The rationality problem in invariant theory.Bohning

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1、TheRationalityProbleminInvariantTheoryChristianB•ohningMathematischesInstitutGeorg-August-Universit•atG•ottingeniiContentsIntroductionv1Therationalityproblemininvarianttheory11.1Introduction............................11.2Therationalityproblem.....................21.2.1Quotientsandeldsofi

2、nvariants............21.2.2Resultsforspecicgroups................111.3Conesandhomogeneousbundles................191.3.1Torusorbitsandthenullcone..............191.3.2Straticationofthenullconeandrationality......231.4Overviewofsomefurthertopics.................352Techniques412.1Introductio

3、n............................412.2Methods..............................412.2.1Slicemethodand2-formtrick..............412.2.2Doublebundlemethod..................492.2.3Methodofcovariants...................512.2.4G-bundlesandcongurationspacesofpoints......533RationalityofM3573.1Introduction

4、............................573.2Aremarkable(SL3C;SO3C)-section..............583.2.1(G;H)-sectionsandcovariants..............583.2.2Binaryquartics......................603.2.3Fromquintictocubicequations.............633.2.4Fromcubictoquadraticequations............673.3Furthersectionsandinne

5、rprojections..............78iiiivCONTENTS3.3.1Binaryquarticsagainanda(PSL2C;S4)-section...783.3.2DividingbytheactionofH...............823.3.3Innerprojectionsandthe"no-name"method.....863.3.4AppendixA:Collectionofformulasforsection2....943.3.5AppendixB:Collectionofformulasforsection3....984

6、Rationalityofmoduliofplanecurves1014.1Introduction............................1014.2Proofofrationality........................1044.2.1Outlineofproof......................1044.2.2Notationanddenitionofthecovariants........1074.2.3Speciallinearsubspacesofthebaseloci.........1104.2.4Fiberwise

7、surjectivityofthecovariants.........1124.2.5Sectionsofprincipalbundlesandproofofrationality.1184.3AnotherproofforthedominanceofSd.............1194.3.1CongurationsoflinesandScorzamap.........1195Smalldegreeplanecurves1255.1Introduction.........................