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Unimodal Gorenstein h-vectors without the Stanley-Iarrobino property
The study of the -vectors of graded Gorenstein algebras is an important
topic in combinatorial commutative algebra, which despite the large amount of
literature produced during the last several years, still presents many
interesting open questions. In this note, we commence a study of those unimodal
Gorenstein -vectors that do \emph{not} satisfy the Stanley-Iarrobino
property. Our main results, which are characteristic free, show that such
-vectors exist: 1) In socle degree if and only if ; and 2) In
every codimension five or greater. The main case that remains open is that of
codimension four, where no Gorenstein -vector is known without the
Stanley-Iarrobino property. We conclude by proposing the following very general
conjecture: The existence of any arbitrary level -vector is
\emph{independent} of the characteristic of the base field.Comment: A few minor revisions. Final version to appear in Comm. Algebr
Gorenstein algebras presented by quadrics
We establish restrictions on the Hilbert function of standard graded
Gorenstein algebras with only quadratic relations. Furthermore, we pose some
intriguing conjectures and provide evidence for them by proving them in some
cases using a number of different techniques, including liaison theory and
generic initial ideals
Stanley's nonunimodal Gorenstein h-vector is optimal
We classify all possible -vectors of graded artinian Gorenstein algebras
in socle degree 4 and codimension , and in socle degree 5 and
codimension . We obtain as a consequence that the least number of
variables allowing the existence of a nonunimodal Gorenstein -vector is 13
for socle degree 4, and 17 for socle degree 5.
In particular, the smallest nonunimodal Gorenstein -vector is
, which was constructed by Stanley in his 1978 seminal paper on
level algebras. This solves a long-standing open question in this area. All of
our results are characteristic free.Comment: 9 page
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