4,176 research outputs found
Tetrahedral curves via graphs and Alexander duality
A tetrahedral curve is a (usually nonreduced) curve in P^3 defined by an
unmixed, height two ideal generated by monomials. We characterize when these
curves are arithmetically Cohen-Macaulay by associating a graph to each curve
and, using results from combinatorial commutative algebra and Alexander
duality, relating the structure of the complementary graph to the
Cohen-Macaulay property.Comment: 15 pages; minor revisions to v. 1 to improve clarity; to appear in
JPA
Borel generators
We use the notion of Borel generators to give alternative methods for
computing standard invariants, such as associated primes, Hilbert series, and
Betti numbers, of Borel ideals. Because there are generally few Borel
generators relative to ordinary generators, this enables one to do manual
computations much more easily. Moreover, this perspective allows us to find new
connections to combinatorics involving Catalan numbers and their
generalizations. We conclude with a surprising result relating the Betti
numbers of certain principal Borel ideals to the number of pointed
pseudo-triangulations of particular planar point sets.Comment: 23 pages, 2 figures; very minor changes in v2. To appear in J.
Algebr
Generalizing the Borel property
We introduce the notion of Q-Borel ideals: ideals which are closed under the
Borel moves arising from a poset Q. We study decompositions and homological
properties of these ideals, and offer evidence that they interpolate between
Borel ideals and arbitrary monomial ideals.Comment: 19 pages, 1 figur
On the componentwise linearity and the minimal free resolution of a tetrahedral curve
A tetrahedral curve is an unmixed, usually non-reduced, one-dimensional
subscheme of projective 3-space whose homogeneous ideal is the intersection of
powers of the ideals of the six coordinate lines. The second and third authors
have shown that these curves have very nice combinatorial properties, and they
have made a careful study of the even liaison classes of these curves. We build
on this work by showing that they are "almost always" componentwise linear,
i.e. their homogeneous ideals have the property that for any d, the degree d
component of the ideal generates a new ideal whose minimal free resolution is
linear. The one type of exception is clearly spelled out and studied as well.
The main technique is a careful study of the way that basic double linkage
behaves on tetrahedral curves, and the connection to the tetrahedral curves
that are minimal in their even liaison classes. With this preparation, we also
describe the minimal free resolution of a tetrahedral curve, and in particular
we show that in any fixed even liaison class there are only finitely many
tetrahedral curves with linear resolution. Finally, we begin the study of the
generic initial ideal (gin) of a tetrahedral curve. We produce the gin for
arithmetically Cohen-Macaulay tetrahedral curves and for minimal arithmetically
Buchsbaum tetrahedral curves, and we show how to obtain it for any non-minimal
tetrahedral curve in terms of the gin of the minimal curve in that even liaison
class.Comment: 31 pages; v2 has very minor changes: fixed typos, added Remark 4.2
and char. zero hypothesis to 5.2, and reworded 5.5. To appear, J. Algebr
Splittings of monomial ideals
We provide some new conditions under which the graded Betti numbers of a
monomial ideal can be computed in terms of the graded Betti numbers of smaller
ideals, thus complementing Eliahou and Kervaire's splitting approach. As
applications, we show that edge ideals of graphs are splittable, and we provide
an iterative method for computing the Betti numbers of the cover ideals of
Cohen-Macaulay bipartite graphs. Finally, we consider the frequency with which
one can find particular splittings of monomial ideals and raise questions about
ideals whose resolutions are characteristic-dependent.Comment: minor changes: added Cor. 3.10 and some references. To appear in
Proc. Amer. Math. So
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