The Voisin map via families of extensions

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 ) × F ( Y ) ⤏ 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 ) × 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 .