Geometry of tropical moduli spaces and linkage of graphs
type: Item
instance of: scholarly article
title: Geometry of tropical moduli spaces and linkage of graphs
zbMATH Open document ID: 1234.14043
author: Lucia Caporaso
published in: Journal of Combinatorial Theory. Series A
publication date: 2012-02-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1001.2815
review text: This paper proves a purely combinatorial result about graphs which then translates into a topological property of the moduli space of tropical curves, as we now discuss. The main theorem, from the combinatorial side, is that any pair of graphs which have the same topological genus and the same uniform valency can be related by a finite sequence of edge contractions. The valency condition is simply that all the vertices of the two graphs have the same valency, i.e., number of incidence edges. If this number is \(p\), such graphs are sometimes called \(p\)-regular. The genus is the genus as a 1-dimension CW complex. The finite sequence relating the two graphs is a sequence of operations in which, alternately, an edge of the graph is contracted down to a point, then a different vertex is ``blown up'' to an edge. Thus, this main result can be viewed as a type of combinatorial surgery which transforms any given \(p\)-regular graph into any other \(p\)-regular graph of the same genus. The proof of this result is, not surprisingly, purely combinatorial, but the main application is quite pleasingly of geometrical interest, as explained below. A key step in the proof is to reduce to the case of so-called Hamiltonian graphs by showing that every \(p\)-regular graph is linked by a finite sequence as described above to a \(p\)-regular graph possessing a cycle passing through every vertex exactly once. It has been known for a while now that metric graphs can be viewed as a sort of combinatorial degeneration of algebraic curves. This is the perspective of tropical geometry, in which algebraic varieties of arbitrary dimension are replaced by polyhedral complexes of the same dimension. In the case of curves, one thinks of the graph as being the dual graph of a (nodal) curve being tropicalized, and the lengths of the edges in the tropical curve represent in a sense the relative rates at which a smooth curve degenerates to this nodal curve. For any dimension, it is known that the tro…
reviewed by: Noah Giansiracusa
Mathematics Subject Classification ID: 14T05
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 05C40
Mathematics Subject Classification ID: 05C38
zbMATH DE Number: 6005655
zbMATH Keywords: graph
zbMATH Keywords: connectedness
zbMATH Keywords: tropical curve
zbMATH Keywords: moduli
MaRDI profile type: Publication
OpenAlex ID: W2078095196
arXiv ID: 1001.2815
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DOI: 10.1016/J.JCTA.2011.11.011
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description: scientific article