| Description |
In this dissertation, we apply Bridgeland stability on triangulated categories to study two questions in algebraic geometry. The first question is about Le Potier's strange duality for K3 surfaces. We prove that O'Grady's birational maps between moduli of sheaves on an elliptic K3 surface can be interpreted as intermediate wall-crossing (wall-hitting) transformations at socalled totally semistable walls, studied by Bayer and Macr`ı. Using this, we give new examples of strange duality isomorphisms, based on a result of Marian and Oprea. The second question is about the so-called Voisin map v : F(Y ) F(Y ) 99K Z(Y ), where Y is a cubic fourfold not containing any plane, F(Y ) is the variety of lines, and Z(Y ) is the Lehn-Lehn-Sorger-van Straten hyperk¨ahler eightfold. We prove that this rational map can be resolved by blowing up the incident locus F(Y )F(Y ) endowed with the reduced scheme structure. Moreover, if Y is very general, then this blowup is a relative Quot scheme over Z(Y ) parameterizing quotients in a heart of a Kuznetsov component of Y: |
