Abstract
The gonality sequence $(\gamma_r)_{r\geq1}$ of a finite graph/metric graph/algebraic curve comprises the minimal degrees $\gamma_r$ of linear systems of rank $r$. For the complete graph $K_d$, we show that $\gamma_r = kd - h$ if $r<g=\frac{(d-1)(d-2)}{2}$, where $k$ and $h$ are the uniquely determined integers such that $r = \frac{k(k+3)}{2} - h$ with $1\leq k\leq d-3$ and $0 \leq h \leq k $. This shows that the graph $K_d$ has the gonality sequence of a smooth plane curve of degree $d$. The same result holds for the corresponding metric graphs.
Highlights
Baker and Norine in [BN07] introduced a theory of linear systems on graphs, later generalized by several authors to metric graphs and other combinatorial objects [MZ08, GK08]
With the notation gsr we indicate a linear system of degree s and rank r
The gonality sequencer≥1 of a finite graph is defined as γr := min{s such that there exists a gsr}
Summary
Baker and Norine in [BN07] introduced a theory of linear systems on graphs, later generalized by several authors to metric graphs and other combinatorial objects [MZ08, GK08] It presents strong analogies with the one on algebraic curves. We translate this problem into a property of sequences of integers (αi)i=1,...,d−1 satisfying certain hypotheses. We conclude by remarking that our arguments do not directly extend to complete metric graphs with arbitrary edge lengths A natural question in this setting is whether the complete graph and the smooth plane curve have the same gonality sequence. The first part of Theorem 1 can be deduced from the result on plane curves
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