Abstract
Let Y be a cubic fourfold not containing any plane, F(Y) be the variety of lines in Y, Z(Y) be the Lehn-Lehn-Sorger-van Straten hyperkähler eightfold constructed in Lehn et al. (J für die Reine und Angewandte Math (Crelles J) 2017(731):87–128, 2017). In Voisin (Remarks and questions on Coisotropic subvarieties and 0-cycles of hyper-Kähler varieties. Springer International Publishing, Berlin, 2016), Voisin defined a degree six rational map \(v: F(Y)\times F(Y) \dashrightarrow Z(Y),\) relating the two hyperkähler varieties F(Y) and Z(Y). In this note, we reinterpret this map v using moduli spaces of Bridgeland stable objects in a triangulated category associated with Y, called a Kuznetsov component of Y. We prove that the Voisin map v can be resolved by blowing up the incident locus of intersecting lines in \(F(Y)\times F(Y)\) endowed with the reduced scheme structure. As a consequence of our approach, we also show that the above-mentioned blowup is a relative Quot scheme over Z(Y) parameterizing quotients in a heart of the Kuznetsov component of Y.
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Acknowledgements
The idea of studying the Voisin map using moduli and relative Quot schemes belongs to Arend Bayer, Aaron Bertram, and Emanuele Macrì. I am grateful to them for leaving the question to me. Also, many thanks to Qingyuan Jiang, Sukhendu Mehrotra, Alex Perry, Paolo Stellari, Franco Rota, and Xiaolei Zhao for helpful discussions. Most of these discussions happened during a wonderful CIMI/FRG workshop in Toulouse, I would like to thank the organizers Marcello Bernardara and Emanuele Macrì.
Funding
This research was partially supported by National Science Foundation Focused Research Group grant DMS-1663813.
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Chen, H. The Voisin map via families of extensions. Math. Z. 299, 1987–2003 (2021). https://doi.org/10.1007/s00209-021-02747-1
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DOI: https://doi.org/10.1007/s00209-021-02747-1