Linear Systems on Metric Graphs, Gonality Sequence and Lifting Problems

A combinatorial theory of linear systems on graphs and metric graphs has been introduced in analogy with the one on algebraic curves. The interplay is given by the Specialization Lemma. Let \(X\) be a smooth curve over the field of fractions of a complete discrete valuation ring and let \( \mathfrak{X} \) be a strongly semistable regular model of \(X\). It is possible to specialize a divisor on the curve to a divisor on the dual graph of the special fiber of \( \mathfrak{X} \); through this process the rank of the divisor can only increase.

The complete graph \(K_d\) pops up if we take a model of a smooth plane curve of degree \(d\) degenerating to a union of \(d\) lines. Moreover omitting edges from \(K_d\) can be interpreted as resolving singularities of a plane curve. In this talk we present some results on linear systems on complete graphs and complete graphs with a small number of omitted edges, and we will compare them with the corresponding results on plane curves. In particular, we compute the gonality sequence of complete graphs and the gonality of graphs obtained by omitting edges. We explain how to lift these graphs to curves with the same gonality using models of plane curves with nodes and harmonic morphisms.

This is partially a joint work with Filip Cools.