Elliptic curves in Hyper-Kähler varieties
We show that the moduli space of elliptic curves of minimal degree in a general Fano variety of lines of a cubic four-fold is a non-singular curve of genus $631$. The curve admits a natural involution with connected quotient. We find that the general Fano contains precisely $3,780$ elliptic curves o...
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| Main Authors: | , |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
February 2021
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| In: |
International mathematics research notices
Year: 2021, Issue: 4, Pages: 2962-2990 |
| ISSN: | 1687-0247 |
| DOI: | 10.1093/imrn/rnaa016 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1093/imrn/rnaa016
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| Author Notes: | Denis Nesterov, Georg Oberdieck |
| Summary: | We show that the moduli space of elliptic curves of minimal degree in a general Fano variety of lines of a cubic four-fold is a non-singular curve of genus $631$. The curve admits a natural involution with connected quotient. We find that the general Fano contains precisely $3,780$ elliptic curves of minimal degree with fixed (general) $j$-invariant. More generally, we express (modulo a transversality result) the enumerative count of elliptic curves of minimal degree in hyper-Kähler varieties with fixed $j$-invariant in terms of Gromov-Witten invariants. In $K3^{[2]}$-type this leads to explicit formulas of these counts in terms of modular forms. |
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| Item Description: | Online veröffentlicht: 14. Februar 2020 Gesehen am 14.01.2025 |
| Physical Description: | Online Resource |
| ISSN: | 1687-0247 |
| DOI: | 10.1093/imrn/rnaa016 |