The Voisin map via families of extensions

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 hyperkahler eightfold constructed in Lehn et al. (J fur 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-Kahler 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 hyperkahler 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.