Combinatorics of compactified universal Jacobians. (English) Zbl 1408.14093
This article deals with the combinatorial structure of the compactified universal Jacobian over \(\overline{\mathcal M}_{g}\), the compactified moduli space of smooth curves of genus \(g\ge 2\), in degrees \(g-1\) and \(g\) and its relation with orientations of stable graphs.
The first section, after a short introduction, recalls the basics of graph theory needed in the following.
The second one studies the functorial behaviour of generalized orientations of graphs with respect to edge-contraction, proving, in particular, the functoriality of the posets of totally cyclic and rooted orientations on a stable graph.
Eventually, the last section applies the graph-theoretic results to the algebro-geometric setting, exhibiting, in particular, a graded stratification of the compactified Jacobian of degree \(g-1\) (respectively \(g\)) of a stable curve of genus \(g\) in terms of totally cyclic (resp. rooted) orientations on the dual graph of the curve.
The first section, after a short introduction, recalls the basics of graph theory needed in the following.
The second one studies the functorial behaviour of generalized orientations of graphs with respect to edge-contraction, proving, in particular, the functoriality of the posets of totally cyclic and rooted orientations on a stable graph.
Eventually, the last section applies the graph-theoretic results to the algebro-geometric setting, exhibiting, in particular, a graded stratification of the compactified Jacobian of degree \(g-1\) (respectively \(g\)) of a stable curve of genus \(g\) in terms of totally cyclic (resp. rooted) orientations on the dual graph of the curve.
Reviewer: Michele Savarese (Roma)
MSC:
| 14H10 | Families, moduli of curves (algebraic) |
| 14H40 | Jacobians, Prym varieties |
| 05C22 | Signed and weighted graphs |
Keywords:
compactified Jacobian; moduli of stable curves; graph; orientation; compactified universal JacobianReferences:
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