University of Paderborn Department of Mathematics Diploma Thesis ...

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Glossary of MuPAD procedures 129Bibliography 1312
IntroductionThe main result of this thesis is to present an algorithm to compute all saturated stableideals to a given Hilbert polynomial according to the Ph.D. thesis Combinatorial Structureon the Hilbert Scheme by Alyson Reeves (see [16]).The importance of the study of stable ideals to the topic of algebraic geometry has beenrecognized through the work of several researchers. The contributions of Robin Hartshorne(see [8], [9]) , David Bayer (see [1]) and Mark Green (see [4], [5]) are of particular interest.Within their work, a special class of monomial ideals – the so-called generic initial ideals –has been studied. The reader, who is not familiar with the definition of the generic initialideal to a given homogeneous ideal I of the polynomial ring K[x 0 , . . . , x n ] in n+1 variablesover the field K may think of it as an initial ideal (i.e. a monomial ideal, generated bythe leading monomials of all homogeneous elements of I with respect to some monomialorder), which can be obtained from I after the application of a sufficiently general changeof coordinates to its generators.A theorem proved by Galligo, Bayer and Stillman, which can be found for example in thebook Commutative Algebra with a View Toward Algebraic Geometry by David Eisenbud(see [6], Chapter 15, Theorem 15.20 ), states that if I is a homogeneous ideal in the polynomialring K[x 0 , . . . , x n ] in n + 1 variables over the field K, the generic initial ideal of Iwill be Borel-fixed. Within this thesis, the field K will always be of characteristic zero.In this case, Borel-fixed ideals are stable ideals, i.e. they are characterized by a certaincombinatorial property.Generic initial ideals also played an important role in Hartshorne’s proof of the connectednessof the Hilbert scheme. The Hilbert scheme is an interesting and rather complicatedobject studied in algebraic geometry. With the help of this scheme, one can parametrizethe subschemes of the projective space P n with the same Hilbert polynomial. It consistsof different components, whose number can be estimated by the number of stable idealswith the same Hilbert polynomial. Hence, for further observations of the Hilbert scheme itwould be useful to have an algorithm to compute all such stable ideals. Thus, the readermight see that it is indeed of interest to study the class of stable ideals not only from thepoint of view of commutative algebra, but also from that of algebraic geometry.3
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Glossary of NotationsK a field of c
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[13] F.S. Macaulay, Some properties
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University of Paderborn Department of Mathematics Diploma Thesis ...

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