An elementary proof of Lelli Chiesa's theorem on constancy of second coordinate of gonality sequence

TL;DR

This paper provides an elementary proof of Lelli Chiesa's theorem for the case r=2 on K3 surfaces, showing the constancy of the second coordinate of the gonality sequence along smooth curves, and extends the theorem to r≥3.

Contribution

It offers a simplified, elementary proof for the r=2 case and extends the theorem to higher r under weaker conditions.

Findings

Proved the constancy of the second gonality coordinate for r=2 on K3 surfaces.

Extended Lelli Chiesa's theorem to r≥3 with weaker assumptions.

Provided an alternative proof approach that simplifies understanding.

Abstract

Let $X$ be a K3 surface and $L$ be an ample line bundle on it. In this article we will give an alternative and elementary proof of Lelli Chiesa's Theorem in the case of $r= 2$. More precisely we will prove that that under certain condition the second co-ordinate of the gonality sequence is constant along the smooth curves in the linear system $|L|$. Using Lelli Chiesa's theorem for $r \ge 3$ we also extend Lelli Chiesa's Theorem in the case of $r= 2$ in weaker condition.