TY - JOUR

T1 - Reduced arithmetically Gorenstein schemes and simplical polytopes with maximal Betti numbers

AU - Migliore, Juan C.

AU - Nagel, Uwe

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2003/12/1

Y1 - 2003/12/1

N2 - An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of graded Artinian Gorenstein algebras with the weak Lefschetz property, a property shared by a nonempty open set of the family of all graded Artinian Gorenstein algebras having a fixed Hilbert function that is an SI sequence. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration G of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. Furthermore, we show that G has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. As an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed h-vector contains a polytope with maximal graded Betti numbers.

AB - An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of graded Artinian Gorenstein algebras with the weak Lefschetz property, a property shared by a nonempty open set of the family of all graded Artinian Gorenstein algebras having a fixed Hilbert function that is an SI sequence. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration G of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. Furthermore, we show that G has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. As an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed h-vector contains a polytope with maximal graded Betti numbers.

KW - Arithmetically

KW - Compressed algebra

KW - Gorenstein ideal

KW - Gorenstein liaison

KW - Gorenstein subscheme

KW - Graded Betti numbers

KW - Hilbert function

KW - SI-sequence

KW - Simplical polytopes

KW - Weak Lefschetz property

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U2 - 10.1016/S0001-8708(02)00079-8

DO - 10.1016/S0001-8708(02)00079-8

M3 - Article

AN - SCOPUS:0346269082

SN - 0001-8708

VL - 180

SP - 1

EP - 63

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 1

ER -