Low degree rational curves on quasi-polarized K3 surfaces

Sławomir Rams; Matthias Schütt

doi:10.1016/j.jpaa.2025.107904

Low degree rational curves on quasi-polarized K3 surfaces

Sławomir Rams^*, Matthias Schütt

^*Corresponding author for this work

Jagiellonian University

Research output: Contribution to journal › Article › Research › peer review

Abstract

We prove that there are at most $(24-r_0)$ low-degree rational curves on high-degree models of K3 surfaces with at most Du Val singularities, where $r_0$ is the number of exceptional divisors on the minimal resolution. We also provide several existence results in the above setting (i.e. for rational curves on quasi-polarized K3 surfaces), which imply that for various values of $r_0$ our bound cannot be improved.

Original language	English
Article number	107904
Number of pages	21
Journal	Journal of Pure and Applied Algebra
Volume	229
Issue number	2
E-pub ahead of print	7 Feb 2025
DOIs	https://doi.org/10.1016/j.jpaa.2025.107904 https://doi.org/10.48550/arXiv.2403.08064
Publication status	Published - Feb 2025

Keywords

Elliptic fibration
Hyperbolic lattice
K3 surface
Parabolic lattice
Polarization
Rational curve

ASJC Scopus subject areas

Algebra and Number Theory

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title = "Low degree rational curves on quasi-polarized K3 surfaces",

abstract = " We prove that there are at most $(24-r_0)$ low-degree rational curves on high-degree models of K3 surfaces with at most Du Val singularities, where $r_0$ is the number of exceptional divisors on the minimal resolution. We also provide several existence results in the above setting (i.e. for rational curves on quasi-polarized K3 surfaces), which imply that for various values of $r_0$ our bound cannot be improved. ",

keywords = "Elliptic fibration, Hyperbolic lattice, K3 surface, Parabolic lattice, Polarization, Rational curve",

author = "S{\l}awomir Rams and Matthias Sch{\"u}tt",

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AU - Schütt, Matthias

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AB - We prove that there are at most $(24-r_0)$ low-degree rational curves on high-degree models of K3 surfaces with at most Du Val singularities, where $r_0$ is the number of exceptional divisors on the minimal resolution. We also provide several existence results in the above setting (i.e. for rational curves on quasi-polarized K3 surfaces), which imply that for various values of $r_0$ our bound cannot be improved.

KW - Elliptic fibration

KW - Hyperbolic lattice

KW - K3 surface

KW - Parabolic lattice

KW - Polarization

KW - Rational curve

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DO - 10.1016/j.jpaa.2025.107904

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JF - Journal of Pure and Applied Algebra

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M1 - 107904

ER -

Low degree rational curves on quasi-polarized K3 surfaces

Abstract

Keywords

ASJC Scopus subject areas

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