21 research outputs found
The Horrocks-Mumford bundle restricted to planes
We study the behavior of the Horrocks-Mumford bundle when restricted to a
plane P^2 in P^4, looking for all possible minimal free resolutions for the
restricted bundle. To each of the 6 resolutions (4 stable and 2 unstable) we
find, we then associate a subvariety of the Grassmannian G(2,4) of planes in
P^4. We thus obtain a filtration of the Grassmannian, which we describe in the
second part of this work.Comment: 19 pages, typos removed, added details in Propostions 2.1 and 3.
On the codimension of permanental varieties
In this article, we study permanental varieties, i.e. varieties defined by
the vanishing of permanents of fixed size of a generic matrix. Permanents and
their varieties play an important, and sometimes poorly understood, role in
combinatorics. However, there are essentially no geometric results about them
in the literature, in very sharp contrast to the well-behaved and ubiquitous
case of determinants and minors. Motivated by the study of the singular locus
of the permanental hypersurface, we focus on the codimension of these
varieties. We introduce a -action on matrices and prove a number
of results. In particular, we improve a lower bound on the codimension of the
aforementioned singular locus established by von zur Gathen in 1987.Comment: 20
Uniform determinantal representations
The problem of expressing a specific polynomial as the determinant of a
square matrix of affine-linear forms arises from algebraic geometry,
optimisation, complexity theory, and scientific computing. Motivated by recent
developments in this last area, we introduce the notion of a uniform
determinantal representation, not of a single polynomial but rather of all
polynomials in a given number of variables and of a given maximal degree. We
derive a lower bound on the size of the matrix, and present a construction
achieving that lower bound up to a constant factor as the number of variables
is fixed and the degree grows. This construction marks an improvement upon a
recent construction due to Plestenjak-Hochstenbach, and we investigate the
performance of new representations in their root-finding technique for
bivariate systems. Furthermore, we relate uniform determinantal representations
to vector spaces of singular matrices, and we conclude with a number of future
research directions.Comment: 23 pages, 3 figures, 4 table
Secants of Lagrangian Grassmannians
We study the dimensions of secant varieties of the Grassmannian of Lagrangian
subspaces in a symplectic vector space. We calculate these dimensions for third
and fourth secant varieties. Our result is obtained by providing a normal form
for four general points on such a Grassmannian and by explicitly calculating
the tangent spaces at these four points
Linear spaces of matrices of constant rank and instanton bundles
We present a new method to study 4-dimensional linear spaces of
skew-symmetric matrices of constant co-rank 2, based on rank 2 vector bundles
on P^3 and derived category tools. The method allows one to prove the existence
of new examples of size 10x10 and 14x14 via instanton bundles of charge 2 and 4
respectively, and provides an explanation for what used to be the only known
example (Westwick 1996). We also give an algorithm to construct explicitly a
matrix of size 14 of this type.Comment: Revised version, 22 pages. Brief intro to derived category tools and
details to proof of Lemma 3.5 added, some typos correcte
A note on secants of Grassmannians
Let G(k,n) be the Grassmannian of k-subspaces in an n- dimensional complex vector space, k ≥ 3. Given a projective variety X, its s-secant variety σs(X) is defined to be the closure of the union of linear spans of all the s-tuples of independent points lying on X. We classify all defective σs(G(k, n)) for s ≤ 12