36 research outputs found
On the Castelnuovo-Mumford regularity of products of ideal sheaves
In this paper we give bounds on the Castelnuovo-Mumford regularity of
products of ideals and ideal sheaves. In particular, we show that the
regularity of a product of ideals is bounded by the sum of the regularities of
its factors if the corresponding schemes intersect in a finite set of points.
We also show how approximations of sheaves can be used to bound the regularity
of an arrangement of two-planes in projective space.Comment: 9 pages, Minor additions to introduction, references added, typos
corrected, minor change in exposition of main theore
Subspace arrangements defined by products of linear forms
We consider the vanishing ideal of an arrangement of linear subspaces in a
vector space and investigate when this ideal can be generated by products of
linear forms. We introduce a combinatorial construction (blocker duality) which
yields such generators in cases with a lot of combinatorial structure, and we
present the examples that motivated our work. We give a construction which
produces all elements of this type in the vanishing ideal of the arrangement.
This leads to an algorithm for deciding if the ideal is generated by products
of linear forms. We also consider generic arrangements of points in
and lines in Comment: 20 pages; AMSLatex; v.2: proof of Proposition 5.1.3 corrected; proof
of Proposition 5.1.6 shortened; references added, v.3: minor corrections;
final version; to appear in the Journal of the London Mathematical Societ
Multigraded regularity: coarsenings and resolutions
Let S = k[x_1,...,x_n] be a Z^r-graded ring with deg (x_i) = a_i \in Z^r for
each i and suppose that M is a finitely generated Z^r-graded S-module. In this
paper we describe how to find finite subsets of Z^r containing the multidegrees
of the minimal multigraded syzygies of M. To find such a set, we first coarsen
the grading of M so that we can view M as a Z-graded S-module. We use a
generalized notion of Castelnuovo-Mumford regularity, which was introduced by
D. Maclagan and G. Smith, to associate to M a number which we call the
regularity number of M. The minimal degrees of the multigraded minimal syzygies
are bounded in terms of this invariant.Comment: 20 pages, 1 figure; small corrections made; final version; to appear
in J. of Algebr
Hyperbanana Graphs
A bar-and-joint framework is a finite set of points together with specified
distances between selected pairs. In rigidity theory we seek to understand when
the remaining pairwise distances are also fixed. If there exists a pair of
points which move relative to one another while maintaining the given distance
constraints, the framework is flexible; otherwise, it is rigid.
Counting conditions due to Maxwell give a necessary combinatorial criterion
for generic minimal bar-and-joint rigidity in all dimensions. Laman showed that
these conditions are also sufficient for frameworks in R^2. However, the
flexible "double banana" shows that Maxwell's conditions are not sufficient to
guarantee rigidity in R^3. We present a generalization of the double banana to
a family of hyperbananas. In dimensions 3 and higher, these are
(infinitesimally) flexible, providing counterexamples to the natural
generalization of Laman's theorem
Geometric aspects of the Jacobian of a hyperplane arrangement
An embedding of the complete bipartite graph in
gives rise to both a line arrangement and a bar-and-joint framework. For a
generic placement of the six vertices, the graded Betti numbers of the
logarithmic module of derivations of the line arrangement are constant, but an
example due to Ziegler shows that the graded Betti numbers are different when
the points lie on a conic.
Similarly, in rigidity theory a generic embedding of in the plane
is an infinitesimally rigid bar-and-joint framework, but the framework is
infinitesimally flexible when the points lie on a conic. In this paper we
develop the theory of weak perspective representations of hyperplane
arrangements to formalize and generalize the striking connection between
hyperplane arrangements and rigidity theory that this example suggests. In
particular, we seek to understand how the interplay of combinatorics and
geometry influence algebraic structures associated to an arrangement, such as
the saturation of the Jacobian ideal of the arrangement. We make connections
between examples and constructions from rigidity theory and interesting
phenomena in the study of hyperplane arrangements.Comment: 34 pages, 11 figures. Changes made primarily to expositio