Details
| Original language | English |
|---|---|
| Article number | 2250008 |
| Number of pages | 21 |
| Journal | Communications in Contemporary Mathematics |
| Volume | 25 |
| Issue number | 6 |
| Early online date | 10 Mar 2022 |
| Publication status | E-pub ahead of print - 10 Mar 2022 |
Abstract
Given d ∈ N, we prove that all smooth K3 surfaces (over any field of characteristic p ≠ 2, 3) of degree greater than 84d2 contain at most 24 rational curves of degree at most d. In the exceptional characteristics, the same bounds hold for non-unirational K3 surfaces, and we develop analogous results in the unirational case. For d ≥ 3, we also construct K3 surfaces of any degree greater than 4d(d + 1) with 24 rational curves of degree exactly d, thus attaining the above bounds.
Keywords
- elliptic fibration, hyperbolic lattice, K3 surface, parabolic lattice, polarization, rational curve
ASJC Scopus subject areas
- Mathematics(all)
- Mathematics(all)
- Applied Mathematics
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In: Communications in Contemporary Mathematics, Vol. 25, No. 6, 2250008, 10.03.2022.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - 24 rational curves on K3 surfaces
AU - Rams, Sławomir
AU - Schütt, Matthias
N1 - Acknowledgement: We are grateful to the anonymous referee for valuable comments. S. Rams would like to thank J. Byszewski for inspiring remarks
PY - 2022/3/10
Y1 - 2022/3/10
N2 - Given d ∈ N, we prove that all smooth K3 surfaces (over any field of characteristic p ≠ 2, 3) of degree greater than 84d2 contain at most 24 rational curves of degree at most d. In the exceptional characteristics, the same bounds hold for non-unirational K3 surfaces, and we develop analogous results in the unirational case. For d ≥ 3, we also construct K3 surfaces of any degree greater than 4d(d + 1) with 24 rational curves of degree exactly d, thus attaining the above bounds.
AB - Given d ∈ N, we prove that all smooth K3 surfaces (over any field of characteristic p ≠ 2, 3) of degree greater than 84d2 contain at most 24 rational curves of degree at most d. In the exceptional characteristics, the same bounds hold for non-unirational K3 surfaces, and we develop analogous results in the unirational case. For d ≥ 3, we also construct K3 surfaces of any degree greater than 4d(d + 1) with 24 rational curves of degree exactly d, thus attaining the above bounds.
KW - elliptic fibration
KW - hyperbolic lattice
KW - K3 surface
KW - parabolic lattice
KW - polarization
KW - rational curve
UR - http://www.scopus.com/inward/record.url?scp=85159636012&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1907.04182
DO - 10.48550/arXiv.1907.04182
M3 - Article
VL - 25
JO - Communications in Contemporary Mathematics
JF - Communications in Contemporary Mathematics
SN - 0219-1997
IS - 6
M1 - 2250008
ER -