Planes of matrices of constant rank and globally generated vector bundles

Abstract

We consider the problem of determining all pairs (c1, c2) of Chern classes of rank 2 bundles that are cokernel of a skew-symmetric matrix of linear forms in 3 variables, having constant rank 2c1 and size 2c1 + 2. We completely solve the problem in the "stable" range, i.e. for pairs with c 2 1 − 4c2 < 0, proving that the additional condition c2 ≤ c 1 +1 2 is necessary and sufficient. For c 2 1 − 4c2 ≥ 0, we prove that there exist globally generated bundles, some even defining an embedding of P 2 in a Grassmannian, that cannot correspond to a matrix of the above type. This extends previous work on c1 ≤ 3.2010 Mathematics Subject Classification. 14J60, 15A30. Key words and phrases. Skew-symmetric matrices, constant rank, globally generated vector bundles. Research partially supported by MIUR funds, PRIN 2010-2011 project "Geometria delle varietà algebriche", and by Università degli Studi di Trieste -FRA 2013 project "Geometria e topologia delle varietà".

Quadric surfaces in the Pfaffian hypersurface in $\mathbb{P}^{14}$

“…An efficient method to construct spaces of matrices of constant rank consists in building bigger size matrices of a given rank, and then projecting them to smaller size matrices of the same rank. This technique was introduced in [FM11] for the case of P 2 , and later used in [BM15], but the results hold in more generality. Indeed, they were already extended to linear spaces of matrices of any size in [BFL18, Proposition 5.1]; here, we wish to apply these results to the case of quadrics.…”

Truncated Modules and Linear Presentations of Vector Bundles

“…This technique was used for example in [FM11,BM15]. So what is the advantage of our method over that of projecting bigger matrices?…”

“…In [EH88,IL99,Syl86] the relation between matrices of constant rank and the study of vector bundles on P n and their invariants was first studied in detail. This interplay was pushed one step further in [BFM13,BM15], where the matrix of constant rank was interpreted as a 2-extension from the two vector bundles given by its cokernel and its kernel. This allowed the construction of skewsymmetric matrices of linear forms in 4 variables of size 14 × 14 and corank 2, beyond the previous "record" of [Wes96].…”

A different perspective on H-like Lie algebras

“…This kind of problem-finding sets of matrices so that nonlinear properties such as rank or spectrum are preserved under linear combinations-has been studied since at least as early as Hurwitz and Radon ([Hur22,Rad22]). For example, see [EM16, IL99, EH88, Wes87, Bea81, Fla62] for subspaces of matrices having fixed or bounded rank, see [BM15,FM11,MM05] for subspaces of skew-symmetric matrices, and see [Skr02] for subspaces of cones over conjugacy classes.…”