Simone Marchesi

Branes in the moduli space of framed sheaves

Abstract In the physicists language, a brane in a hyperkahler manifold is a submanifold which is either complex or lagrangian with respect to three Kahler structures of the ambient manifold. By considering the fixed loci of certain involutions, we describe branes in Nakajima quiver varieties of all possible types. We then focus on the moduli space of framed torsion free sheaves on the projective plane, showing how the involutions considered act on sheaves, and proving the existence of branes in some cases.

Moduli of autodual instanton bundles

We provide a description of the moduli space of framed autodual instanton bundles on projective space, focusing on the particular cases of symplectic and orthogonal instantons. Our description will use the generalized ADHM equations which define framed instanton sheaves.

Covered by lines and Conic connected varieties

We study some properties of an embedded variety covered by lines and give a numerical criterion ensuring the existence of a singular conic through two of its general points. We show that our criterion is sharp. Conic-connected, covered by lines, QEL, LQEL, prime Fano, defective, and dual defective varieties are closely related. We study some relations between the above mentioned classes of objects using basic results by Ein and Zak.

p-Buchsbaum rank 2 bundles on the projective space

It has been proved by various authors that a normalized, 1-Buchsbaum rank 2 vector bundle on P 3 is a nullcorrelation bundle, while a normalized, 2-Buchsbaum rank 2 vector bundle on P 3 is an instanton bundle of charge 2. We find that the same is not true for 3-Buchsbaum rank 2 vector bundles on P 3 , and propose a conjecture regarding the classification of such objects.

Varieties connected by chains of lines

In this paper we give for any integer

A Horrocks' theorem for reflexive sheaves

l\geqslant 2

Jumping pairs of Steiner bundles

a numerical criterion ensuring the existence of a chain of length