We prove that minimal instanton bundles on a Fano threefold X of Picard rank one and index two are semistable objects in the Kuznetsov component Ku(X), with respect to the stability conditions constructed by Bayer, Lahoz, Macrì and Stellari. When the degree of X is at least 3, we show torsion free generalizations of minimal instantons are also semistable objects. As a result, we describe the moduli space of semistable objects with same numerical classes as minimal instantons in Ku(X).

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Via wall-crossing, we study the birational geometry of Beauville-Mukai systems on K3 surfaces with Picard rank one. We show that there is a class of walls which are always present in the movable cones of Beauville-Mukai systems. We give a complete description of the birational geometry of rank two Beauville-Mukai systems when the genus of the surface is small.

We study wall-crossing for the Beauville-Mukai system of rank three on a general genus two K3 surface. We show that such a system is related to the Hilbert scheme of ten points on the surface by a sequence of flops, whose exceptional loci can be described as Brill-Noether loci. We also obtain Brill-Noether type results for sheaves in the Beauville-Mukai system. Contents 1. Introduction 1 2. Preliminaries 4 3. Brill-Noether loci in S [10] and M 8 4. Wall-crossing for S [10] and M 19 5. Appendix 29 References 32

Suppose S is a smooth projective surface over an algebraically closed field k, L = {L 1 , . . . , Ln} is a full strong exceptional collection of line bundles on S. Let Q be the quiver associated to this collection. One might hope that S is the moduli space of representations of Q with dimension vector (1, . . . , 1) for a suitably chosen stability condition θ: S ∼ = M θ . In this paper, we show that this is the case for del Pezzo surfaces. Furthermore, we show the blow-up at a point can be recovered from an augmentation of exceptional collections (in the sense of L. Hille and M.Perling) via morphism between moduli of quiver representations.