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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
Glicci ideals
A central problem in liaison theory is to decide whether every arithmetically
Cohen-Macaulay subscheme of projective -space can be linked by a finite
number of arithmetically Gorenstein schemes to a complete intersection. We show
that this can be indeed achieved if the given scheme is also generically
Gorenstein and we allow the links to take place in an -dimensional
projective space. For example, this result applies to all reduced
arithmetically Cohen-Macaulay subschemes. We also show that every union of fat
points in projective 3-space can be linked in the same space to a union of
simple points in finitely many steps, and hence to a complete intersection in
projective 4-space.Comment: 8 page
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