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Gieseker D. - Lectures on moduli of curves

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Название: Lectures on moduli of curves
Автор: Gieseker D.
Категория: Математика
Тип: Книга
Дата: 30.12.2008 23:43:34
Скачано: 26
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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(g-l), then the m Hilbert point of С is stable. Our second main result in Chapter 1 is that if the m Hilbert point is semi-stable, then the curve is semi-stable 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 quasi-projective 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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