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Brill--Noether Generality of Curves and K3 Surfaces

Irina Shatova

TL;DR

The paper proves that if a quasi-polarized K3 surface $(X,H)$ is Brill-Noether general in Mukai’s sense, then every smooth curve $C$ in the linear system $|H|$ is Brill-Noether general. The authors develop a framework based on Lazarsfeld bundles $F_{A,C}$ and their harmonic filtrations, linking BN properties of curves to decompositions of $H$ into divisors and to Mukai-vector inequalities. They establish sharp necessary conditions on such decompositions (e.g., forbidding long sums of positive divisors with certain square conditions) and show that, in all admissible configurations, a hypothetical non-general curve would force a contradiction via the harmonic filtration data. Consequently, BN generality of $(X,H)$ enforces BN generality for all curves in $|H|$, extending Lazarsfeld’s classical results to the BN-general quasi-polarized K3 setting; the methods also illuminate the structure of Lazarsfeld bundles and their filtrations in this geometric context.

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.

Brill--Noether Generality of Curves and K3 Surfaces

TL;DR

The paper proves that if a quasi-polarized K3 surface is Brill-Noether general in Mukai’s sense, then every smooth curve in the linear system is Brill-Noether general. The authors develop a framework based on Lazarsfeld bundles and their harmonic filtrations, linking BN properties of curves to decompositions of into divisors and to Mukai-vector inequalities. They establish sharp necessary conditions on such decompositions (e.g., forbidding long sums of positive divisors with certain square conditions) and show that, in all admissible configurations, a hypothetical non-general curve would force a contradiction via the harmonic filtration data. Consequently, BN generality of enforces BN generality for all curves in , extending Lazarsfeld’s classical results to the BN-general quasi-polarized K3 setting; the methods also illuminate the structure of Lazarsfeld bundles and their filtrations in this geometric context.

Abstract

Lazarsfeld proved Brill--Noether generality of any smooth curve in the linear system where is a polarized K3 surface with . 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 where is a Brill--Noether general quasi-polarized K3 surface.
Paper Structure (7 sections, 27 theorems, 95 equations)

This paper contains 7 sections, 27 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Divisors on Brill--Noether general K3 surface
  4. Linear systems on K3 surfaces
  5. Necessary conditions for Brill--Noether generality
  6. Harmonic filtrations of Lazarsfeld bundles
  7. Proof of Theorem \ref{['main theorem']}

Key Result

Theorem 1.2

For any $g \ge 0$ a general curve of genus $g$ is Brill--Noether general.

Theorems & Definitions (58)

  • Definition 1.1
  • Theorem 1.2: MR563378lazarsfeld1986brill
  • Definition 1.3
  • Definition 1.4: Mukai
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Proposition 2.3
  • ...and 48 more