The canonical strategy of modern mathematics when studying an object is to put this object into a collection, and see what properties they have in common. Most commonly, the objects depend on some parameter(s), and the goal is to find out how the objects vary with these parameters. The authors of this book take this approach to studying algebraic curves, with the parametrization being called the moduli space, and it enables one to gain information about the geometry of a family of objects from the moduli space and vice versa. The objects are typically schemes, sheaves, or morphisms parametrized by a scheme called the base. Putting an equivalence relation on the families gives a functor, called the moduli functor, which acts on the category of schemes to the category of sets. The functor is representable in the category of schemes if there is an isomorphism between the functor and the functor of points of a scheme. This particular scheme is called the fine moduli space for the functor, as distinguished from the coarse moduli space, where the functor is not representable, i.e. only a natural transformation, and not an isomorphism exists.
The authors clarify the distinction between a moduli space and a parameter space, the former used for problems that involve intrinsic data, the latter for problems involving extrinsic data. An example of the latter, the Hilbert scheme, is discussed in detail in the first chapter, and an example due to Mumford of a component of a Hilbert scheme of space curves that is everywhere nonreduced is given to illustrate the pathologies that can arise in the extrinsic case, and to motivate the use of intrinsic moduli spaces to eliminate these difficulties. Severi varieties are discussed as objects that are more well-behaved than Hilbert schemes but still do not permit a scheme structure to be defined on them so that they represent the functor of families of plane curves with the correct geometric properties.
The second chapter gives a general overview of the approaches taken in the construction of moduli spaces of curves. The authors first study the case of genus 1 (elliptic) curves to illustrate the difficulties involved in constructing fine moduli spaces. The role of automorphisms on the curves as an obstruction to the existence of fine moduli spaces is outlined, as well as approaches to deal with these automorphims, particularly the role of marked points. The authors briefly discuss the role of algebraic spaces and algebraic stacks in the moduli problem. They explain also the various approaches to the construction of the moduli space of smooth curves of genus g, namely the Teichmuller, Hodge-theoretic, and geometric invariant theoretic approaches. The local properties of the moduli space are outlined, along with a discussion of to what extent the moduli space deviates from being a projective or affine variety. The rational cohomology ring of the moduli space is also treated, in low dimensions via the Harer stability theorem, and for high dimensions via the Mumford conjecture. Most interestingly, Witten's conjectures and the Kontsevich formulas are introduced, as a theory of moduli spaces of stable maps. The famous Gromov-Witten invariants of a projective scheme and the quantum cohomology ring are briefly discussed. These have generated an enormous amount of research, the results of which show the power of viewing mathematical constructions from a "quantum" point of view.
The next chapter gives a very specialized overview of the techniques used to study moduli spaces. The authors are very meticulous in their explanations of where the names of the concepts come from, and this is an enormous help to those seeking an in-depth understanding of the topics. One of the first is the dualizing sheaf of a nodal curve, which is the analogue of the canonical bundle of a smooth curve. The authors then describe, by taking a point as the base, the scheme-theoretic automorphism group of a stable curve, and show that it is finite and reduced. Deformation theory is introduced first as over smooth varieties. Readers will appreciate the discussion more if they have a background in the deformation theory of compact, complex manifolds. The authors then tackle the stable reduction problem, and give several beautiful examples, with lots of diagrams, to illustrate the concepts. This is one of the best discussions I have seen in print on these topics. After a brief interlude on the properties of the moduli stack, the authors treat the generalization of the Riemann-Roch formula due to Grothendieck. This section is very important to physicists working in superstring theory. The Porteous formula is also stated and applied to the determination of the class in the rational Picard group of hyperelliptic curves. The determination of the class of the locus of hyperelliptic stable curves of genus 3 is continued in two more sections using the method of test curves and admissible covers.
The actual construction of the moduli space is the subject of chapter 4, from the viewpoint of geometric invariant theory. A nice example of this approach is given for the case of the set of smooth curves of genus 1. The numerical criterion for stability is discussed in detail, with Giesecker's criterion given the main focus. The case of the moduli space of curves with genus greater than two is tackled via the potential stability theorem.
The authors show indeed in the next chapter that the moduli space can be used to prove results about a single curve. As one would expect intuitively, the taking of limits must be justified, and indeed this is the case here, where limits of line bundles and linear series are considered. Then in the last chapter they show the reverse, that the properties of various moduli spaces can be proven using the techniques introduced in the book, such as the irreducibility of the moduli space, the Diaz result that complete subvarieties of the moduli space have dimension at most genus - 2, and moduli of hyperelliptic curves and Severi varieties.