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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| Hauptverfasser: | , |
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| Dokumenttyp: | Article (Journal) |
| Sprache: | Englisch |
| Veröffentlicht: |
February 2021
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| In: |
International mathematics research notices
Year: 2021, Heft: 4, Pages: 2962-2990 |
| ISSN: | 1687-0247 |
| DOI: | 10.1093/imrn/rnaa016 |
| Online-Zugang: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1093/imrn/rnaa016
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| Verfasserangaben: | Denis Nesterov, Georg Oberdieck |
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| 520 | |a 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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