曲線模

內(nèi)容概要

　  《曲線模》是Springer數(shù)學(xué)研究生教材系列之一，全面而深入地講述了曲線模這個(gè)科目，即代數(shù)曲線及其在族中是如何變化的?！肚€模》對(duì)曲線模的講述，符合學(xué)習(xí)理解的規(guī)律，也是對(duì)該領(lǐng)域的廣泛而簡(jiǎn)潔的概述，使得具有現(xiàn)代代數(shù)幾何背景的讀者很容易學(xué)習(xí)理解。書中包括了許多技巧，如Hilbert空間，變形原理，穩(wěn)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應(yīng)用。文中繼而討論了曲線?？臻g的構(gòu)成，通過(guò)有限線性系列說(shuō)明了Brill-Noether和Gieseker-Petri定理證明的典型應(yīng)用，也講述了一些有關(guān)不可約性，完全子變量，豐富除子和Kodaira維數(shù)的重要幾何結(jié)果。書中也包括了該領(lǐng)域相當(dāng)重要的重要定理幾何開(kāi)放性問(wèn)題，但只是做了簡(jiǎn)明引入，并沒(méi)有展開(kāi)討論。書中眾多的練習(xí)和圖例，使得內(nèi)容更加豐富，易于理解。

書籍目錄

preface1 parameter spaces： constructions and examplesa parameters and modulib construction of the hfibert schemec tangent space to the hilbert schemed extrinsic pathologiesmumford's exampleother examplese dimension of the hilbert schemef severi varietiesg hurwitz schemesbasic facts about moduli spaces of curvesa why do fine moduli spaces of curves not exist？b moduli spaces we'll be concerned withc constructions of mgthe teichmiiller approachthe hodge theory approachthe geometric invariant theory （g.i，t.） approachd geometric and topological propertiesbasic propertieslocal propertiescomplete subvarieties of mgcohomology of mg： hater's theoremscohomology of the universal curvecohomology of hfibert schemesstructure of the tautological ringwitten's conjectures and kontsevich's theoreme moduli spaces of stable mapstechniquesa basic facts about nodal and stable curvesdualizing sheavesautomorphismsb deformation theoryoverviewdeformations of smooth curvesvariations on the basic deformation theory planuniversal deformations of stable curvesdeformations of mapsc stable reductionresultsexamplesd interlude： calculations on the moduli stackdivisor classes on the moduli stackexistence of tautological familiese grothendieck-riemann-roch and porteousgrothendieck-riemann-rochchern classes of the hodge bundlechern class of the tangent bundleporteous' formulathe hyperelliptic locus in m3relations amongst standard cohomology classesdivisor classes on hilbert schemesf test curves： the hyperelliptic locus in m3 begung admissible coversh the hyperelliptic locus in m3 completed4 construction of m3a background on geometric invariant theorythe g.i.t. strategyfinite generation of and separation by invariantsthe numerical criterionstability of plane curvesb stability of hilbert points of smooth curvesthe numerical criterion for hilbert pointsgieseker's criterionstability of smooth curvesc construction of mg via the potential stability theoremthe plan of the construction and a few corollariesthe potential stability theoremlimit linear series and brill-noether theorya introductory remarks on degenerationsb limits of line bundlesc limits of linear series： motivation and examplesd limit linear series： definitions and applicationslimit linear seriessmoothing limit linear serieslimits of canonical series and weierstrass pointslimit linear series on flag curvesinequalities on vanishing sequencesthe case p = 0proof of the gieseker-petri theoremgeometry of moduli spaces： selected resultsa irreducibility of the moduli space of curvesb diaz' theoremthe idea： stratifying the moduli spacethe proofc moduli of hyperelliptic curvesfiddling aroundthe calculation for an （almost） arbitrary familythe picard group of the hyperelliptic locusd ample divisors on mgan inequality for generically hilbert stable familiesproof of the theoreman inequality for families of pointed curvesample divisors on mge irreducibility of the severi varietiesinitial reductionsanalyzing a degenerationan examplecompleting the argumentf kodaira dimension of mgwriting down general curvesbasic ideaspulling back the divisors drdivisors on mg that miss j（m2，1 \ w）divisors on mg that miss i（m0，g）further divisor class calculationscurves defined over qbibliographyindex

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