when $\delta \ge g$ the residue functionals at the nodes span $H^0(C,\omega_C)^\vee$, so canonical differentials are completely determined by their residue data. This provides a concrete, linear description of $H^0(C,\omega_C)$ and yields powerful applications to deformation theory, Severi varieties, and moduli problems. \vspace{0.1cm} We then develop the residue--balancing principle, showing that global residue conditions on each irreducible component of a singular curve are equivalent to local balancing conditions at the singular points. This equivalence clarifies the local--to--global structure of dualizing sheaves and extends naturally to $k$--differentials. Finally, we address the case of arbitrary singularities, where nodes no longer suffice to describe local geometry. Using normalization and the conductor ideal, we formulate a refined balancing principle that replaces simple residue cancellation by higher--order and conductor--level constraints. Together, these results provide a unified framework for understanding how local singular behavior governs global differentials and their deformations.