Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers

By Juan Migliore, Uwe Nagel. 2001

88 citations recorded by the source corpus.

Abstract 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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b974cdda68196dbb42a650531df4b31b2f5d14c0

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Citations in the corpus

The Weak and Strong Lefschetz properties for Artinian K-algebras (2002, methodology): “This is false in higher codimension.”
Open Problems on Syzygies and Hilbert Functions (2009, background): “(Geramita-Harima-Shin) [58] Does there exist an ideal which has greatest graded Betti numbers among all Gorenstein artinian graded ideals with a fixed Hilbert function? The problem is solved under the additional hypothesis that the weak Lefschetz property holds in [92].”
On the first infinitesimal neighborhood of a linear configuration of points in P2 (2006, background): “Diesel [22] and Geramita and Migliore [33], while for Gorenstein rings with the Weak Lefschetz Property see, e.g., Geramita, Harima, and Shin [30], Migliore and Nagel [ 51 ], where there are sharp upper bounds.”
Graded Betti numbers of Cohen–Macaulay modules and the multiplicity conjecture (2006, background): “The conjecture has been verified in a number of interesting cases, including the ones mentioned above where the structure of the minimal free resolution is known (see Herzog and Srinivasan [6], Migliore-Nagel [13], Migliore, Nagel and Römer [14],[15], Kubitzke and Welker [9]).”
Combinatorial Lefschetz theorems beyond positivity (2018, methodology): “and Σ |Wthe induced subcomplex on W. For a numerical measure, one often uses either the 1-norm |(Σ)| k−1,1,m:= X W⊂Σ(0) |W|=m b k−1(Σ |W) or the ∞-norm |(Σ)| k−1,∞:= max W⊂Σ(0) b k−1(Σ |W). Following [MN03] and using TheoremI(1) (and its generalization Theorem9.1), it follows that |(Σ)| k−1,1,m is m”
Survey Article: A tour of the weak and strong Lefschetz properties (2013, background): “We point out that, if this question has an affirmative answer, then, by the main result of [41], we have a classification of the Hilbert functions of reduced, arithmetically Gorenstein schemes: their h-vectors are precisely the SI-sequences, meaning that they are symmetric, with the first half itsel”
On the shape of a pure O-sequence (2010, background): “Recall that an ideal is called glicci if it is in the Gorenstein liaison class of a complete intersection.”
The Hilbert Function of a Level Algebra (2007, background): “i) What can HA be if A is a domain? (see [65]); ii) What can HA be if I = IX is the ideal of a reduced set of points, X, in Pn−1? (see [25]); iii) What can HA be if I = IX is the ideal of a set of points, X, which is the generic hyperplane section of a curve in P? (see [26], [34], [52]); iv) What ca”