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.

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