1,039 research outputs found
Simplicial ideals, 2-linear ideals and arithmetical rank
In the first part of this paper we study scrollers and linearly joined
varieties.
A particular class of varieties, of important interest in classical Geometry
are Cohen--Macaulay varieties of minimal degree.
They appear naturally studying the fiber cone of of a codimension two toric
ideals.
Let be an ideal defining a linearly joined arrangement of
varieties:
- We compute the depth, and the cohomological dimension. is the connectedness
dimension.
- We characterize sets of generators of , and give an effective algorithm
to find equations, as an application we compute arithmetical rank. in the case
if defines a union of linear spaces, (ara =projective dimension), in
particular this applies to any square free monomial ideal having a linear
resolution.
- In the case where is a union of linear spaces, the ideal , can be
characterized by a tableau, which is an extension of a Ferrer (or Young)
tableau.
- We introduce a new class of ideals called simplicial ideals, ideals
defining linearly-joined varieties are a particular case of simplicial ideals.Comment: 31 pages, 5 figure
Generalized multiplicities of edge ideals
We explore connections between the generalized multiplicities of square-free
monomial ideals and the combinatorial structure of the underlying hypergraphs
using methods of commutative algebra and polyhedral geometry. For instance, we
show the -multiplicity is multiplicative over the connected components of a
hypergraph, and we explicitly relate the -multiplicity of the edge ideal of
a properly connected uniform hypergraph to the Hilbert-Samuel multiplicity of
its special fiber ring. In addition, we provide general bounds for the
generalized multiplicities of the edge ideals and compute these invariants for
classes of uniform hypergraphs.Comment: 24 pages, 6 figures. The results of Theorem 4.6 and Theorem 9.2 are
now more general. To appear in Journal of Algebraic Combinatoric
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