Gieseker D.  Lectures on moduli of curves

Название: 
Lectures on moduli of curves 
Автор: 
Gieseker D. 
Категория: 
Математика

Тип: 
Книга 
Дата: 
30.12.2008 23:43:34 
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Описание: 
These notes are based on some lectures given at TIFR during January and February 1980. The object of the lectures was to construct a projective moduli space for stable curves of genus g > 2 using Mumford's geometric invariant theory.
The general plan of the notes is as follows: Chapter 0 consists of preliminaries. In particular, for m » 0, We review how to attach to each space curve С С HP a point in some projective space called the m Hilbert point of с We then consider the question of the stability of the m • Hilbert point in the sense of geometric invariant theory. Our first main result in Chapter 1 is that if С is smooth and d > 20(gl), then the m Hilbert point of С is stable. Our second main result in Chapter 1 is that if the m Hilbert point is semistable, then the curve is semistable as a curve. In Chapter 2, we use the results of Chapter 1 to give an indirect proof that the n canonical embedding of a stable curve is stable if n £ 10, and to construct the projective moduli space for stable curves. As corollaries, we obtain proofs of the stable reduction theorem for curves, and of the irreducibility of the moduli space for smooth curves.
Historically speaking, Mumford used his theory to construct a quasiprojective moduli space for smooth curves by studying the stability of the Chow points of space curves. Himford and Deligne ( 1 ) introduced the concept of stable curve in their proof of the irreducibility of the moduli space of curves of genus g > 2, and 
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