Computational aspects of gonal maps and radical parametrization of curves

Abstract

We develop in this article an algorithm that, given a projective curve C, computes a gonal map, that is, a finite morphism from C to P 1 of minimal degree. Our method is based on the computation of scrollar syzygies of canonical curves. We develop an improved version of our algorithm for curves with a unique gonal map and we discuss a characterization of such curves in terms of Betti numbers. Finally, we derive an efficient algorithm for radical parametrization of curves of gonality ≤ 4. C ֒→ P g−1 = P(Γ(C, ω C ))

Recognizing Hyperelliptic Graphs in Polynomial Time

“…We study all three kinds of gonality of graphs from the point of view of computational complexity. The analogous problem of computing the gonality of an algebraic curve is decidable [33]. From the definition of divisorial gonality, it follows that divisorial gonality is computable.…”

Point counting on curves using a gonality preserving lift

“…In genus five we used the discriminant curve for this, but in general the desired information should be traceable from (the first few steps of) a minimal free resolution See [44,Prop. 4.11].…”

A Version of Green’s Conjecture in Positive Characteristic

“…This scroll X is swept out by the unique g 1 k on C and contributes with an Eagon-Northcott complex of length (g − k) (with β g −k−1,g −k+1 (X) = g −k) to the minimal free resolution of the curve C ⊂ P g −1 (see e.g., [Sch86]). In [SSW13] the following conjecture is made.…”