Combinatorial aspects of nodal curves

Details Combinatorial aspects of nodal curves Combinatorial aspects of nodal curves

2006-02-24

arxiv.org/abs/math/0602553v2

`Le Matematiche'', vol. LXI (2006), Fascicolo I, pp. 109-141

Mathematics

Algebraic Geometry

28 pages, to appear in Le Matematiche. Revised version: minor changes, references added

Scientific paper

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.

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