Geometry of tropical moduli spaces and linkage of graphs
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 tropicalization of an algebraic variety is connected in codimension one, which means that it remains topologically connected after any codimension at least two locus has been removed. The set of all tropical curves fits together to form what is known as the moduli space of tropical curves, and this topological space is stratified by the number of edges in the graphs underlying the tropical curves it parameterizes. The dimensions of these strata are dual to the dimension of the corresponding strata in the Deligne-Mumford moduli space. The author of the present paper points out that although the moduli space of tropical curves appears to be very close to being a tropical variety, it is not literally a tropicalization of a subvariety of a toric variety as defined by Sturmfels et al., and hence one cannot immediately apply the above-mentioned codimension one connectivity result. In fact, Caporaso, with Abramovich and Payne, have recently shown that the moduli space of tropical curves is the tropicalization of the Deligne-Mumford moduli space, when one uses a type of tropicalization coming from the theory of toroidal embeddings, but this is a slightly different framework than the purely algebraic approach Sturmfels. The main punchline of the paper then is that the combinatorial linkage theorem described above (for the case of trivalent curves, namely \(p=3\)) implies that the moduli space of tropical curves is connected in codimension one, lending further evidence that these various theories of tropicalization ought to be united in some way. In addition to the linkage of graphs and its topological manifestation in the tropical world, the author shows that if the graphs in question are ``highly'' connected in the sense that they remain connected after any pair of edges is removed, then they can be linked by a finite sequence of edge contractions as above such that all the intermediate graphs share this connectivity property. Earlier work of the author and coauthors showed that the tropical Schottky locus, namely the image in the moduli of tropical abelian varieties of the tropical Torelli map, coincides with the restriction to tropical curves whose underlying graph has this connectivity property. Consequently, the author concludes that the tropical Schottky locus is also connected in codimension one.
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