On the gonality of graph curves
A graph curve \(X\) of genus \(g>2\) is a nodal curve, with \(2g-2\) smooth and rational, irreducible components, such that each of them intersects exactly three other components. These curves are stable, and they appear in the boundary of moduli spaces of smooth curves. Thus, the behaviour of linear series on graph curves is relevant for our knowledge of the Brill-Noether theory of general curves. The author studies the gonality of these reducible curves. As usual, when linear series on reducible curves are considered, the first task is to provide good definitions, to work with. The author defines the gonality of \(X\) as the minimum of the set \(\{d:\) there exists a strongly positive, spanned sheaf \(F\) of rank \(1\) and depth \(1\) on any component of \(X\), such that \(\deg(F)=d\}\). Then, the author shows that the gonality of a graph curve is \(2g-2\). It is interesting to notice that the gonality is computed by a sheaf \(F\) whose restriction to every component has degree \(1\). The sheaf \(F\) provides a map \(X\to\mathbb P^n\) which sends every component to a line.
scientific article; zbMATH DE number 5598924