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内容简介

《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。《曲线模》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。

目录

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
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
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
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
limit linear series on flag curves
inequalities on vanishing sequences
the case p = 0
proof of the gieseker-petri theorem
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

前言/序言

曲线模 [Moduli of Curves] 电子书 下载 mobi epub pdf txt

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