Brill–Noether theory of curves on $\protect \mathbb{P}^1 \times \protect \mathbb{P}^1$: tropical and classical approaches

Abstract

The gonality sequence (dr) r≥1 of a smooth algebraic curve comprises the minimal degrees dr of linear systems of rank r. We explain two approaches to compute the gonality sequence of smooth curves in P 1 × P 1 : a tropical and a classical approach. The tropical approach uses the recently developed Brill-Noether theory on tropical curves and Baker's specialization of linear systems from curves to metric graphs [1]. The classical one extends the work [11] of Hartshorne on plane curves to curves on P 1 × P 1 .

On the slope inequalities for extremal curves

“…These results motivate the discussion of slope inequalities for specific classes of curves. Some sporadic examples of curves violating some slope inequality can be found by in [2], [3], [12], [11], [8] and various families have been detected in [12]. Among such examples there are extremal curves, that is curves attaining the Castelnuovo bound for the genus.…”

Gonality sequences of graphs

“…In this paper, we study the gonality sequence of a graph G, which is the sequence of r th gonalities of G as r ranges from 1 to ∞: gon 1 (G), gon 2 (G), gon 3 (G), gon 4 (G), • • • Recent progress has been made towards determining the gonality sequences of various families of graphs, including the complete graphs K n [10] and the complete bipartite graphs K m,n [8]. Work has also been done to study higher gonalities of Erdös-Renyi random graphs [25].…”