On Green and Green-Lazarfeld conjectures for simple coverings of algebraic curves

TL;DR

This paper proves that for large genus, smooth curves with simple morphisms to certain base curves satisfy Green and Green-Lazarsfeld conjectures, using Aprodu's method, and explores cases with infinite gonality pencils.

Contribution

It extends the verification of Green and Green-Lazarsfeld conjectures to a broad class of algebraic curves with simple coverings, employing a novel application of Aprodu's technique.

Findings

For large genus, curves satisfy Green and Green-Lazarsfeld conjectures.

The method applies to curves with base curves of finite gonality.

Partial results are obtained for curves with infinite gonality pencils.

Abstract

Let X be a smooth genus g curve equipped with a simple morphism f: X -> C, where C is either the projective line or more generally any smooth curve whose gonality is computed by finitely many pencils. Here we apply a method developed by Aprodu to prove that if g is big enough then X satisfies both Green and Green-Lazarsfeld conjectures. We also partially address the case in which the gonality of C is computed by infinitely many pencils.