In the first chapter we proved some elementary properties of coherent sheaves related to semistability. The main topic of this chapter is the question how these properties vary in algebraic families. A major technical tool in the investigations here is Grothendieck's Quotscheme. We give a complete existence proof in Section 2.2 and discuss its infinitesimal structure. As an application of this construction we show that the property of being semistable is open in flat families and that for flat families the Harder-Narasimhan filtrations of the members of the family form again flat families, at least generically. In the appendix the notion of the Quot-scheme is slightly generalized to Flag-schemes. We sketch some parts of deformation theory of sheaves and derive important dimension estimates for Flag-schemes that will be used in Chapter 4 to get similar a priori estimates for the dimension of the moduli space of semistable sheaves. In the second appendix to this chapter we prove a theorem due to Langton, which roughly says that the moduli functor of semistable sheaves is proper (cf. Chapter 4 and Section 8.2).

Flat Families and Determinants

Let f : X → S be a morphism of finite type of Noetherian schemes. If g : T → S is an S scheme we will use the notation XT for the fibre product T ×sX, and gx : XT → X and fT : XT → T for the natural projections.