On the existence of primitive pencils for smooth curves

TL;DR

This paper proves the expected dimension and primitivity of certain linear systems on smooth curves with high gonality and genus, extending understanding of primitive pencils and their existence.

Contribution

It establishes the existence and properties of primitive pencils on smooth curves with specified gonality and genus, including new existence results for certain linear systems.

Findings

W^1_d(C) has the expected dimension for specified d

General elements of irreducible components are primitive under certain conditions

Existence of complete and primitive g^1_{g-k+3} in specific cases

Abstract

Let $C$ be a smooth curve with gonality $k\ge 6$ and genus $g\ge 2k^2+5k-6$. We prove that $W^1_d({C})$ has the expected dimension and that the general element of any irreducible component of $W^1_d({C})$ is primitive if either $g-k+4\le d\le g-2$ or $d=g-k+3$ and either $k$ is odd or $C$ is not a double covering of a curve of gonality $k/2$ and genus $k-3$. Even in the latter case we prove the existence of a complete and primitive $g^1_{g-k+3}$.