Linear spaces of matrices of constant rank

Given a complex vector space V of dimension n, one can look at
d-dimensional linear subspaces A in \wedge^2 V, whose (nonzero)
elements have constant rank r. The natural interpretation of A as a
vector bundle map yields restrictions on the values that r, n, and d
can attain.
In this talk we will deal with the case r=n-2. I will mention a
classification result for the 3-dimensional case, and then try to
convince you that 5-dimensional examples cannot exist. Then we will
concentrate on the 4-dimensional case (the most interesting one), for
which I will give a method to construct new examples, based on the
derived category of P^3, as well as an algorithm that explains the
technique in terms of the cohomology graded modules of the bundles
involved.
This talk summarizes joint works (some in progress) with J. Buczynski,
D. Faenzi, G. Kapustka, P.Lella, and E. Mezzetti.