**University Lecture Series**

Volume: 39;
2006;
151 pp;
Softcover

MSC: Primary 14;

Print ISBN: 978-0-8218-2983-7

Product Code: ULECT/39

List Price: $38.00

AMS Member Price: $30.40

MAA Member Price: $34.20

**Electronic ISBN: 978-1-4704-2183-0
Product Code: ULECT/39.E**

List Price: $38.00

AMS Member Price: $30.40

MAA Member Price: $34.20

#### Supplemental Materials

# The Moduli Problem for Plane Branches

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*Oscar Zariski*

with an appendix by Bernard Teissier

Translated by Ben Lichtin

Moduli problems in algebraic geometry date back
to Riemann's famous count of the \(3g-3\) parameters needed to
determine a curve of genus \(g\). In this book, Zariski studies the
moduli space of curves of the same equisingularity class. After
setting up and reviewing the basic material, Zariski devotes one
chapter to the topology of the moduli space, including an explicit
determination of the rare cases when the space is compact. Chapter V
looks at specific examples where the dimension of the generic
component can be determined through rather concrete methods.
Zariski's last chapter concerns the application of deformation theory
to the moduli problem, including the determination of the dimension of
the generic component for a particular family of curves.

An appendix by Bernard Teissier reconsiders the moduli problem from
the point of view of deformation theory. He gives new proofs of some
of Zariski's results, as well as a natural construction of a
compactification of the moduli space.

#### Readership

Graduate students and research mathematicians interested in algebraic geometry, especially moduli questions, and singularities.

#### Table of Contents

# Table of Contents

## The Moduli Problem for Plane Branches

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Introduction vii8 free
- Chapter I. Preliminaries 110 free
- Chapter II. Equisingularity Invariants 514
- Chapter III. Parametrizations 1928
- Chapter IV. The Moduli Space 2938
- Chapter V. Examples 4554
- Chapter VI. Applications of Deformation Theory 7382
- §1. Introduction 7382
- §2. Viewing the problem from the perspective of Deformation Theory 7382
- §3. On the dimension of the generic component of the moduli space 8089
- §4. The dimension of the generic component of the moduli space for the equisingularity class (n; n + 1) 8594
- §5. Remarks and open problems concerning the generic component Mi of the moduli space ( characteristic (n; n + 1)) 94103
- §6. An approach to the problem from the point of view of parametrizations 96105

- Bibliography 109118
- Appendix 111120
- Back Cover Back Cover1161