Brill--Noether Generality of Curves and K3 Surfaces

TL;DR

This paper proves that smooth curves in linear systems on Brill--Noether general quasi-polarized K3 surfaces are themselves Brill--Noether general, extending previous results from polarized to quasi-polarized cases.

Contribution

It establishes Brill--Noether generality for smooth curves on quasi-polarized K3 surfaces, broadening the class of surfaces where this property holds.

Findings

Brill--Noether generality holds for smooth curves on quasi-polarized K3 surfaces.

Extends Lazarsfeld's result from polarized to quasi-polarized K3 surfaces.

Confirms Mukai's notion of Brill--Noether generality in this broader setting.

Abstract

Lazarsfeld proved Brill--Noether generality of any smooth curve in the linear system $|H|$ where $(X,H)$ is a polarized K3 surface with $\mathrm{Pic}(X) = \mathbb{Z}\cdot H$. Mukai introduced the notion of Brill--Noether generality for quasi-polarized K3 surfaces. We prove Brill--Noether generality of any smooth curve in the linear system $|H|$ where $(X,H)$ is a Brill--Noether general quasi-polarized K3 surface.