哈里斯编著的这本《曲线模》是Springer数学研究生教材系列之一，全面而深入地讲述了曲线模这个科目，即代数曲线及其在族中是如何变化的。本书对曲线模的讲述，符合学习理解的规律，也是对该领域的广泛而简洁的概述，使得具有现代代数几何背景的读者很容易学习理解。书中包括了许多技巧，如Hilbert空间，变形原理，稳定约化，相交理论，几何不变理论等，曲线模型的讲述涉及从例子到应用。文中继而讨论了曲线模空间的构成，通过有限线性系列说明了Brill-Noether和Gieseker-Petri定理证明的典型应用，也讲述了一些有关不可约性，完全子变量，丰富除子和Kodaira维数的重要几何结果。书中也包括了该领域相当重要的重要定理几何开放性问题，但只是做了简明引入，并没有展开讨论。书中众多的练习和图例，使得内容更加丰富，易于理解。

preface

1 parameter spaces: constructions and examples

 a parameters and moduli

 b construction of the hfibert scheme

 c tangent space to the hilbert scheme

 d extrinsic pathologies

mumford's example

other examples

 e dimension of the hilbert scheme

 f severi varieties

 g hurwitz schemes

2 basic facts about moduli spaces of curves

 a why do fine moduli spaces of curves not exist?

 b moduli spaces we'll be concerned with

 c constructions of mg

the teichmiiller approach

the hodge theory approach

the geometric invariant theory (g.i,t.) approach

 d geometric and topological properties

basic properties

local properties

complete subvarieties of mg

cohomology of mg: hater's theorems

cohomology of the universal curve

cohomology of hfibert schemes

structure of the tautological ring

witten's conjectures and kontsevich's theorem

 e moduli spaces of stable maps

3 techniques

 a basic facts about nodal and stable curves

dualizing sheaves

automorphisms

 b deformation theory

overview

deformations of smooth curves

variations on the basic deformation theory plan

universal deformations of stable curves

deformations of maps

 c stable reduction

results

examples

 d interlude: calculations on the moduli stack

divisor classes on the moduli stack

existence of tautological families

 e grothendieck-riemann-roch and porteous

grothendieck-riemann-roch

chern classes of the hodge bundle

chern class of the tangent bundle

porteous' formula

the hyperelliptic locus in m3

relations amongst standard cohomology classes

divisor classes on hilbert schemes

 f test curves: the hyperelliptic locus in m3 begun

 g admissible covers

 h the hyperelliptic locus in m3 completed

4 construction of m3

 a background on geometric invariant theory

the g.i.t. strategy

finite generation of and separation by invariants

the numerical criterion

stability of plane curves

 b stability of hilbert points of smooth curves

the numerical criterion for hilbert points

gieseker's criterion

stability of smooth curves

 c construction of mg via the potential stability theorem

the plan of the construction and a few corollaries

the potential stability theorem

5 limit linear series and brill-noether theory

 a introductory remarks on degenerations

 b limits of line bundles

 c limits of linear series: motivation and examples

 d limit linear series: definitions and applications

limit linear series

smoothing limit linear series

limits of canonical series and weierstrass points

 e limit linear series on flag curves

inequalities on vanishing sequences

the case p = 0

proof of the gieseker-petri theorem

6 geometry of moduli spaces: selected results

 a irreducibility of the moduli space of curves

 b diaz' theorem

the idea: stratifying the moduli space

the proof

 c moduli of hyperelliptic curves

fiddling around

the calculation for an (almost) arbitrary family

the picard group of the hyperelliptic locus

 d ample divisors on mg

an inequality for generically hilbert stable families

proof of the theorem

an inequality for families of pointed curves

ample divisors on mg

 e irreducibility of the severi varieties

initial reductions

analyzing a degeneration

an example

completing the argument

 f kodaira dimension of mg

writing down general curves

basic ideas

pulling back the divisors dr

divisors on mg that miss j(m2,1 \\ w)

divisors on mg that miss i(m0,g)

further divisor class calculations

curves defined over q

bibliography

index