Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers.
An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a growth condition. From an arbitrary SI-sequence the authors construct a reduced, arithmetically Gorenstein configuration of linear varieties, \(G\), of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. It is shown that \(G\) has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of the projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. Finally, it is shown that over a field of characteristic zero every set of simplicial polytopes with fixed \(h\)-vector contains a polytope with maximal graded Betti numbers.
Gorenstein configuration of linear varieties
scientific article