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.

The topic of this book is the theory of semistable coherent sheaves on a smooth algebraic surface and of moduli spaces of such sheaves. The content ranges from the definition of a semistable sheaf and its basic properties over the construction of moduli spaces to the birational geometry of these moduli spaces. The book is intended for readers with some background in Algebraic Geometry, as for example provided by Hartshorne's textbook [98].

There are at least three good reasons to study moduli spaces of sheaves on surfaces. Firstly, they provide examples of higher dimensional algebraic varieties with a rich and interesting geometry. In fact, in some regions in the classification of higher dimensional varieties the only known examples are moduli spaces of sheaves on a surface. The study of moduli spaces therefore sheds light on some aspects of higher dimensional algebraic geometry. Secondly, moduli spaces are varieties naturally attached to any surface. The understanding of their properties gives answers to problems concerning the geometry of the surface, e.g. Chow group, linear systems, etc. From the mid-eighties till the mid-nineties most of the work on moduli spaces of sheaves on a surface was motivated by Donaldson's ground breaking results on the relation between certain intersection numbers on the moduli spaces and the differentiable structure of the four-manifold underlying the surface. Although the interest in this relation has subsided since the introduction of the extremely powerful Seiberg-Witten invariants in 1994, Donaldson's results linger as a third major motivation in the background; they throw a bridge from algebraic geometry to gauge theory and differential geometry.

This chapter provides the basic definitions of the theory. After introducing pure sheaves and their homological aspects we discuss the notion of reduced Hilbert polynomials in terms of which the stability condition is formulated. Harder-Narasimhan and Jordan-Hölder filtrations are defined in Section 1.3 and 1.5, respectively. Their formal aspects are discussed in Section 1.6. In Section 1.7 we recall the notion of bounded families and the Mumford-Castelnuovo regularity. The results of this section will be applied later (cf. 3.3) to show the boundedness of the family of semistable sheaves. This chapter is slightly technical at times. The reader may just skim through the basic definitions at first reading and come back to the more technical parts whenever needed.

Some Homological Algebra

Let X be a Noetherian scheme. By Coh(X) we denote the category of coherent sheaves on X. For E ∈ Ob(Coh(X)), i.e. a coherent sheaf on X, one defines:

Definition 1.1.1 — The support of E is the closed set Supp(E) = {x ∈ X|Ex ≠ 0}. Its dimension is called the dimension of the sheaf E and is denoted by dim(E).

The annihilator ideal sheaf of E, i.e. the kernel of Ox → εnd(E), defines a subscheme structure on Supp(E).

Definition 1.1.2 — E is pure of dimension d if dim(F) = d for all non-trivial coherent subsheaves F ⊂ E.

Equivalently, E is pure if and only if all associated points of E (cf. [172] p. 49) have the same dimension.

Moduli spaces of bundles with fixed determinant on algebraic curves are unirational and very often even rational. For moduli spaces of sheaves on algebraic surfaces the situation differs drastically and, from the point of view of birational geometry, discloses highly interesting features. Once again, the geometry of the surface and of the moduli spaces of sheaves on the surface are intimately related. For example, moduli spaces associated to rational surfaces are expected to be rational and, similarly, moduli spaces associated to minimal surfaces of general type should be of general type. We encountered phenomena of this sort already at various places (cf. Chapter 6).

There are essentially two techniques to obtain information about the birational geometry of moduli spaces. First, one aims for an explicit parametrization of an open subset of the moduli space by means of Serre correspondence, elementary transformation, etc. Second, one may approach the question via the positivity (negativity) of the canonical bundle of the moduli space. The first step was made in Section 8.3. The best result in this direction is due to Li saying that on a minimal surface of general type with a reduced canonical divisor the moduli spaces of rank two sheaves are of general type. This and similar results concerning the Kodaira dimension are presented in Section 11.1. The use of Serre correspondence for a birational description is illustrated by means of two examples in Section 11.3.

In this chapter we take up a problem already discussed in Section 3.1. We try to understand how µ-(semi)stable sheaves behave under restriction to hypersurfaces. At present, there are three quite different approaches to this question, and we will treat them in separate sections. None of these methods covers the results of the others completely.

The theorems of Mehta and Ramanathan 7.2.1 and 7.2.8 show that the restriction of a µ-stable or µ-semistable sheaf to a general hypersurface of sufficiently high degree is again µ-stable or µ-semistable, respectively. It has the disadvantage that it is not effective, i.e. there is no control of the degree of the hypersurface, which could, a priori, depend on the sheaf itself. However, such a bound, depending only on the rank of the sheaf and the degree of the variety, is provided by Flenner's Theorem 7.1.1. Since it is based on a careful exploitation of the Grauert-Mülich Theorem in the refined form 3.1.5, it works only in characteristic zero and for µ-semistable sheaves. In that respect, Bogomolov's Theorem 7.3.5 is the strongest, though one has to restrict to the case of smooth surfaces. It says that the restriction of a µ-stable vector bundle on a surface to any curve of sufficiently high degree is again µ-stable, whereas the theorems mentioned before provide information for general hypersurfaces only. Moreover, the bound in Bogomolov's theorem depends on the invariants of the bundle only.