Harris J., Morrison I.  Moduli of curves

Название: 
Moduli of curves 
Автор: 
Harris J., Morrison I. 
Категория: 
Математика

Тип: 
Книга 
Дата: 
31.12.2008 00:07:01 
Скачано: 
23 
Оценка: 

Описание: 
Before we take up any of the constructions that will occupy us in this chapter, we want to make a few general remarks about moduli problems in general.
What is a moduli problem? Typically, it consists of two things. First of all, we specify a class of objects (which could be schemes, sheaves, morphisms or combinations of these), together with a notion of what it means to have a family of these objects over a scheme B. Second, we choose a (possibly trivial) equivalence relation ~ on the set S(B) of all such families over each B. We use the rather vague term "object" deliberately because the possibilities we have in mind are wideranging. For example, we might take our families to be
1. smooth flat morphisms C—+B whose fibers are smooth curves of genus g, or
2. subschemes С in V* x B, flat over B, whose fibers over В are curves of fixed genus g and degree d,
and so on. We can loosely consider the elements of S(Spec(Q) as the objects of our moduli problem and the elements of S(B) over other bases as families of such objects parameterized by the complex points ofB.1
The equivalence relations we will wish to consider will vary considerably even for a fixed class of objects: in the second case cited above, we might wish to consider two families equivalent if
'More generally, we may consider elements of 5(Spec(k)) for any field к as objects of our moduli problem defined over k. 
Файл: 
3.91 МБ 
Скачать
