TABLE OF CONTENTS 1 . GORENSTEIN ARTIN ALGEBRAS AND DUALITY 1 Intersection of the m1 and the Loewy filtrations 1 1A. Introduction 1 IB. The Gorenstein Artin algebras 8 1C. The associated graded algebra 10 ID. The Hilbert function decomposition 12 IE. Linking and the induced Q(a) decomposition 19 IF. R-module structure 21 2 . THE INTERSECTION OF TWO PLANE CURVES 23 3 . EXTREMAL DECOMPOSITIONS 31 3A. Upper bounds for H (a) 32 3B. Relatively compressed modifications of A 36 3C. The family of algebras of decomposition D 40 4 . COMPONENTS OF THE HILBERT SCHEME STRATA 4 6 4A. Hilbert functions of mu and 0:mv and semicontinuity 47 4B. Components of GorTR when T = (1,3,3,2,1,1) 49 4B.i. The variety GorTR 49 4B.i.a. Finding f in II determining an algebra with Hilbert function T 50 4B.i.b. First parametrization 53 4B.ii. Fibration GorTR to GT: Parametrization as fibred varieties 55 4C. Finding Q(a) from a generator of the dual module. 58 4C.i. Algorithm for finding Q(a) from the dual module Qi 61 5. WHAT DECOMPOSITIONS D AND SUBQUOTIENTS Q(a) CAN OCCUR? .62 5A. Hilbert function of a graded Gorenstein Artin algebra 64 5B. Numerical conditions on the decomposition D 69 5C. Gorenstein Artin algebras with given decomposition D 73 5D. Applications 81 5D. i . Higher dimensions 81 5D.ii. Multiplicity and Orders of Generators. ...82 5D. iii. Finite mapping germs 85 5E. Problems 88 5F. Appendix: Hilbert function decompositions for lengths n 21 when e 3, and for n 16 90 5F.i. Hilbert functions of unique decomposition. 90 5F.ii. Hilbert functions having several decompositions 94 5F.iii. Discussion 99 5F.iii.a. Non-unimodal H 99 5F.iii.b. Nearly symmetric H 101 5F.iii.c. Unimodal H 101 6 . RELATIVELY COMPRESSED ARTIN ALGEBRAS 103 BIBLIOGRAPHY 105 LIST OF THEOREMS, DEFINITIONS, AND EXAMPLES 109 INDEX 112 vii

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