Embedding pencils of matrices in Grassmannians

Pencils of matrices are 2-dimensional linear subspaces in a space of matrices. One can identify the space of squared matrices with an affine open subset of a Grassmannian, and embed a given pencil L into it. In this talk we describe the closure Y_L of the pencil inside the Grassmannian as a blow-up of the complex projective plane at finitely many points, and we relate points in the exceptional locus to a C*-action on the Grassmannian. This is a joint work in progress with F. Gesmundo and H. Keneshlou.