Combinatorial aspects of nodal curves

TL;DR

This paper investigates the degree class group of nodal curves, constructing graphs with cyclic degree class groups, providing recursive formulas for their size, and analyzing how this invariant changes under geometric operations.

Contribution

It introduces a broad family of graphs with cyclic degree class groups and derives recursive formulas for their cardinality, enhancing understanding of this invariant's behavior.

Findings

Constructed a wide family of graphs with cyclic degree class groups

Provided recursive formulas for the size of the degree class group

Analyzed the effect of blow-up and normalization on the degree class group

Abstract

To any nodal curve $C$ is associated the degree class group, a combinatorial invariant which plays an important role in the compactification of the generalised Jacobian of $C$ and in the construction of the N\'eron model of the Picard variety of families of curves having $C$ as special fibre. In this paper we study this invariant. More precisely, we construct a wide family of graphs having cyclic degree class group and we provide a recursive formula for the cardinality of the degree class group of the members of this family. Moreover, we analyse the behaviour of the degree class group under standard geometrical operations on the curve, such as the blow up and the normalisation of a node.