圖書信息
出版社: 世界圖書出版公司; 第1版 (2011年4月1日)
外文書名: Moduli of Curves
平裝: 366頁
正文語種: 英語
開本: 24
ISBN: 9787510032974
條形碼: 9787510032974
尺寸: 22.2 x 14.8 x 1.8 cm
重量: 458 g
作者簡介
作者：（美國）哈里斯（Joe Harris） （美國）Ian Morrison
內容簡介
《曲線模(英文版)》是Springer數學研究生教材系列之一，全面而深入地講述了曲線模這個科目，即代數曲線及其在族中是如何變化的。《曲線模》對曲線模的講述，符合學習理解的規律，也是對該領域的廣泛而簡潔的概述，使得具有現代代數幾何背景的讀者很容易學習理解。書中包括了許多技巧，如Hilbert空間，變形原理，穩定約化，相交理論，幾何不變理論等，曲線模型的講述涉及從例子到套用。文中繼而討論了曲線模空間的構成，通過有限線性系列說明了Brill-Noether和Gieseker-Petri定理證明的典型套用，也講述了一些有關不可約性，完全子變數，豐富除子和Kodaira維數的重要幾何結果。書中也包括了該領域相當重要的重要定理幾何開放性問題，但只是做了簡明引入，並沒有展開討論。書中眾多的練習和圖例，使得內容更加豐富，易於理解。
目錄
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
